Joszef S<ndor

Transcription

1 Smarandache 问题 新进展 陈国慧著 High American Press 007

2 Smarandache 问题 新进展 陈国慧 海南师范大学数学系 Chen Guohui Department of Mathematics, Hainan Normal University Haikou, Hainan, 5758 P. R. China High American Press 007

3 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning University of Microfilm International 300 N. Zeeb Road P.O. Box 346, Ann Arbor MI , USA Tel.: Customer Service Peer Reviewers: Wenpeng Zhang, Department of Mathematics, Northwest University, Xi an, Shannxi, P.R.China. Wenguang Zhai, Department of Mathematics, Shangdong Teachers University, Jinan, Shandong, P.R.China. Guodong Liu, Department of Mathematics, Huizhou University, Huizhou,Guangdong, P.R.China. Copyright 007 by High Am. Press, translators, editors, and authors for their papers Many books can be downloaded from the following Digital Library of Science: ISBN: Standard Address Number: Printed in the United States of America

4 ,.,.,.,., :,.,,.,,,., Smarandache Sn, F.Smarandache., 99, F.Smarandache 05.,,!,, Smarandache, Smarandache, Smarandache,,,,.,,.,.,! I

5 Smarandache Smarandache Sn dn F.Smarandache Sm n SZn Sd d n.6 Sn Sn k Smarandache Sa k n Smarandache Sn Smarandache Smarandache Smarandache Smarandache S n SMn Smarandache SLn SLn SLn SLn Sn SLn SLn SLn Smarandache F.Smarandache Smarandache LCM Smarandache LCM II

6 Ln Ln Ln k Ln Ln Smarandache LCM F.Smarandache Smarandache S p n S p n Smarandache Smarandache ceil Smarandache III

7 IV Smarandache

8 Smarandache Smarandache n, y y fn.,,,.,,, φn, dn, µn, fn x. Smarandache, Smarandache..,. n, Smarandache Sn m n m!, Sn min{m : n m!, m N}.. n, Dirichlet dn n, dn. d n.3 Zn k n kk + /, Zn min{k : n kk + /}. Jozsef Sandor..4 n, SMn n, SM n > n p α pα pα k k n, SMn max{α p, α p, α 3 p 3,, α k p k }. SMn Smarandache..5 n k, n k a k n m, m n k. Smarandache., n p α pα pα k k n.. n, Sn max i k {Spα i i }.

9 Smarandache Sp p.. P n n, P n max{p, p,, p k }, P n > n Sn P n. : P n P n > n, SP n P n. n p i i k p i P n, Sp α i i α i p i. Sp α i i α i, Sp i p i n. α i, Sp i p i n 4 n. 3 α i 3, Sp α i i α i p i α i n α i n α i ln n n, ln p i p α n α ln n ln p. -3, Sp α i i n., Sn max i k {Spα i i..3 Sp α } SP n P n. p α + Sp α p [α + + log p α] +..4 Sn Sn π x x ln x + O ln. x : A B: A {n n x, P n n}, B {n n x, P n > n}.

10 Smarandache Euler. n ln n Sn n A x x t ln tdt + t [t] t ln t dt + x ln xx [x] x 3 ln x., Abel Sn P n n B P n> n n x x p x n x x n n π x n n x n p x n p + O p n x n p x n xπ x x n πx x. πx x x ln x + O ln, x, x p x n n x x x πsds xπ x n x πsds x p x x n π n x n ln x/n x ln x + O x n ln x/n x +O ln x + O x n ln x/n x ln x x n ln x/n x n ln x/n + O n ln x π 6 x x ln x + O ln x ln x n x + O x 3 ln x, x. n ln x n x x x n ln x/n O ln. x 3

11 Smarandache, Sn Sn Sn + n A n B π x x ln x + O ln x...5 p k < p k fx x n k + x n k + + x n n k > n k > > n, Sp fp p fp + pf., S p kpn k φp n + p, k φn Euler..6 P n n, x >, Sn P n ζ 3 3 x 3 ln x 3 x + O ln, x ζs Riemann zeta-..7 k x >, ΛnSn x k i0 c i x ln i x + O ln k+, x Λn Mangoldt, c i i,,, k, c 0.. Sn dn Smarandache Sn Dirichlet dn,... k, x >, Sn dn π4 36 x ln x + k c i x x ln i x + O ln k+ x dn Dirichlet, c i i, 3,, k. :, Sn dn - 4 i,

12 Smarandache, n x n A B, A p p n p > n n; B [, x] A.., Sn dn n A p n, n<p np x n<p Sn dn np x n<p p dn n x πx p x, Abel dn Snp dnp n<p x n p. - πx k i c i i,,, k c. n<p x n p x n π x n x n ln x + a i. c i x x ln i x + O ln k+, x x n n πn πydy n k a i x ln i n i n ln i x + O x n ln k+ x, -3 n n π n dn n n π4 36, n n A Sn dn x ln x b i. dn k a i x dn ln i n n n + x n n ln i x x i x +O ln k+ x x k ln x + b i x x ln i x + O ln k+ x π4 36 i, -4 5

13 Smarandache B, - B n B, n n p α pα r r, -5 Sn max i r {Spα i i n B Sn dn n B dn n ln n } max i r {α ip i } n ln n. -5 dn x ln x + Ox. A B -4-6 i dn n ln n x 3 ln x, -6 Sn dn Sn dn Sn dn + n A n B π4 36 x k ln x + b i x x ln i x + O ln k+, x b i i, 3,, k...3 F.Smarandache Sm n S p n Sn, Sp n S p n. Sm n,..3. m > m p α pα pα k k, n, m Sm n p αn + O ln m ln n, p α max i k {p i α i }. : m p α pα pα k k. Sm n Sp α n p α n p α kn max S p n Sn 6 m, mn p α n p α n p α kn k. k Sp n S p n, i k {Spα in i }. -7

14 Smarandache Sp α in i S pi α i n. Sp n S p n. p S p n p n + O ln p ln n Sp α in pi i S pi α i n p i α i n + O lnnα i. -8 ln p i Sp αn max i k {Spα in i } max {S p i α i n}. -9 i k -8 n, p i α i, Sp α in i. pi m O lnnα i O ln p i ln m ln n, -7, -8-9 p Sm n Sp αn S p αn p αn + O ln p lnαn m p αn + O ln m ln n. p α max {p i α i }.. i n.3. m > m p α pα pα k k, Sm n lim p α, n n p α max i k {p i α i }..4 SZn S Zn,. 7

15 Smarandache.4. k, x >, S Zn π 8 x 3 ln x + c i i, 3,, k., k i c i x 3 3 x ln i x + O ln k+, x S Zn -0 m m, Zn m m + n mm + Zn m, n m m +, m m +,, mm +. n x, Zn Zn m, m m 8x +. Sn n, S Zn Sm m Sm + Ox Znm m x m 8x+ m Sm + Ox. - m x A B, A p p m p > m m; B [, x] A.., m Sm n A m x p m, m<p mp x m<p m Sm mp p m 4 x mp x m<p m mp Smp m<p x m πx p x, Abel p. - 8 m<p x m p x m π x m x m m πm y πydy m

16 Smarandache a i. 3 x 3 k m 3 ln x + a i x 3 ln i m m 3 ln i x i x 3 +O m 3 ln k+, -3 x n π, n S Zn 3 x 3 ln x n A 3 x +O ln k+ x π m 4 x 8 x 3 ln x + b i. B, k i m + Sn max i k {Spα i i } m 4 x i k a i x 3 ln i m m ln i x b i x 3 3 x ln i x + O ln k+, -4 x B, m B, m m p α pα r r Sm max m B m Sm m B i r {Spα i i, } max i r {α ip i } m ln m. -5 m m ln m m x A B -, -4-6 S Zn m x n A π m Sm + Ox m Sm + n B m Sm + Ox 8 x 3 ln x + k i m 3 ln m x 5 4 ln x. -6 b i x 3 3 x ln i x + O ln k+, x b i i, 3,, k.. 9

17 Smarandache, k.4. x >, S Zn π 8 x 3 ln x + O x 3 ln x..5 d n Sd Smarandache Sn,. n, d n Sd -7, n -7.,.,,.5. n, -7., : n, n > p, p n p n. n, n p p p k n, p < p < < p k. Sn Sn Sp p p k p k. k, n p, d n Sd Sd S + Sp p d p p >, -8. k >, d p p p k Sdp k p k. 0 d n Sd Sd d p p p k d p p p k Sd + Sd + k p k d p p p k Sd + d p p p k d p p p k p k Sdp k d p p p k

18 Smarandache k i. p i -9 i -9. k i i p i., d n k i k >, p k, i p i m. Sd, -9 k i i p i m k p k, p p p k k i i p i m p p p k k p k, p k,, ,.5. n >, C n. C n n, :. n >, C n. C n m i αi l i, l i, i,,, n. α max{α, α,, α k }. α i,,, n,., r, s n α r α s α. r s, l r l s, l r l s, l. < α l α+ l n, m α r+ l n α. α. α. u α l u, M α l l l n. M C n M u + u M n u

19 Smarandache M u + u n + l l l n. - -, M C n, M M u + u n, l l l n,. C n,..5. p α, n p α α p, -7. : p α, n p α, α p. Sd Sd S + Sp + Sp + + Sp α d p α d n + p α n p α α p, n, n p α pα pα k k p k Sn p k, -7. : n p α pα pα k k p k u p k Sn p k, d n Sd du. d u d u Sd + d u Sdp k d u Sd + d u Sd + du, -3 p k -3 d u, Sd < p k. d u p k. d n Sd, du p k p k Sd, p k du α + α + α k +. p k, p k α i +., α i + p k. -4

20 Smarandache.3-4 Sp α i i.5.3. Sp α i i p i α i + α i + p k -5 p k, p i Sp α i i, Sp α i i > p k. Sn p k, n n, 8. Sd d n.6 Sn n, n Sd n + + Sn, -6 d n n Smarandache., Ashbacher [], : n 0 6, Smarandache...6. p r, Sp r pr..6. dn, dn. n p r pr pr k k n, dn r + r + r k n,, 3, 4, 6, n <, n N -7 dn. : n, n p r pr pr k k fn n dn. n. f, f, f3 3/, f4 4/3, f6 3/, -7 n,, 3, 4, 6. n -7, n,, 3, 4 6. f5 5 >, n > 6. n, k, r, n p 7, 3

21 Smarandache > fn p 7,. k, r, n p, p 3. > fn pr r + 3,. k, n > 6, 4. k 3, > fn pr r + p r r + > fn pr r + p r 5 p, r, p, r >, 5 p >. 4 p r 3 3 r + r ,., n,, 3, Smarandache. : n Smarandache. [] n > , n p r pr pr k k n, -9, n, Sn Sp r, -8 p p j, r r j, j k. -9 n p r m, m N, gcdp r, m. -30 gn d n Sd. -3-6, Smarandache n gn n + + Sn n Sp r!., n d, Sd Sp r. -33, -3, dnsp r > n

22 Smarandache dn.,.6.3-8, r + Sp r p r > fm. -35 r, Sp p, -35 > fm.,.6.4, m,, 3, 4 6. m, -3 gn gp S + Sp + p p + + Sp + p,. m, p > gn gp S + S + Sp + Sp 3 + p 3p +, -36 p,., r, m 3, 4 6, -3. r, Sp p, > fm. -37 p n > 0 6, 8 Sp Sn 0, p 5., -37 fm < 6 5.,.6.4 m 6. n p m 6p, p r 3, 4, 5 6, -35. r 7, -35 Sp r pr r + r p r > r + Spr p r > fm, r + r > p r r r 0 + r + r + r 3, 0 > r 6r + 5 r r 5 > 0,. 0 6 Smarandache...7 Sn k n Sn k,. 5

23 Smarandache.7. n, k, x, n Sn k x Ux π + O x x k k ln x + O ln. x n A A n Sn k. : n p α pα pα k k n, Sn max i k {Sp i α i } Sp α. n Sn k, I k, n Sn, n n p. II k, n Sn, Sn Sp α n. n, n >, a α, Sn Sp n n p, n, p, p >, Sp p n p, n, p >, n p. b α, Sn Sp 4 n n p, n, p, p > 4, Sp 4 4p n p, n p 4, p > 4,. p, n 4. p 3, p 3 n 9. c α 3, p >, Sp 4 S 4 6, Sp 4 S , Sn Sp α αp < p α n, p α > αp, Sn n,., k, n Sn n, n 9, n p. III k 3, n Sn k. n, n >, n p α pα pα t t n, Sn max i k {Sp i α i } Sp α l l. l k p l k 6 Sn k Sp kα l l p α pα pα t t,

24 Smarandache Sp kα l l kα l p l, p α l l Sp kα l l p α pα pα t p α pα pα t t Sp kα l l p α pα pα t t kα l p l. > α l p l, kp α l l > kα l p l, p α pα pα t t > kp α t t > kα l p l, t kα l p l. kα l p l, α l, n kp l. p l p, p, n Sn k n n kp,., n Sn k Ux n A p x k kp A + O k kp x kp A + O k x π + O k k x x k ln x + O ln. x..8 Smarandache n, P n n, x >, Sn P n ζ 3 3 x 3 ln x 3 x + O ln, x ζs Riemann zeta-., Smarandache Sn,..8. k, n > k, i P n > n, Sn k SMn k kp n ii n mp P n, n 3 < p < P n n, Sn k SMn k kp n; 7

25 Smarandache iii n mp n n 3 < P n n, Sn k SMn k kp n. :, Sn k. SMn k. i. n p α pα pα r r P n n. P n > n, p kα p kα p kα r r < n k < P k n, p kα i i kp n!, i,,, r. n k kp n!. P k n kp n!. Sn k SP k n kp n..8. i., n 3 < p < P n n Sn k kp n..8. ii. n mp n, n 3 < P n n, m < P n. P k n kp n!, m k kp n!. P k n kp n!,. Sn k kp n..8. k x 3, Sn k kp n k x 4 3 ln x P n n 3 SMn k kp n k x 4 3 ln x. P n n 3 :, n p α pα pα r r Sn k max i r {Spkα i i }. n, 8

26 Smarandache kαp max {kα ip i }, Sn k kp ln n. α, kp kp n, i r Sn k kp n 0, Sn k kp n k P n ln x P n n 3 np x p x 3 k p ln x p x 3 p P n n 3, P n n n x p k ln x O k x 4 3 ln x p m, m x 3,. m p x m p x 3 3m 3 ln x ln m + O x 3 m 3 ln x m : [] k x 3, Sn k kp n k ζ 3 3 x 3 ln x + O k x 3 ln x ζs Riemann zeta-, O k O k. :.8. P n > n, Sn k SMn k kp n Sn k kp n., P n> n P n n P n n 3 n 3 <P n n Sn k kp n + Sn k kp n P n n Sn k kp n + n 3 <P n n Sn k kp n Sn k kp n Sn k kp n + Ok x 4 3 ln x. -39 n 3 < P n n, 9

27 Smarandache a n m P n, m < P n. b n m p P n, m < n 3 < p < P n. c n m P n P m n 3. Sn k kp n, b c, Sn k kp n, p p α m, α αp > P n., p < n iii Sn k kp n + n 3 <P n n mp x mp 3 <p mp mp p x mp p 3 <p <p mp p mp x mp 3 <p mp Sm k p k kp m k p k Smp p kp mp p + O k x 4 3 ln x k p + O k x 4 3 ln x k p + O k x 4 3 ln x. -40 m<x 3 m <p x m.8.3 m x 3 m <k p x m x 3 3 ln x k ζ 3 3 k m e m 3 lnx x 3 ln x + O k k p + O k m x 3 x 3 ln x e ln x m x 3 k x 3 3m 3 ln x ln m + O k -39, , Sn k kp n k ζ 3 3 x 3 ln x x 3 3 x ln + O k x ln x. -4 x 3 3 x ln x + O k ln. x 0

28 Smarandache.8. x 3, SMn k kp n k ζ 3 3 x 3 3 x ln x + O k ln. x.9 Sa k n a k n n k, F.Smarandache,, [3], [4] [5]. Sa k n..9. k. n, i P n > n, S a k n SM a k n k P n; ii n mp P n n 3 < p < P n n, S a k n SM a k n k P n; iii n mp n n 3 < P n n, k > S a k n SM a k n k P n; k, S a k n SM a k n kn 3. : Sa k n., SMa k n. i. n p α pα pα r r n, P n > n, P n p r, α r. Sa k P n S P k n k P n. S a k p α i i S p k i k pi k P n, i,,, r. S a k n k P n. ii. m p p < n 3. n p α pα pα r r n, a k n p β pβ pβ r r β i k, i,,, r. n 3 < p < P n n S a k n k P n.

29 Smarandache iii. n mp n n 3 < P n n, m < P n, k >, S a k P n S P k n k P n, S a k n k P n. k, a k P n, m n 3.. S a k n S P k m kn k, x 3, S a k n k P n k x 4 3 ln x P n n 3 P n n 3 SM a k n k P n k x 4 3 ln x. : n p α pα pα r r n, a k n p β pβ pβ r r a k n. βp max i r {β ip i }, S a k n max i r {Spβ i i }. S a k n βp kp. P n n, β k S a k n k P n 0, Sa k n k P n k P n P n n 3 P n n 3, P n n

30 Smarandache np x p x 3 k p p x 3 p k k x 4 3 ln x. n x p p, m, m p x m p x 3 3m 3 ln x ln m + O x 3 m 3 ln x m. : [] k, x 3, S a k n k P n ζ 3 3 x 3 3 x ln x + O k x 4 3 ln + O, x ln x ζs Riemann zeta-. :.9. P n > n, Sa k n k P n P n> n P n n P n n 3 n 3 <P n n Sa k n SMa k n k P n. Sa k n k P n + Sa k n k P n Sa k n k P n + P n n n 3 <P n n Sa k n k P n k x O ln x n n 3 < P n n, a n m P n m < P n. Sa k n k P n Sa k n k P n. -4 3

31 Smarandache b n m p P n m < n 3 < p < P n. c n m P n P m n 3. b c n, n Sa k n k P n, n a n, k > Sa k n k P n. -4 Sa k n k P n n 3 <P n n mp x mp 3 <p mp mp x mp 3 <p mp Sp k k p p p. -43 m<x m 3 <p x m a n, k 4 n 3 <P n n mp x mp 3 <p mp mp x mp 3 <p mp m<x m 3 <p x m Sa k n k P m k n 3, Sa n P n S a n psa n + p p + O p + O mp x mp 3 <p mp mp 3 + mp 3 p x

32 Smarandache -43, , x 4 x 3 3 ln, k x Sa k n k P n n 3 <P n n m x m 3 <p x m m x 3 x 3 3 ln x 3 ζ 3 p + O x 4 3 x 3 3m 3 ln x ln m + O m e m 3 ln x x 3 ln x + O + O x ln x e ln x m x 3 x 3 m 3 ln x x 3 m + O m 3 ln x x 3 ln x. -45 Sa k n k P n 3 3 ζ x 3 3 x ln x + O k x 4 3 ln + O. x ln x..9. x 3, SM a k n k P n ζ 3 3 x 3 3 x ln x + O k x 4 3 ln + O. x ln x.0 Smarandache nn + n + S + S + + Sn S 6,,.0. n, nn + n + S + S + + Sn S 6 5

33 Smarandache n,. : n p α pα pα k k n, Sn max i k {Sp i α i } Sp α, Sp α i i Sp α, i k. nn + n + S + S + + Sn S, 6 n. n >, I n, S + S S + S4 5 nn + n + S S5 5, 6 n. II n 3, Sn 3. Sn S + S + + Sn n 3n. n, n +, n, n +, n +, n +. nn + n + S max{sn, Sn +, Sn + } n n n +. n.,., nn + n + S + S + + Sn S 6, n, k, Sm + Sm + + Sm k Sm + m + + m k. : k, 6 Sm + Sm + + Sm k Sm + m + + m k.

34 Smarandache I k,. II k p, p, m m m k, S, Sm + Sm + + Sm k p Sp Sm + m + + m k. III k > k, p k p k. p k l, l k. m m m l, m l+ m l+ m k S, S, Sm + Sm + + Sm k l + k l l + k p Sm + m + + m k Sk + l Sp p Sm + Sm + + Sm k. m m m l, m l+ m l+ m k,., Sn.. k, x >, [Sn SSn] 3 3 ζ x 3 k i 3 c i x ln i x + O ln k+, x ζs Riemann zeta-, c i i,,, k, c. : n >, n p α pα pα s s n, Sn max{sp α, Spα,, Spα s s } Sp α. -46 [Sn SSn] [Sn SSn] + [Sn SSn], -47 n A n B A B [, x]. A {n Sn Sp, n [, x],p?}; B {n Sn Sp α, α α 3, n [, x], }. n A, 7

35 Smarandache n p m P m < p, P m m. Sn, p >, Sn Smp Sp p SSn Sp p. -46 A [Sn SSn] n A p n, n<p p n<p, p, n p n<p, p, n n x n<p x n n x p x n [ Sp SSp ] + p n, p n [ Sp SSp ] + p + p n p, p, n p + O 5 x p 4 + O ln x p n p n, p 3 n. : πx m x 4 p m x 3 p p n, p, n p + O p + O [ Sp SSp ] [ Sp SSp ] p p x p 4 n x p, -48 k i a i x x ln i x + O ln k+, x a i i,,, k, a. p x n p x n π x n 3 x 3 n 3 k i x n y πydy 3 b i ln i x n + O n 3 x 3 ln k+ x, -49 n x, b i, b. 3 ln i n ζ, i,, 3,, k. n n 3 n [Sn SSn] 8 n A n 3

36 n x 3 ζ 3 [ 3 x 3 n 3 Smarandache x 3 k i k i b i ln i x n + O n 3 x 3 ln k+ x ] + O x 5 4 ln x 3 c i x ln i x + O ln k+, -50 x c i i,, 3,, k, c. B. n B, Sn Sp p, [Sn SSn] [Sp SSp] 0; Sn Sp α, α 3, α ln x. [Sn SSn] [Sp α SSp α ] α p [Sn SSn] n B np α x α 3-47, [Sn SSn] 3 3 ζ x 3 c i i,, 3,, k, c.. α p x ln x. -5 k i 3 c i x ln i x + O ln k+, x 9

37 Smarandache Smarandache Smarandache S n, Smarandache Sn. Sn, Smarandache, Smarandache... n, Smarandache S n m m! n, S n max{m : m! n, m N}.. n, S n n, S n m m!! n n, S n m m!! n. S n. n, S n, n, S n.. k, S k!k +! q, k, q k +..3 Res > ζs.4 n n S n n s ζs Riemann zeta-. ns n S n n! s, ln S x n e x + O ln ln x.. Smarandache 30, n S n n s -

38 Smarandache, - S n, S n... s > ζs 3.3., n n S n k n k n k n s ζs n! s n S nn s ζs n n Riemann zeta-. ns nn + n +! s,, m, Stirling s [6] lnm! m ln k m ln m m + O. - k S n : m! n, m! n, lnm! ln n. S n m ln n S n ln ln n n s ln n n s ln ln n. S n s > Dirichlet n s. S n m, m! n. n s >, S n n m! n, m + n. n S n k n s m n S nm m m k m! s m k m k n s n m+ n m! s m n n s n m m n m+ n n s n s n m k m! s m k m! s n s m + s n s m m k m +! s 3

39 n ζs... Smarandache n s + n m n k n k n! s. m + k m +! s m m k m +! s ζ π 6, lims ζs, s n n! e, ζs n µn n s, µn Möbius...,.. n, { n µds, n m!, m ; d 0,. d n n : ζs n n! s n S n n π 6 n n!. µn n s,.., µn n s n n µus v n u vn n s S n n s n d n µm S n mn s m n n µds d n s Dirichlet, { n µds, n m!, m ; d 0,. d n 3..,.. lims s n S n n s e,

40 Smarandache e Perron s [6] 6.5., S n.,,.. x >, ln S x n e x + O ln ln x. : x >, k k! x < k +!. - k S n S n m! x S nm n x m! ln x ln x ln ln x + O ln ln x. m m! m+ n m m! x x x x x m! x m ln x ln ln x m m e x + O n0 n m m +! + O x m+! m m +! + O m m! x m n x m! m+ n m m! x m m! x ln x ln ln x m m +! + O m x m +! m!>x m! m! + + O m +! ln x ln ln x, e.. n! x m m +! + O ln x ln x + O ln ln x ln ln x ln x + O ln ln x ln x ln ln x.3 Smarandache Smarandache, n 33

41 Smarandache S d n. -4 d n n S d n. n -4? d n, n,. : n -4. n > -4, i n k +, d n! n, S d. n > -4 n d n S d d n dn, -5 dn Dirichlet. n 3 n > dn, -5, -4. ii n m, m. m, 3, 5 n -4. m 7. 3 m -4 n m d n S d d m S d + d m S d d m + d m 3dm. m 3dm, m 7 m > 3dm,. 3 m n m -4, n m 6 m 3 S d S d + S d d n d m d m d m + d m 3 dm + d m 3 S 6d + d m 3 S d 3 + 3d m 3 dm + 6dm 3. m m + 9 m 3 dm + 6dm 3. 34

42 Smarandache m 3 3. m 3 n 8, 3 S d S + S + S 3 + S 6 + S 9 + S 8 d , -4. n m m, -4. iii n m, m. m n 4-4. m 3 n, S d S + S + S 3 + S 4 + S 6 + S d , n -4. m > 3, 3 m m 9, n -4, n m d n S d d m S d + d m S d + d m S 4d d m + d m 3 dm + 4 d m 3 S 6d + d m 3 S d + d m 3 3 dm + d m 3, 4m m + 9 m 3 dm + dm 3, S d + d m 3 S 6d m 3 > 3. m 3 3 n 36, S d S + S + S 3 + S 4 + S 6 + S + S 9 d 36 +S 8 + S n m, m 3 m, n -4, n 4m d n S d d m S d + d m S d + d m S 4d d m + d m + d m 5dm, 35

43 Smarandache 4m 5dm. n m, m n -4. iv n α m, m, α 3. m, n α, n α d α S d + d α S d m 3 + d α α + α. n α 3 S d S d + S 3d d α d α d α 3 α d α 3 + d α 3α +, α 3 3α +. n α 3-4. S n S α m u, m > 3. u, n -4 n α m d n S d d m S d + d n S d d m + d n n dm + d. α m dm + d α m. m > 3 α m > dm + d α m. u 3, 3 m. n -4 n α m S d S d + S d d n d m d n n dm + 6d. 6 d m d n 6 d n 6 n α m dm + 6d. m 3 6 n n α m > dm + 6d u 4, 3 m. u! n α m,

44 Smarandache α i u 4 n -4 n α m d n d m + d m [ u i ]. -6 S d α S i d i0 d m 3 + α d m 4 4dm + 4α dm, α m 4αdm. α 3, m > 3. u 5, 3 m 5 m, m 5, n -4 n α m d n d m 5dm 4m d m S d α S i d i0 d m 3 + α d m u 4dm + uα dm uα dm. -6 α u -7 α m uα dm. + u 4 α 5 4uα 4 α 3.. 3u 3. 4 [ ] 4α α Smarandache S n S n S n., S n, n s,. n 37

45 Smarandache.4., n S n n s s >, S n n s n ζs s + ζs Riemann zeta-. :, m m +!! s + ζs m s >, S n << ln n Dirichlet n S n n s n n S n n s + n n S n n s. m!! s, n S n n s S n n, S n m m!! n. n m!! n m + n. s >, 38 n n S n n s m m m n n S nm m m!! s n s n n m n s n n m+ n m m!! s n s n n n n m + + n s n n m n s n n m+ n n s n n m m!! s m m m m + m +!! s m +!! s m m!! s n s m + s n s m m +!! s m m m +!! s

46 Smarandache ζs s + m m +!! s. -8 n, S n m, m!! n. n m!! n m + n. s >, n n n S n n s m m n n S nm m m!! s m m n s n m+ n m n m+ n n s m m!! s n s -9 m m!! s n s m + s n s m n n m n s m!! s m m +!! s n m m n s + m +!! s n m ζs + m +!! s ζs m -8-9 S n S n n s n s + n n ζs m!! s. -0 n n s + S n n s m..4. s, 4, n n S n n S n n 4 π 4 π4 48 m m m +!! s m +!! + π 3 m +!! 4 + π4 45 m m + ζs m m!! + π 8, m!! 4 + π4 96. m!! s. 39

47 Smarandache :.4. s, 4, ζ π 6 [4] S n n ζ m +!! + ζ n m π + 8 m +!! + π 3 n S n n 4 π 4. π4 48 m m m m +!! + π 3 m +!! 4 + π4 45 m m m ζ4 π4 90 m m!! m!! + π 8 m!! m!! 4 + π SMn SMn Smarandache.,, [] SMn P n ζ 3 3 x 3 ln x 3 x + O ln. x SMn, SMd n - d n, d n n. n d n SMd > n. n p, d n n d n SMd + p > p. SMd < n. n pq, p q p < q, SMd + p + q < pq. d n n -?, 40

48 Smarandache.5. n, - n, 8. :, i n, SMd SM, n -. d n ii n p α -. -, SMn SMd SMd + p + p + + αp p α. - d p α d n - p, p,. n -. iii n > n, n p p α pα k k p n -, ii k. SMn SMd SMd + SMp d SMd + p p n.-3 d n d n d n d n -3,. -. iii : n, n -.. n > -, ii iii n, n. n p α pα pα k k, α >, k. SMn αp. A α. p n, n n p, d n SMd p, SMd n d n n p n SMd SMd + SMdp d n d n d n p SMd + p + p + pdn d n d n d n d> + p dn p, -4 n + < dn, -5 dn Dirichlet. -5 n 7. n 6. n, n 4. n n p 4p, p > 3. 4p d 4p SMd SM + SM + SM4 + SMp + SMp + SM4p 4

49 Smarandache p, p 7 n 8. B SMn αp α >. n n p α, n, p. n -, α n p α n SMp i d. i0 d n < n < 8, -. a n, n p α p >, iii, n p α - ; b n 3, n 3p α. n, p, p 3. p, n 3 α -, SMd SMd + SM3d SMd α, d 3 α d α d α d α SMd + 3, 3 α., n 3 α - d α ; p > 3, n 3 p α -, n, iii, n 3 p α -. n 3 p α p 3 -. c n 4, n 4 p α p 3, SMd SMd + SMd + SM4d, d 4 p α d p α d p α d p α p 3, n 4 3 α -, SMd SMd+ SMd+ SM4d 3 SMd+ 4 3 α, d 4 3 α d 3 α d 3 α d 3 α d 3 α d> 3 3 SMd, α, 3.. d 3 α d> p > 3, n 4 p α -, SMd SMd + SMd + SM4d 3 SMd + 8 d 4 p α d p α d p α d p α d p α 3 αα + p + 4 pα, 4 3 α 3 αα + p

50 Smarandache α, fx 4 x α αα + x +, x 3, fx, fx f3 4 3 α 3 αα + + gα. α, gα α. fx f3 gα g > 0,, x 3, fx 0. p > 3, -. d n 5, n 5 p α p 5. p > 5, iii, n 5 p α - ; p, SMd SMd + SM5d SMd α, d 5 α d α d α d α d> Sd, 5 α, 0,. n 5 α d α d> -; p 3, SMd SMd + SM5d SMd + 6, d 5 3 α d 3 α d 3 α d 3 α d 3 α SMd + 6, 5 3 α., n 5 3 α -. e n 6 n 3 p α, iii, n -. f n 7, n 7 p α p 7. p > 7, iii, n 7 p α - ; p, α 4. SMd SMd + SM7d SMd + 5, d 7 α d α d α d α SMd + 5, n 7 α. n 7 α d α -; p 3, SMd SMd + SM7d SMd + 3, d 7 3 α d 3 α d 3 α d 3 α d> 43

51 Smarandache 3 SMd, α, -, 3 3,. d 3 α d> n 7 3 α - ; p 5,. SMd SMd + SM7d SMd + 8, d 7 5 α d 5 α d 5 α d 5 α d 5 α SMd+8, 7 5 α. n 7 5 α - g n 8, n n p α, p α > αα + p, d n p α SMd < SMp α dn p α αα + pdn αα + pn < p α n n, n 8, n n p α -. -, n, Smarandache Smarandache. n n p α pα pα k k, ωn n,, ωn k. Ωn n, Ωn α + α + + α k. n S d ωnωn. -6 d n,..6. d n S d ωnωn. n p α p n p p β. < p < p, α, β. n p p p 3 n p p p 3 n p p p 3 3. n p p p 3 p 4, p < p < p 3 < p 4. : n, S d S, d n 44

52 Smarandache ωnωn 0. -6, n -6. n >, -6 i n, n p α pα pα k k, d n! n, S d. n > -6 S d dn + α + α + α k, d n d n dn Dirichlet. k a k, ωnωn kα + α + + α k, S d α + α +, ωnωn α + α. d n α + α + α + α α α. n p α p n p p β, < p < p, α, β. b k 3, -6 α + α + α 3 + 3α + α + α α i, i α i, α, -7 α + α α + α 3. -7, α i, α, α, -7 α n p p p 3 n p p p 3 n p p p 3. α i, α i,. α + α + α 3 + > 3α + α + α 3 α α α 3 + α α + α α 3 + α α 3 + > α + α + α 3. 45

53 Smarandache α α α 3 + α α + α α 3 + α α 3 + α + α + α 3 α α α 3 + α α + α α 3 + α 3 α + α α α 3 + > 0. -7, -6. c k 4, -6 α + α + α 3 + α 4 + 4α + α + α 3 + α α i, i α i, n p p p 3 p α i, α, α, α 3 α 4 >, α i, α, α, α 3 >, α 4 >, α i, α, α >, α 3 >, α 4 >, α -8 α + α 3 + α α + α 3 + α 4 α α 3 α 4 + α α 3 + α α 4 + α 3 α 4 + α + α 3 + α 4. α >, α 3 >, α 4 >, α i, α >, α >, α +α + > α + α., α 3 >, α 4 >, α 3 + α 4 + > α 3 + α 4. α + α + α 3 + α 4 + > 4α + α α 3 + α 4 4α α 3 + α α 4 + α α 3 + α α 4 > 4α + α + α 3 + α 4., d k > 4, i k. -6. α + α α k + > kα + + α k. 46

54 Smarandache -6.. k i,. k i + α + α α i + > iα + + α i α + α α i + α i+ + > iα + + α i α i+ +. iα + + α i α i+ + i + α + + α i + α i+ iα + + α i α i+ + iα + + α i iα i+ α + + α i α i+ iα i+ α + + α i i + α i+ > iα i+ α + + α i i + iα i+ iα i+ α + + α i i > 0., k i +,. ii n, n α m, m, m p α pα pα k k. S d S d + S d d n d n d n d n 3d n. + d n ωnωn k + α + α + α + + α k. 3d n 3αα + α + α k + > k + α + α + + α k. -9 a k, -9. b k 3, -9, -9. c k > 3, -9, n k > 4,.,.. 47

55 Smarandache SLn SLn,. SLn., Smarandache SLn, n, Smarandache LCM SLn k, n [,,, k], [,,, k],,, k. 3. n, Smarandache SLn : SL, n > n p α pα pα k k n SLn min{p α, pα,, pα k k }. 3.3 n, Smarandache LCM SL n k, [,,, k] n, [,,, k],,, k. SLn., n p α pα pα k k n. 3. n,, SLp α p α. 3. p, SLn max i k {pα i i }. 3- SLp Sp n n p α pα pα r r p SLn Sn, Sn n, 3-3 p, p,, p r, p, α, α,, α r p > p α i i, i,,, r. 3. SLn SLn,. 48

56 SLn 3.. k, x >, SLn π x ln x + k c i x x ln i x + O ln k+ x i c i i, 3,, k. n >, n p α pα pα s s n, [7], SLn max{p α, pα,, pα s s }. 3-4 SLn n A SLn + n B SLn, 3-5 [, x] A B. A n [, x] : p p n p > n; B [, x] n / A. 3-4 A SLn n A Abel n<p x n n p x n π x n x n ln x + p n, n<p p p n<p SLn SLpn p n<p n x n<p x n x n n πn πydy n k b i x ln i n i n ln i x + O p. 3-6 x n ln k+ x, 3-7 n x, b i. n π 6, ln i n n i, 3,, k n A n x n x k SLn n ln x + b i x ln i n n ln i x i + O x n ln k+ x 49

57 π x ln x + Smarandache k c i x x ln i x + O ln k+ x i, 3-8 c i i, 3,, k. B. α,, 3-4 B n n α+ α SLn n B SLnp, p n p n, p n 3-5, p + p + α ln x p min{n, x n } p + SLnp α, α> p x α ln x np α x p α p α p x n α x 3 ln x + x 3 ln x ln x x SLn π x ln x + k c i x x ln i x + O ln k+ x c i i, 3,, k x >, x SLn π i x ln x + O ln x., p α 3.3 SLn Smarandache LCM, SLd n 3-0 d n 50

58 SLn n, d n n. n d n SLd > n., n p α, SLd SLd + p + + p α > p α n. d p α d n n, SLd < n. n d n, n p q, 3 p < q, SLd SLd + p + q < p q n. d p q d n n 3-0,.,., 3.3. n > 600, dn n 9 0, dn Dirichlet, dn d n. :. fn n 9 0 dn., n > 600 fn >. fn, α, f α α ; p > q 3, f p β β fp > fq. α 7, f α >, α 3, f 3 α >, α, f 5 α >, p 7 α, f p α >., f 3 5 f n 8 ; 3 5 ; α 3 β, α + β 9; p 7; p 3 p 5; p p 7; 7 p 3 ; p 3 ; 5 p 3 p 3 p, p, p, p 3, fn >, d, fd 60 >, n d n fd n >. n, n 4 3,, n fn >.. 5

59 Smarandache n, 8. : n, , 3-0. n > n p α pα pα r r p < p < < p r n. n 3-0, α r. n, r. α n 3-0, n p n, d n d > SLd p SLd. SLd SLd + SLp d p + SLd p n. d n d n d n d n, p,,. p >,,,. n 3-0 α, r. SLn max{p α, pα,, pα r r } p α. n n p α 3-0. α i α. n n p, p n, SLd SLd + SLd p SLd + p d n d n d n d n d n d n SLd + p dn n p. 3-3-, d n, SLd < p. 3- p dn > n p. dn > n. n 5, 7, 8, 9, 0, n. n. n 4. n 4 dn 3, p 4 p. p 7, n 8. ii α. n : A p α < n 0. n 3-0 : n d n SLd < d n n 0 n 0 dn. 3- n 9 0 < dn. n > 600 n 8 ; 3 5 ; γ 3 β, γ +β 9; p 7; p 3 p 5; p p 7; 7 p 3 ; p 3 ; 5 p 3 p 3 p, p, p, p 3, n 9 0 < dn. n 600, n, 8, n

60 SLn, n : SLn 5 ; 3 ; 3 3 ; 3 4 ; α, α 7. SLn 5, n n δ 3 β 5 7 γ, δ 0,, 3, 4; β 0,, ; γ 0, δ + β + γ 0. n 3-0; SLn 3 α, α 4, n n β 3 α 5 γ p δ, β < 3 α ; 5 γ < 3 α ; p 7, 3, δ 0, β + γ + δ 0. SLn α, α 7, n n α 3 β 5 γ p δ, α > 3 β ; α > 5 γ ; α > p δ β + γ + δ 0, p 7, 3., n, 8, n 3-0. B p α n 0, p 9α n. d n, SLd n p 9α. n α n p α SLd SLd p i d n i0 d n SLd p i + SLd p i 0 i 9α d n 9α <i α d n [ ] 9α < + p 9α dn + p i dn p α α n < 0 i α [ 9α ] [ 9α ]+ i α 9α + < 5 p α 9α p i + + dn < 4 dn. n 36. n 35, n, n 3-0, n n 4, 8, 9,, 6, 0, 4, 5 7. n 4 p α, n 8 p α, 9 p α, n p α, n 6 p α, n 0 p α, n 4 p α, n 5 p α n 7 p α. : p 3, SLd SLd + SLd + SL4d d 4 p α d p α d p α d p α p α d p α SLd 4 3 α. 3,,. n 4 3 α 3-0. n 4 p α p 5, SLd SLd + SLd + SL4d d 4 p α d p α d p α d p α 53

61 Smarandache d p α SLd 4 p α. SLd SLd SLd 4 p α. d 4 p α d p α d> d p α d> d p α d>p p SLd, p SLd. p 7 p 8,. d p α d>p n 4 p α p 5, n 3-0. n 8 p α 3-0. p 3, SLd SLd + SLd + SL4d + SL8d d 8 3 α d 3 α d 3 α d 3 α d 3 α SLd 8 3 α. d 3 α 4 SLd 8 3 α 4, 4 7,. d 3 α p 5, SLd SLd + SLd + SL4d + SL8d d 8 5 α d 5 α d 5 α d 5 α d 5 α SLd 8 5 α. d 5 α 4 SLd 8 5 α 4, 4 4. d 5 α p 7, SLd SLd + SLd + SL4d + d 8 7 α d 7 α d 7 α d 7 α SLd 8 7 α. d 7 α d> d 7 α SL8d 4 SLd 8 7 α 7, 7 6,. 54 d 7 α d> p, d 8 p α SLd SLd + SLd + SL4d + SL8d d p α d p α d p α d p α + 4 SLd 8 p α. d p α

62 SLn 4 SLd 8 p α 4, 4,. d p α n 8 p α 3-0. n 9 p α 3-0. p, α 4. d α SLd SLd + SL3d + SL9d d 9 α d α d α d α SLd 9 α. d α d 6 SLd 9 α, 69,. n 9 α. d 6 p 5, d 5 α SLd SLd + SL3d + SL9d d 9 5 α d 5 α d 5 α d 5 α SLd 9 5 α. d 5 α d>5 SLd 9 5 α 5, 3 5,. d>5 n 9 5 α. p 7, SLd 9 7 α. d 7 α d>7 7. n 9 7 α. p, SLd 3 + 3p + 3 SLd 9 p α. d p α d> d p α d>p p p. n 9 p α. n p α 3-0. p d p α SLd p α.,. n p α. 55

63 Smarandache n 6 p α 3-0. p 3, α 3 SLd SLd 6 3 α. d 6 3 α d 3 α d>9 5 SLd 6 3 α 7, 7 7,. d 3 α d>9 p 5, SLd 6 5 α. d 5 α d>5 5 SLd 6 5 α 5, 5 80,. d 5 α d>5 p 7, SLd 6 7 α. d 7 α d> 5 SLd 6 7 α 7, 7 4,. d 7 α d> p, 5 d α d> SLd 6 α. d α d> SLd 6 α, 36,. p 3, 5 d 3 α d> SLd 6 3 α. d 3 α d> SLd 6 3 α 3, 3 34,. p 7, SLd 3 + 5p + 5 SLd 6 p α. d p α d> p p. 56 d p α d>p

64 SLn n 0 p α 3-0. p 3 p 7, p 3, d 3 α SLd 0 3 α.,.. p 7, + 6 SLd + 6p + 6 SLd 0 p α. d p α d> d p α d>p p p. n 4 p α 3-0. p 5, p 5, 7 p, SLd SLd 4 5 α, d 4 5 α d 5 α SLd SLd 4 7 α, d 4 7 α d 7 α SLd SLd 4 p α. d 4 p α d p α,. n 4 p α. n 5 p α 3-0. p, α SLd 5 α. d α d>6,. n 5 α. p SLd 5 3 α. d 3 α d>5 3, 3,. n 5 3 α. 7 p p + 3 SLd 5 p α. d p α d>p p p. n 5 p α. 57

65 Smarandache p p + 3 SLd 5 p α. d p α d>p p p. n 5 p α. n 7 p α 3-0. p, α SLd 7 α. d α d> 4 SLd 7 α 8, 8 3,. d α d> p 5, 7,, 3, 7, 9, 3 p 9, SLd SLd 7 5 α, d 7 5 α d 5 α SLd SLd 7 7 α, d 7 7 α d 7 α SLd SLd 7 α, d 7 α d α SLd SLd 7 3 α, d 7 3 α d 3 α SLd SLd 7 7 α, d 7 7 α d 7 α SLd SLd 7 9 α, d 7 9 α d 9 α SLd SLd 7 3 α, d 7 3 α d 3 α SLd SLd 7 p α. d 7 p α d p α,. n 7 p α SLn d n SLd,

66 SLn d n n, n > n n, 36, n, 3-4.,., n, n p α pα pα r r n p < p < < p r. α,. : n >, n p α pα pα s s n, SLn SLn max{p α, pα,, pα r r }. 3-5 α n d n SLd m. n p n, d n d >, SLp d SLd, m d n SLd d n SLd + d n SLp d d n SLd + d n SLd + p d n SLd + p, n m d n n SLd + n p p. 3-6 n d n, SLd n m, n p, 3-6 p., α, n >, SLn,. : n p α pα pα s s n. SLn, SLn p s α s. n p α pα p s n p s. 3-4 m, d n SLp s d p s, m d n SLd SLd + SLp s d d n d n d n SLd + d n p s d n SLd + dn p s,

67 Smarandache dn n Dirichlet. d n, SLd, p s. 3-7 p s dn α + α + α s +., p s α i +, i s. α i + p s α i p s. p α i i p p s i + ps > p s, SLn p s p α. n p α,. : p n p α. d n SLd α i0 SLp i + p + p + + p α + p + p + + p α p α. 3-8 p α, + p + p + p α, n n >, p n p n,. 3.5 Sn SLn, Sd SLd 3-9 d n d n,, , n, α p p p k, k, α 0,, < p < < p k. :, Sn SLn n 3-9. n >, n p α pα pα k k p < p < < p k n. Sn SLn, Sn max{sp α, S pα, Spα k k } Spα i i α i p i SLn max{p α, pα,, pα k k } pα j j, 60

68 SLn p α j j α p α pα pα k k p α i i α i p i., n >, n < p < < p k, n > α 0,, a α α α k., n p p p k n p p p k, n d, Sd SLd, 3-9. b α i, Sp α i i 3-9. α, α α α k, c p 3, n 4 3n n, d n Sd α i p i, SLp α i i p α i i. d n Sd + d n Sd + d n S4d + d n S3d + d n S6d + d n Sd Sd + + Sd Sd + d n d n d n 3 + Sd Sd Sd d n d n d n + 6 d n Sd, d n SLd +6 d n SLd, d n Sd d n SLd d p > 3, n 4 n 4 n, d n Sd d n Sd SLd SLd, 3-9. d n d n 3 α 3, 3-9 S α α, SL α α > α, 3-9., 3-9 n, α p p p k α 0,, < p < < p k.. A 3-9, 3.5. s Res >, n A ζs Riemann zeta-. n s ζs 4 s + s + ζs 4 s + s, 6

69 Smarandache : n A n s Euler [4].7 Möbius, n µn n s + 4 s + 4 s + s n n µn n s p + p s x, p 7 π x + O x. n A + p s + 4 s ζs 4 s + s + ζs 4 s + s. : Möbius µn µn + µn µd + µd n x 4 d n n x 4 d n n A n n µd + µd µd + d l x d x l x d d x µd d l x 4 d l l x d + d x 4 d µd l x 4d l x µd d + O + µd d x x d x x d x µd d + O d x µd d + x 8 d x d d x 4 d µd + x µd d d x 4 d + O x, d x 4 d x 8d + O µd 8d + p s p + O + s l x 4d l d x 4 d µd µd µn n 6 π + O x n x n µn n 8 π + O x, 6

70 SLn x n A 6 π + O. + x 8 x 8 π + O x + O x 7 π x + O x. ζ π 6 π4 π8, ζ4, ζ , s, 4, n A n 63 4π n π 4. n A 3.6 SLn SLn,., 3.6. n > n p α pα...pα k k n, SLn max i k {pα i i }. 3-0 : [7] p n, n p n a p α + a p α + + a s p α s α s > α s > > α 0, s a i p i,,, s, an, p a i α p n αn + i i [ ] n p i n an, p, p [x] x. : [x] [ ] [ n a p α + a p α + + a s p α s p i p i ] 63

71 Smarandache s a j p αj i, α k < i α k jk 0, i α s. αn + i [ ] n p i + i α s j a j p αj k j k s j [ a p α + a p α ] + + a s p α s p i s a j + p + p + + p αj j a j pα j p p n an, p. p s a j p α j a j j n, [ ] n ln n SLn! + O. n : n! p α pα...pα k k n!. 3.6., SLn! max i k {pα i i } p α., p n, n a p α + a p α + + a s p α s α s > α s > > α 0, a i p i,,, s, an, p a i [8], i α p n αn i [ ] n p i n an, p. 3- p 3-, k p k n < p k+, 3- [ ] n s [ ] n α p i p i, i i SLn! p α p n an,p p e n an,p p ln p. 64

72 SLn [9] [8] s i [ ] n p i < s i n p i n p, αn n an, p, p an, p an, p p Oln n ln p, α i αn, p αp s i α i αp i p ln n, ln p [ ] n p i n p + O n p i + O ln n ln p i ln n, ln p,, p i < p j p α i i p α i i p n p i +O ln n ln p i i, e n ln pi p i +Oln n. ln p i p i < ln p j p j., p i, α ln n n+o ln. x x + Ox. 3-0 SLn! max p n pαp α n ln n +O ln n+oln n [ ln n O n [ + O ] n ] n ln n n 65

73 Smarandache 3.6. n, lim [SLn!] n lnsln! lim ln. n n n 3.7 SLn SLn, x >, SLn P n 5 5 ζ x 5 ln x + O x 5 ln x ζs Riemann zeta-, P n n. :, n >, n p α pα pα s s n, [7], SLn max{p α, pα,, pα s s }. 3- SLn P n. 3-3 n [, x] A, B, C D : A: P n n, n m P n, m < P n; B: n 3 < P n n, n m P n, m < n 3 ; C: n 3 < p < P n n, n m p P n, p ; D: P n n 3., n A, SLn P n. P n P n n A SLn P n n A, n C, SLn P n. SLn P n P n P n n C n C B. Abel SLn P n SLmp P mp n B mp x m<p

74 m x 3 m x 3 m x 3 SLn m<p x m p p [ x m π x m 5 ζ 5 x 5 5m 5 ln x m x 5 ln x + O x ] m 4 y 3 πydx + O m 5 + x m m x 5 + O m 5 ln x m 5 x ln, 3-6 x ζ s Riemann zeta-., D. n D, SLn p α. α, SLn p P n, SLn P n 0. α. P n n 3, x SLn P n n D n D p α + n 3 mp α x α, p<x 3 p α x α, p x 3 SL n + P n p α x α, p x 3 3-4, 3-5, p α m x p α + x p α + x 5 3 x SLn P n n A SLn P n + n B. + SLn P n + n C n D 5 5 ζ x 5 5 x ln x + O ln. x SLn P n SLn P n 3.8 Smarandache SLn Smarandache, SLn SLn. n, SLn SLn. 67

75 Smarandache SLn n SLd > n., n p α d n, SLd SLd + p + + p α > p α n. d p α d n n, SLd < n. n d n, n p q, 3 p < q, SLd SLd + p + q < p q n. d p q d n, n, SLd n? d n n, d n SLd n 3-8 d n n m 3, m 3dm, dm. : m 3 m p α pα pα k k m. i α i 4 i k, m k α dm p i i α i > 3, i m 3dm. ii max i k {α i} α j 3, p j 3 m k α dm p i i α i > 3, i p j, m, 68 m k dm i p i α i α i ,

76 SLn m 3dm. iii max i k {α i} α j i,,, k, p j 3, m k α dm p i i α i , i p j, n 3 q 5, m k dm i p i α i α i > 3, m 3dm. iv α i i,,, k, m k, p 3, dm 3 + > 3; k, m q 7, m dm > 3; m 3 q 5, m 5, k 3, m dm + 3dm m dm m dm > 3; 7 + > 3; > 3, m 3.8. SLd n n, 0. d n : n. n > n p α pα pα k k n, n p α, n k. SLn min{p α, pα,, pα k k } pα. n mp α, SLd d n α SLdp i i0 d m d m SLd + d m, SLdp i p i, mp α d m SLd + α i α SLdp i mp α. i d m d m p i d m SLd + dm α i p i 69

77 Smarandache d m SLd + ppα dm. p p α, d m SLd p α, SLd m p α + ppα p dm < dm + dm 3dm. p pα p d m n mp α, m < 3dm, m 3dm, n mp α, m 3. m < 3, n mp α. m 7, 9,, m 3dm, n mp α. m, n p α, p 3. SLd SLd + SLd SLd + α + d p α d p α d p α d p α. + p + p + + p α + α +. α, + p + + p p 5, n p 0 α, p 3, d 3 SLd > n. d 3 SLd > 3, p 5, d p SLd + p + p p + p < p, d p SLd < n. α 3, α 3, 70 + p + p + + p α + α + < p α SLd < n. d p α 3 m 3, n 3p α, p, α p 5, p, SLd SLd + SL3d α+ + 3α +. d 3 α d α d α α, 3, d 3 α SLd < n. d 3 α SLd > n; α 4, α+ + 3α + < 3 α,

78 SLn p 5, d 3p α SLd SLd + SL3d d p α d p α + p + p + + p α + 3α + < 3p α, SLd < n. d 3p α 4 m 4, n 4p α, p 3, p 3, SLd α +6α+ < 4 3 α, SLd < d 4 3 α d 4 3 α n. p 5, SLd +p+p + +p α +6α+ < 4p α, SLd < d 4 p α d 4p α n. 5 m 5, n 5p α, p, α 3, SLd α+ + 5α 5 α n. d 5 α p 3, α, SLd α + 5α + 3 < 5 3 α n. d 5 3 α p > 5, SLd + p + p + + p α + 5α + < 5p α n. d 5 p α 6 m 6, p 5, SLd + p + p + + p α + 7α + < 6p α n. d 6p α 7 m 8, p 3, α, SLd α + 4α + 8 < 8 3 α n. d 8 3 α p 5, α, SLd α + 4α + < 8 5 α n. d 8 5 α p 7, α, SLd α + 4α + 3 < 8 7 α n. d 8 7 α p > 7, SLd + p + p + + p α + 4α + < 8p α n. d 8p α 8 m 0, p 3, α, SLd α + 9α + 7 < 0 3 α n. d 0 3 α p > 5, SLd + p + p + + p α + 9α + < 0p α n. d 0p α 9 m, p 5, SLd + p + p + + p α + 4α + < p α n. d p α 7

79 Smarandache, n, F.Smarandache SLn x > k, SLn x k i a i x ln i x + O ln k+, x a, a i i, 3,, k. : x >, πx x, πx p x. [0] k, πx x k i c i x ln i x + O ln k+, 3-9 x c, c i i, 3,, k. [, x] n : n ; A; B. n, n p α, α ; SLn n. n, n, SLn SLn min{p α, pα,, pα s s } n. 3-9, Abel, SLn + SLn + p + O n p x 7 SLn + n A x πx x x x k 3 i [ k i y x 3 n B πydy + O c i ln i x + O k i a i ln i x + O ln k+ x x 3 c i y ln i y + O x ln k+ x ln k+ y, ] dy n B

80 SLn a, a i i, 3,, k x > k, SLn pn x 5 k i 5 b i x ln i x + O ln k+, x pn n, b 5, b i i, 3,, k. : [, x] n : n ; ωn C; ωn D; ωn 3 E, ωn n,. n p α pα pα s s n, ωn s. pn n. p 0 SL, SLn pn + SLn pn SLn pn + n C SLn pn. n D + n E, n C n p α SLn SLp α p α pn pslp pp Abel SLn pn p α p p α p n C p x x π x p α x p 4 p 3 + p + O x 3 3 α ln x p x α 4y 3 πydy + O x 7 3 p α x α p α [ k ] x 5 c i ln i x + O ln k+ x i [ x k ] 4y 3 c i y y 3 ln i y + O ln k+ dy y i k 5 x 5 b i x ln i x + O ln k+ x i,

81 Smarandache b 5, b i i, 3,, k. n E, n p α pα pα s s, SLn [ SL p α SL pα SL pα s s ] ωn n ωn n SLn pn n E n 3 x n D, n, n p α pα p < p. SLn n. α, SLn p pn, SLn pn 0. SLn pn n D p α x p α x p α α p α + p α pα x p x p α x p α 3-30, SLn pn x 5 k i [ SL p α pα p ] p x b i x ln i x + O ln k+, x pn n, b 5, b i i, 3,, k n, Smarandache Un U, n >, n p α pα pα s s n, Un max{α p, α p,, α s p s } V n : V, n >, n p α pα pα s s n, V n min{α p, α p,, α s p s }. 74

82 SLn V n V, V, V 3 3, V 4 4, V 5 5, V 6 3, V 7 7, V 8 6, V 9 6, V 0, V, V 3, V 3 3, V 4, V 5 3,. V n,. V n Un, V n Un., V n, x > k, V n pn x 3 k i 3 c i x ln i x + O ln k+, x pn n, c i i,,, k, c 3. :, k n >, n p α pα pα s s n, n [, x] A B A: ωn. A p α x, p, α. B: ωn, ωn n. A B V n pn + n A V n pn + n B n A, n p α, V n αp. V n pn V n pn α p p n A p α x Abel p x k i x 3 p x π x a i x 3 ln i x + O k i x 3 ln k+ x p x x 3 3 c i x ln i x + O ln k+ x p + <α ln x p x α y πy dy α ln x p x α αp p αp p [ x k ] a i y y 3 ln i y + O ln k+ dy y i,

83 Smarandache c i i,,, k, c 3., <α ln x p x α x 3 π x , αp p V n pn x 3 n A <α ln x p x 3 α p ln 3 x x ln x k i 3 c i x ln i x + O ln k x n B, n p α pα pα s s n, s. α, V n pn p p 0. V n pn 0, α, V n α p., V n pn n B α ln x α ln x p x α+ p x α+ p <m x p α α p p α m x α, pm>p x p α αp p V p α m p x 4 3 ln x , V n pn x 3 k i 3 c i x ln i x + O ln k+, x c i i,,, k, c x > k, V n x k i d i x ln i x + O ln k+, x 76

84 SLn d i i,,, k, d. : V n p x p, V p p. V p + V p α + V p α m p + O p x x πx p α x α α ln x p x α x 3 p α m x α, pm>p α p + O πy dy + O x 3 ln x k a i x x ln i x + O ln k+ x i k x d i x ln i x + O ln k+ x i x 3 [ k i α ln x p x α+ d i i,,, k, d.., α p p α m x ] a i y y ln i y + O ln k+ dy y 3. Smarandache LCM Smarandache LCM SL n : SL, SL, SL 3, SL 4, SL 5, SL 6 3, SL 7, SL 8, SL 9, SL 0., n, SL n,. SL n Dirichlet SL n,.,. 3.. n, p, α. SL n p α, : SL n k, SL n : [,,, k] n k + n. [,,, k, k + ] n, SL n k +, SL n k. k + p α pα pα s s k +, p i, p < p < < p s, α i, i,,, s. 77

85 Smarandache s >, p α k, p α pα s s k, p α, pα pα s s, p α [,,, k], p α pα s s [,,, k], p α pα pα s s [,,, k]. k + [,,, k], k + n,. s. k + p α, k p α, SL n p α Ln,,, n, Ln [,,, n], 3 cln n 5 lnln n + O n exp, ln ln n 5 c. : s >,, n SL n n s n SL n n s ζs α p p α p s [,,, p α ] s, ζs Riemann zeta-, p. : SL n : [,,, k] n, [,,, k] n, ln[,,, k] ln n., 3.. SL n k ln n, s >, Dirichlet n SL n p α, [,,, p α ] n. p m, s >, SL n n s ln n n s. SL n n s. 3.., n [,,, p α ] m n SL n n s α p n SL np α p α n s α p m p m p α [,,, p α ] s m s 78

86 SLn p α [,,, p α ] s α α p p α [,,, p α ] s p α p m ζs m p m m m s p α [,,, p α ] s m m s α p α p m s m s p α p s [,,, p α ] s p α p s [,,, p α ] s. m p s m s p s 3.. s, ζ π 6, SL n 3.. n, n n SL n n π 6 α p p α p [,,, p α ], p. 3.. x >, SL n c x + Oln x, c : α p SL n p α p [,,, p α. ] SL n, , x [,,,p α ] m x p m [,,,p α ] x [,,,p α ] x p α p α [,,,p α ] x p α m x [,,,p α ] p m x [,,, p α ] x [,,, p α ] + O p α p [,,, p α ] + O [,,,p α ] x p α 79

87 Smarandache c x α c x + O ln x, α. p p p α p [,,, p α. ] p α p [,,, p α ] + O ln x 3. Smarandache LCM 3.. k, x >, [SLn Sn] 3 3 ζ x 3 k i 3 c i x ln i x + O ln k+, x ζs Riemann zeta-, c i i,,, k. :, n >, n p α pα pα s s, n Sn max{sp α, Spα,, Spα s s } Sp α 3-40 SLn max{p α, pα,, pα s s }. 3-4 [SLn Sn] [SLn Sn] + [SLn Sn], 3-4 n A n B A B [, x]. A {n SLn p, p, n [, x]}; B {n SLn p α, α α 3, n [, x]}. n A n p α pα pα s s, n p m, p m, p α i i p, i,, s. Sp α i i α i p i α i ln n, SLn Sn [SLn Sn] [ p Smp ] n A mp x SLm<p mp x SLm<p p 4 p Smp + S mp 80

88 SLn mp x SLm<p m x m x m x p 4 + O m<p x m p x m p x m mp x p 4 + O p p ln x + O p x p 4 + O m x p <m x p p 4 p m p 4 mp x + O x + O x p ln x p 4 + O x πx k i a i x x ln i x + O ln k+, x a i i,,, k, a., p x m p 4 x x m π m x x m π m 5 x 5 m 5 k i x m 3 x m 3 b i ln i x m + O 4y 3 πydy 4y 3 [ k i m 5 ] a i y y ln i y + O ln k+ dy y x 5 ln k+, 3-44 x m x, b i, b. 5 ln i m ζ, i,, 3,, k. m m [SLn Sn] n A m x 5 ζ 5 [ 5 x 5 m 3 x 5 k i k i m b i ln i x m m 5 + O m 5 x 5 ln k+ x ] + O x 5 c i x ln i x + O ln k+, 3-45 x 8

89 Smarandache c i i,, 3,, k, c. B. n B. n B SLn p, Sn p. [SLn Sn] [p p] 0. SLn p α with α 3, [SLn Sn] [p α Sn] p α + α p α ln n. [SLn Sn] n B np α x α 3 3-4, [SLn Sn] 5 5 ζ x 5 c i i,, 3,, k, c.. p α + p ln n x k i 5 c i x ln i x + O ln k+, x 8

90 Ln Ln Ln,., n, Ln n n. Ln [,,, n]. 4. Ωn n, n p α pα pα r r n, Ωn Ωp α pα pα r r α + α + + α r. 4.3 Mangoldt Λn, { ln p, n p α, p, α ; Λn 0,. 4.4 Smarandache LCM SLS L, L,, Ln,. Smarandache LCM Smarandache SLOS n, SLOSn [, 3, 5,, n ]. 4. Ln Ln, Ln. 4.. x > 0, θx p x 3 cln x 5 ln p x + O x exp, ln ln x 5 c > 0, p x p x p. :, [6]. 83

91 Smarandache 4.. p n : n, Ln p p n n e + O exp c ln n 3 5, ln ln n 5 p n p. Ln [,,, n ] p α pα pα s s 4- Ln, α i αp i,, 3,, n p i. Ln p p n n exp n ln Ln exp n ln Ln n ln p n p p n p. 4- ln Ln ln p n p ln p α pα pα s s ln p n p αp ln p ln p αp ln p p n p n p n αp ln p + αp ln p p n 3 + n 3 <p n n<p n αp ln p. 4-3, n < p i n, αp i. n 3 < p i n, αp i, αp i 3, p 3 i > n. p i n. p i n 3, αp i , αp ln p ln p n 3 <p n n 3 <p n n 3 <p n ln p 84

92 Ln θn θ n n + O exp c ln n 3 5 ln ln n p n 3 n<p n αp ln p p n 3 αp ln p O ln n O 4-3, 4-4, ln Ln ln p n p O n<p n ln p αp ln p + p n 3 n 3 ln n + n + O ln n n 3 O n 3 ln n. 4-6 ln n n 3 <p n n exp 3 cln n 5 n + O n exp. ln ln n 5 ln p cln n 3 5 ln ln n 5 Ln p p n n exp ln Ln ln n [ exp n exp p n p [ 3 cln n 5 n + O n exp ln ln n 5 [ + O exp [ e + O exp e + O exp cln n 3 5 ] ln ln n 5 ] cln n 3 5 ln ln n 5 3 cln n 5 ln ln n 5. ]]., n, 85

93 Smarandache 4.. lim n Ln p p n n e. 4.3 Ln k Ln, Ln k,., 4.3. n k, p n k : Ln k p p n k n k c k e + O exp ln n 3 5, ln k ln + ln ln n 5 p n k p. Ln k [,,, n k ] p α pα pα s s 4-7 Ln k. α i : αp i,, 3,, n k p i. Ln k p p n k n k exp n k Ln k ln exp n k ln Ln k ln p n k p ln Ln k ln p n k p ln p α pα pα s s ln αp ln p ln p p n k p n k αp ln p p n k p n k p p n k p. 4-8

94 Ln p n k+ k : M 0 + αp i. 4.. M i i ln p n k i+ <p n k i i ln p i p n k i n k i αp ln p + k i n k i+ <p n k i αp ln p k M i. 4-9 i n k i+ < p n k i, p n k i+ n k i+ + O n k i ln p c k exp i ln n ln k ln i + ln ln n 5 k M i n k i c k + O n k exp ln n ln k ln + ln ln n 5, M 0 O k k + ln n O k k + ln n n k k k+ p n k+ k k+ ln n O n k k+ ln n , 4-4- ln Ln k c k ln p n k + O n k exp ln n ln k ln + ln ln n p n k Ln k p p n k n k 87

95 Smarandache exp n k ln Ln k ln exp n k exp + O p n k p c k n k + O n k exp ln n 3 5 ln k ln + ln ln n 5 c k exp ln n 3 5 ln k ln + ln ln n 5 c k e exp O exp ln n 3 5 ln k ln + ln ln n 5 c k e + O exp ln n 3 5 ln k ln + ln ln n 5 c k e + O exp ln n 3 5. ln k ln + ln ln n n k, lim n Ln k p p n k n k e. 4.4 Ln Ln Smarandache LCM, Smarandache LCM SLOS. Smarandache LCM SLES: n, SLESn [, 4, 6,, n], SLES, 4, 4, 40,. T n n, T n+ SLES n T n SLOS. Ln SLOSn Smarandache.,,. Ln SLOSn, 88

96 Ln, n. [lnln] α 4.4. a, b, a b [a, b] ab a, b, a, b a b., a, b, [a, b] ab. : [] n, Ln Ln SLOSn SLOS [ ], n+ [x] x. Ln SLOSn Ln [,,, n] [[, 3,, n ], [, 4,, n]] [[, 3,, n ], [,,, n]] [SLOSn, Ln] Ln SLOSn Ln, SLOSn Ln SLOSn Ln, SLOSn Ln Ln SLOSn Ln, SLOSn. 4-4 Ln, SLOSn SLOS [ n + T i, T i α i t i i, t i. Ln [,, n] [ α t, α t,, α n t n ] [ ] ] n + [, α 3, 5,,, ]. 4-5 α max{α, α,, α n }. [ [ ] ] n + Ln, SLOSn α, 3,,, [, 3,, n ] 89

97 Smarandache [, 3,, [, 3,, SLOS [ n + [ n + [ n + ] ], [, 3,, n ] ] ] ] ,, Ln Ln SLOSn SLOS [ ]. n n, c. lnln n + O n exp cln n 3 5 ln ln n 5, Ln [,,, n] p α pα pα n n, p i Ln, α i,, 3,, n p i. lnln ln p α pα pα n n p n αp ln p ln p + ln p p n p nαp θn + αp ln p + αp ln p, n p n p n p i > n, α i. α i, p i > n, pα i i. αp ln p 0. n p n > n, p < n p α i i 90 lnln θn + αp ln p. p n < n, α i < ln n, p < ln n, 4.. lnln θn + O ln n p n

98 Ln θn + O ln n n + O n exp n + O n exp n α >, α., 4.4. [lnln] α n ln n 3 cln n 5 ln ln n 5 3 cln n 5 ln ln n 5 [lnln] α + O n ln n. [ 3 cln n 5 n + O n exp ln ln n 5 n [ α + O exp cln n 3 5 ln ln n 5 ]α ]α, n [lnln] α n n [ α + O exp cln n 3 5 ln ln n 5 ] α. n α >. lim n n [ α + O exp n α cln n 3 5 ln ln n 5 ] α, α >, α. nα n [lnln] α, α. 4.5 Ln Ln,, Ln,. 9

99 Smarandache 4.5. n >, Ln exp θ n k exp Λk, k k n expy e y, θx p x Mangoldt. ln p, p x p x p, Λn Ln [,,, n] p α pα pα s s p n p αp 4-6 Ln. i s, < k n, p α i i k., 4-6 s Ln [,,, n] p α pα pα s s exp α t ln p t exp αp ln p exp p n k t n k+ <p n k αp ln p. 4-7 n k+ < p n k, p k n, p k+ > n αp k., 4-7 Ln exp k ln p θx p x k exp k exp exp exp k k k k n k+ <p n k [ k θ n k+ <p n k n k ln p ] θ n k+ [ ] kθ n k k + θ n k+ + θ n k+ θ n k exp k n Λk ln p, Λn Mangoldt.., 9

100 Ln dn Dirichlet, n p α pα pα k k 4.5. n >, Ω Ln k π n k. n., Ωn 4.5., Ω Ln p n αp k k k k k [ k π k n k+ <p n k n k n k+ <p n k π ] n k+ αp k n k+ <p n k [ ] kπ n k k + π n k+ + π n k+ π n k, k k i i p i m k p k. πx p x n >, d Ln exp ln + π n k, k k expy e y, πx p x. Dirichlet dn, d Ln 93

101 Smarandache αp + exp ln[αp + ] p n exp k exp k n k+ <p n k n k+ <p n k exp lnk + k exp lnk + exp exp k k k p n ln[αp + ] lnk + n k+ <p n k [ π ] n k π n k+ [ lnkπ n k lnk + π n k+ + ln + ] π n k k ln + π n k. k., 4.5. n >, lim [Ln] n e lim [d Ln] ΩLn, n n e θx ln p x + O x exp c ln x 3 5 ln ln x 5 p x πx x x ln x + O ln, x c > 0 [7], 4.5. n >, n 94 Ω Ln n ln n + O ln n.

102 Ln 4.6 Smarandache LCM x, x,, x n [x, x,, x n ] x, x,, x n r T r, n [n, n +,, n + r ], [,,, r] T r, n r Smarandache LCM. T n, n, lim [T n, n] n, n Smarandache LCM Ln n, [,,,, n ] [n, n +,, n ]. : i n, n n, n +,, n i, i [n, n +,, n ].,,, n [n, n +,, n ], [,,, n] [n, n +,, n ]. a b ab [a, b]a, b, a b [a, b] b, a, b a, [,, 3,, s, s +,, t] [[,,, s], [s, s +,, t]], [,, 3,, n ] [[,,, n], [n, n +,, n ]] [,,, n] [n, n +,, n ] [,,, n], [n, n +,, n ] [,,, n] [n, n +,, n ] [,,, n] [n, n +,, n ] n, 3 T n, n cln n 5 n e + O n exp, ln ln n 5 95

103 Smarandache c, expy e y. n, 4.6. T n, n n [ ] [n, n +,, n ] n [,,, n] [ ] [,,, n ] n [,,, n] [ ] Ln n. 4-8 Ln 4-8 Ln [,,, n ] p α pα pα i i p α s s Ln [,,, n] p β pβ pβ j j p β t t Ln Ln, α i αp i β j βp j,,, n p i,,, n p j. Ln Ln n lnln lnln ln p α pα pα i i p n p n exp n exp p α s s ln βp ln p Ln ln Ln ln Ln ln Ln n p β pβ pβ j j p β t t. 4-9 αp ln p p n p n ln p + αp ln p ln p + ln p p nβp ln p + αp ln p + αp ln p p n p n n <p n ln p βp ln p βp ln p. 4-0 p n n<p n p n p n 96

104 Ln n < p i n, α i. α i, p i > n, p α i i > n, p α i i n., n < p j n, β j. 4.. αp ln p βp ln p n <p n n<p n n<p n 3 cln n 5 ln p θn θn n + O n exp. 4- ln ln n 5 α i, α i < ln n, p i n, αp ln p O ln n ln p O n ln n. 4-3 p n p n p n βp ln p O ln n ln p O n ln n. 4-4 p n 4-0, 4-, 4-, , lnln lnln ln p + αp ln p βp ln p n<p n p n p n θn θn + O n ln n 3 cln n 5 n + O n exp. 4-5 ln ln n [ T n, n Ln n Ln exp [ exp n exp ] n ln Ln ln Ln n [ 3 cln n 5 n + O n exp ln ln n 5 [ + O exp [ e + O exp cln n 3 5 ] ln ln n 5 ] cln n 3 5 ln ln n 5 ]] 97

105 Smarandache 3 cln n 5 e + O exp, ln ln n 5 expy e y , lim n [T n, n] n lim n [Ln] n e n lim n T n, n n, [ 3 ] cln n 5 lim e + O exp e. n ln ln n 5 lim [Ln] n e. n [Ln] n exp n ln Ln lnln ln p β pβ pβ t t p n βp ln p ln p + ln p p n p nβp ln p + βp ln p + βp ln p p n p n n p n 3 cln n 5 n + O n exp. ln ln n 5 [Ln] n exp n [ exp n e + O ln Ln [ 3 cln n 5 n + O n exp ln ln n 5 exp cln n 3 5 ln ln n 5. ]] 98

106 Ln. lim [Ln] n e. n 4.7 F.Smarandache, Ln, SLn. SLn p n n, d n p n+ p n, p n x d n x 3 8 +ε, ε. : D.R.Heath-Brown, [3] x, SLn x + O x 3 8 +ε, ε. : p β p β p s β s Ln, n < p i n, β i βp i. βp Ln [,,, n] p βp p βp p s s p βp p. n<p n p n p i βp i n, n, n p πn n βp i, Sn Sp βp i i βp i p i βp i n n ln n p πn, Sn, n SLn SLn max{sp β, Sp β,, Sp s β s } p πn. SLn p πn + O 99

107 Smarandache p n<p p πn + p n<p 3 p πn + + p πx n<x p πn + O p p p + p p 3 p + + p πx x p πx + O p p + p p p 3 p + p 3 p x p πx + x p πx + O x p n+ p n + O. p n x 4.7. SLn x + O x 3 8 +ε.. 00

108 , :, Smarandache,.,,. 5.,, Dirichlet fs a s,. n n 5.. m, n mm, n [4]., 6, 5, 8,., 5, 9, 3,, 4n +,. an max{mm : mm n}, an n. 5.. s >, fs, f n a n 5 3 π 6 ln. :, an an mm m + m + mm 4m +. fs n a s n m n anmm a s n m 4m + m s m s. s >, s >, fs. f. f m 4m + m m 0

109 Smarandache m m m + m + ζ + m 8 mm m m m 4m m m m ζ 4 ζ m 8 mm 8 0ζ mm m 0ζ 6 m m m m 0ζ , 8 mm ζs Riemann zeta-. ln + x ln + x x x + x3 3 x n x n +. n x ln ζ π 6. f 0ζ 6 ln 5 3 π 6 ln. 5., nn,,,. 5.. x >, 0 ln x + γ + 5 ln + O an x,

110 γ Euler. x >, m mm x < m + m +, x + 4 < m + 8x +. 4 m x <. 5.., an. m k 4k + kk + m k m k + 3 k ln m + γ + O ln x + γ + 5 ln + O n mm < kk + O m + 6 ln + O x. ln, nn bn x mm mm m 5.3, an, an n, bn bn n an an. bn, bn σn, bn Euler n >, an mm, n m + O. 03

111 Smarandache : n >, an mm. an mm n mm n < m + m +. 8n + 4 m + 8n +, 4 n < m + m + m < 3 8n + 4 m > 3 + 8n +. 4 m 3 + 8n + 4 < m + 8n +. 4 n m + O fx n fm > 0 m β d m x σfm βa n n + xn+ + Ox n log n x, Nd d, a n fx x n. Nm,. : [4] x >, ϕn 3 π x + Ox log x. : [4], [] x > n<x bn 3 x 3 + Ox. 04

112 : x, M MM < x < M + M +. n [mm, m + m +, bn [0, 4m] bn bn + bn n MM MM < i + i m M i 4m i x MM x MM m4m + + O m M 4 MM + M + + Ox 3 3 x 3 + Ox x >, β. : d σfan 3 n+ βa n n + n + x n+ + Ox n+ log n x, Nd d, a n fx x n. Nm, an x MM < M + M + MM 4M +, σfan n MM m M m M i 4m σfan + i 4M MM < σfi + σfi σfan βan n + 4mn+ + O4m n log n 4m + βa n n + 4Mn+ + O 4M n log n 4M 4n+ βa n m n+ + OM n+ log n M n + m M 4 n+ βa n n + n + M n+ + OM n+ log n M 05

113 Smarandache 3 n+ βa n n + n + x n+ + Ox n+ log n x x >, ϕan 4 π x 3 + Ox log x Smarandache Smarandache, n K s n n, Smarandache Kn Kn m, m nn + + k, k n m s >, K s n a b n n Kn 3 ln ; K n 08 π + ln. 7 n,, 5-., n, nn + 3 Kn n, Kn ; n, Kn nn +., n < Kn < n + 3 n + 3 s K s n n s. 06

114 , s >, 5-. a, Kn n Kn Kn + Kn n n n n + + nn + n n 3 n + n + n n + n n 3 lim N n + n + n N n N 3 lim N n N n. 5- n N n N N >, [4] 3. n ln N + γ + O, 5-3 N n N γ Euler , n Kn [ 3 lim lnn + γ + ] N ln N γ + O + N 3 ln a. b, Kn, n K n n n K n + K n n n n + + n n n n n + 7 n + n + 9 n + 9 n + ;

115 Smarandache n n + n + + n n + n + ; 5-6 n n π 8 n n + π , 5-4, 5-5, , n K n 7 n + + n 9 n n + n + + n 4 n + n + n n [ 7 lim ] N n + n n N n N [ + lim ] N n + + π n 4 + π 4 4 n N n N [ 7 lim N + ln N lnn + O N + π ln π. 7 n + + π 7 + π 6 36 ] + π n + b.., s, 5-. s,. 5.5 S p n S p n n, p S p n m p n m. S p n min{m : p n m!}, p., 5.5. p >, 08 β β p β p p, 5-8

116 β : 5-8. T T β β pp + pβ p β β p β, β p β p + p + 3 p 3 +, T T p p + p p 4 +. p p + p + p 3 + p T p p S S β β p β p + 4 p + 9 p p 4, S p p + 4 p p p 5 +. β β p β, 5-8 S p p n + n p n+ p + p p p p + p, n n S pp + p p, x, S p n [ ln x + lnp + γ p ] p p ln p + O ln xp x ln p, 09

117 Smarandache γ Euler. x >, x. p, [8] p S p x p x + O ln p ln x. 5-0 y > Euler [4] n ln y + γ + O, 5- y n y γ Euler. n, S p n m, S p n p α m α n S p n m. S p n m S p nm 5-0 m m p x, p O ln p ln x. S p n m, n lnxp, 5- ln p S p n m ln x m + O lnxp x ln p ln p 0 m S p nm α m p x p α m α lnp x ln p α lnp x ln p p p p α O x α p α α p α α m + O m p x S p nm m p x p α m,p α lnp x ln p + O ln xp m p x p α α p α x ln p m + O m p m. ln xp x ln p m p x p α+ { p x ln p α + γ + ln p p + ln xp x ln p ln m + O xp x ln p

118 p p p p p ln x + lnp + γ + ln p p α ln p ln xp p α + O x ln p α ln x + lnp + γ + ln p p ln xp p + p ln p + O p x ln p [ ln x + lnp + γ α p ] p ln p + O α p α ln xp x ln p.., p, x, ln x + γ ln + O S n ln x x S 3 n [ln x + ln + γ 3 ] ln 3 + O ln x x. 5.6 S p n S p n, S p n k,.,, 5.6. n, µd d n [ ] n { n ; 0 n >. µd Möbius. : [4] p, k s, Rs >, Spkn s p ks ζs, n

119 Smarandache µd Möbius. : m S p kn. p a m, S p kn [ a ] n, k m S p kn, m [ a k ]. S pknn,, p, n S s pkn m p a m k [ a k ] m s β γ0 n n,p k β γ0 m p βk+γ m β m s β n s p βk+γs β β k p βks γ0 p γs n n,p n s. n n,p n s n n s n p n n s ζs m p s m s p s ζs, n k γ0 β p γs p sk, p s β p βs p s p s. Spkn s p ks ζs, p k > Res > s, n n A k Spn s ζs d µd p dk s. A k k µd Möbius ζs Riemann zeta-. : S p n n,, p

120 n n A k S s pn µd n d k n m d S s pn µd d µd Spd s k m m S s pd k m ζs µd p dk s d µd ζs p dk s. d., k, A, 5.6. p Res > s, n n A Spn s ζs d µd p d s. 5.7 Smarandache Smarandache, Smarandache nn + S p + S p + + S p n S p,., 5.7. m, m,, m n m, m,, m n, m!m! m n! m + m + + m n!. : m, m,, m n, a, a,, a n, a m + a m + + a n m n. 3

121 Smarandache m + m + + m n! a m m + m + + m n! + a m m + m + + m n! + + a n m n m + m + + m n! m + m + + m n!. 5- m!m! m n! m + m + + m n! m!m! m n! m + m + + m n!m. m!m! m n! m + m + + m n!m. m!m! m n! m + m + + m n!m n. 5- m!m! m n! m + m + + m n! n, p, p α n!. [ ] n α p i. i : [7] p, n. nn + S p + S p + + S p n S p 5-3 [ ] 8p +. n,,,, [x] x. :, S p k, k p [ S p k pk. ] nn + 8p + k > p, S p k < pk. p, n, [x] x. nn + nn + S p p

122 [ ] [ ] 8p + 8p + p, n, nn + S p + S p + + S p n p + p + + np p. 5-5 [ ] 8p , n,,, 5-3. [ ] 8p + nn + < n p, > p, nn + nn + S p < p, 5-3. n p +, S p + S p + + S p n nn + p. S p + S p + + S p n p + p + + p p + p p p, pp + 3 p. S p + S p + + S p n S + S + S p 3, nn + S p S 6 8 < , S p + S p + + S p n S 3 + S 3 + S S , nn + S p S < p > 3, [ ] p +3p [ ] p p i p + 3p p + 3p + p i p + 3p + p + 5

123 Smarandache p + p + S p p + 4p p + p +, p + 3p p < pp + 3 p. nn + S p + S p + + S p n > S p. n p n p +, m i i, i,,, n S p m p, S p m p,, S p n m n p. S p + S p + + S p n m p + m p + + m n p m + m + + m n p. 5-6, m, m,, m p p. S p n, j [ ] mj p, j n. 5-7 i p i, m p+ p, m p+ p+, m p, m p+,, m n, p >, Gauss [x] 5.7., [ ] m + m + + m n p i p i m + m + + m n + m + m + + m n + [ ] m + m + + m n p i p i [ ] m + m + + m n p + p m + m + + m n [ ] pp p + p + m p + m p+ + + m n p + i m + m + + m n + i i p i [ p p + p p i ] p i 6

124 + [ ] mp + m p+ + + m n i p i m + m + + m n + i m p + i [ mp p i ] + [ ] mp + m p+ + + m n i m p+ + +m + m + + m p [ ] m p [ ] m p p i n nn +. i p i i p i [ mp+ p i [ ] mn p i p i ] + + p nn+ m + m + + m n p!, nn + S p 5-6, 5-8 m + m + + m n p m n + i [ mn p i ] < m + m + + m n p. 5-8 S p + S p + + S p n > S p nn + p, S p + S p + + S p n > S p nn +, n p ,. p 3, 5, 7, 5.7. n,. S 3 + S S 3 n S 3 nn S 5 + S S 5 n S 5 nn + 7

125 Smarandache n, n,, 3. S 7 + S S 7 n S 7 nn Smarandache ceil, Smarandache ceil,. Smarandache ceil F.Smarandache, k n, Smarandache ceil S k n min{m N : n m k } n k S k + S k + + S k n S k n, 5-9 n,, 3. : S k, S k, S k 3 3 S k 6 6. n,, nn + n 4,, S k n S k S k n n S k 4, nn + nn +, S k + S k + + S k n < S k n,. nn +, S k n nn

126 A, S k + S k + + S k n S k a a a µa a n a n a n a n a A a A d uµd d µd [x] x {x}, d u n d n S k + S k + + S k n d µd d n n n d n d n d µd d d µd µd d d µd d n n d 4 n d { n d } + n d + n n d n { n d } n d> n { n } µd d + n d n µd d [ n d ] [ n d ] + u n d u, { n } { n } d d d n { n } d µd d n n n n µn n ζ 6 π, µd + n n n n, S k + S k + + S k n n 6 π 5 n 3 3n π 5 n 3. 3n π 5 n 3 n + n 3 > 3n 4 π 5 n 3 nn + >, 4 n > 600, n > π π, S k + S k + + S k n > S k n, a d a µd. nn +, 3, n nn +, 4 n 600,, n 4 600,

127 Smarandache, 5-9 n,, Smarandache 5, Smarandache , 5, Smarandache 5. : 5, 5, 53, 54, 56, 57, 58, 59, 0, 0, 03, 04, Smarandache Smarandache. Smarandache 5, A Smarandache 5, B Smarandache, C Smarandache, [6] fn fn + O n A Mx ln 8 ln 0 fn, M max{ fn }. fn dn Ωn, dn, Ωn n, dn x ln x + γ x + O x ln 8 ln 0 +ε, n A γ Euler, ε, Ωn x ln ln x + Bx + O n A, x, ln x B. Smarandache x, n ln x + γ + O γ Euler. : [4] x,

128 5.9. x, D,, 3, 4, 6, 7, 8, 9. n A + O x ln 8 ln 0, n D A. : D, D,, 3, 4, 6, 7, 8, 9, 0 5., D 8 8 m 8 m., n n D < A. x, n n D k 0 k x < 0 k+. n D n n n D n D n>x 8 k+ A + O 0 k A + O A + O n A + O lg x x x ln 8 ln 0. n k x, n 4 4γ + ln 5 ln x + A + O 5 5 n A A. n A 8 n+ 0 n 0 + x ln 8 ln 0, : Smarandache n n n x 5 5n n D n

129 ln x + γ + O ln x + 4γ + ln 5 5 ln x + 4γ + ln 5 5 Smarandache [ ln x ] x γ + O A + O x ln 8 ln 0 x A + O x A + O x ln 8 ln 0. + O x ln 8 ln x, B. n B n γ + ln ln x + B + O x lg, x, n C n γ + ln ln x + C + O x lg, C. 5.0 n p α pα pα k k n, Ωn Ωn α + α + + α k. Ωn., n k, kn k, n, C k kn kn! n! k. C k kn, m, m p m ln p m ln m + O p. : Abel [4], p m ln p πm ln m + m [5 ] Ωn ln 3 m, 5-0 πt t ln t dt,

130 πm m. πm m m ln m + O ln, m p m p m ln p m m ln m + O ln 3. m m, ln ln m + A + O, 5- p ln m A. : [7] A, B. k : m! p α i i, m, m Ωm! m ln ln m + A + Bm + O ln m i α i [ ] m, j [m] m. Ωm! i j p j i k [ ] m p j i p m j [ ln m ln p ] m p j + O j p m [ ln m ln p ] [ ] m p j ln m ln p p m m p m p m p m p [ + O ln m ln p ] p p m m p + O ln m ln p p m p m m p + ln m + O. pp ln p p m p m p m, 5- ln m ln p 3

131 Smarandache pp p p m pp p>m pp B + O, 5-3 m B p pp , Ωm! m ln ln m + A + O + B + O m m m ln ln m + A + Bm + O m ln m. m + O ln m n, k, n k, n > k ln k, Ω C k kn kn ln k ln n kn + O. 5-4 ln n : Ω 5.0.3, Ω C k kn Ωkn! kωn! kn kn ln ln kn kn ln ln n + O ln n kn ln + ln k kn + O ln n ln n ln k ln kn ln n + O k kn ln + O. n ln n n k, n > k ln k, ln n > ln k, Ω C k kn kn ln k ln n kn + O. ln n. n k, k,,. 4

132 . n, F.Smarandache Sn m n m!. Sn min{m : m N, n m!}. Sn, n > n 8 d n Sd Sd φn d n, φn Euler.. n, F.Smarandache LCM k n [,,, k], [,,, k],,, k SLn, n > n 36 SLd d n SLd φn d n ln SLn, ln SLn 3. n, n p α pα pα k k n, Sn max{α p, α p,, α k p k }. Sn, n > n 4 Sd d n Sd φn d n 4. n, F.Smarandache S n m m! n. S n max{m : m N, m! n}. S n S d φn d n S d, d n 5. n, F.Smarandache Zn m mm m n. Zn max{m : m N, n}. Zn, Zn, Zn 5

133 Smarandache [] C.Ashbacher, On numbers that are pseudo-smarandache and Smarandache perfect, Smarandache Notions Journal, 5004, [] Xu Zhefeng, On the Value Distribution of the Smarandache Function Acta Mathematica Sinica in Chinese, Vol.49006, No.5, [3] S. Tabirca, About S-multiplicative functions, Octogon, 999,7: [4] Tom.M. Apostol, Introduction to analytic number theory, New York: Springer- Verlag, 976, 77. [5] P.Erdös, Problem 6674, Amer. Math. Monthly, Vol.98, 99, 965. [6],,,,, 999, [7] A.Murthy, Some notions on least common multiples, Smarandache Notions Journal, Vol. 00, [8] Zhang Wenpeng, Liu Duansen, On the primitive numbers of power p and asymptotic property, Smarandache Notions Journal, Vol. 300, [9] Mark Farris and Patrick Mitchell, Bounding the Smarandache function, Smarandache Notions Journal, Vol. 3 00, [0],,, [] Amarnth Murthy, Generalized Partitions and New Ideas On Number Theory and Smarandache Sequences, Hexis, 005, 0-. [],,,,, 99. [3] D.R.Heath-Brown, The differences between consecutive primes. III, Journal, London Math. Soc., 0 979, [4] H.N.Shapiro, Introduction to theory of numbers, John Wiley and Sons, 983, 8. [5],,,, 977, 5. [6] Wang Xiaoying, On the Smarandache Pseudo-multiples of 5 sequence, Research on Smarandache Problems in Number theory, Hexis, 004, 7-9. [7],,,, 975, 05. [8] Liang F. C. and Yi Y, The primitive number of power p and its asymptotic property, Research on Smarandache problems in number theory, Hexis, Phoenix, 004, 9-3. [9] C. Ashbacher, Some Properties of the Smarandache-Kurepa and Smarandache- Wagstaff Functions, Mathematics and Informatics Quarterly, 997, 7: 4-6. [0] Liu Yaming, On the solutions of an equation involving the Smarandache function, Scientia Magna, Vol. 006, No., [] Fu Jing, An equation involving the Smarandache function, Scientia Magna, Vol. 006, No.4,

134 [] F. Smarandache, Only Problems, Not Solutions, Chicago, Xiquan Publishing House, 993. [3] C.Ashbacher, On numbers that are pseudo-smarandache and Smarandache perfect, Smarandache Notions Journal, 5004, [4] Yi Yuan, On the primitive numbers of power p and its asymptotic property, Scientia Magna, Vol., 005: [5] A., Begay, Smarandache Ceil Functions, Bulletin of Pure and Applied Sciences, 997, 6E: 7-9. [6] G H.Hardy, E M.Wright, An introduction to the theory of numbers, Oxford Univ Press, Oxford, 98. [7] Ma Jinping, The Smarandache Multiplicative Function, Scientia Magna, Vol. 005, No., 5-8. [8] Li Hailong and Zhao Xiaopeng, On the Smarandache function and the K-th roots of a positive integer, Research on Smarandache problems in number theory, Hexis, 004, 9-. [9] Shejiao Xue, On the Smarandache dual function, Scientia Magna, Vol , No., 9-3. [30] M.H.Le, A conjecture concerning the Smarandache dual function, Smarandache Notions Journal, 4004, [3] Li Jie, On Smarandache dual functions, Scientia Magna, Vol.006, No., -3. [3] Lv Zhongtian, On the F.Smarandache LCM function and its mean value, Scientia Magna, Vol , No., -5. [33] Xu Zhefeng, Some new arithmetical functions and their mean value formulas, Mathematics in Practice and Theory in Chinese, 36006, No. 8,

135 New Progress on Smarandache Problems Chen Guohui Department of Mathematics Hainan Normal University Haikou, Hainan, 5758 P. R. China High American Press 007

136 责任编辑 : 张沛 封面设计 : 张小蹦 本书主要将目前中国学者关于 Smarandache 问题的部分研究成果汇编成册, 其主要目的在向读者介绍关于 Smarandache 问题的一些最新的研究成果, 主要包括 Smarandache 函数及相关函数的渐近性质 级数收敛问题 特殊方程的解等一系列问题, 并提出了关于这些函数的一些新的问题, 有兴趣的读者可以对这些问题进行研究, 从而开拓读者的视野, 引导和激发读者对这些领域的研究兴趣. This book includes part of the research results about the Smarandache problems written by Chinese scholars at present, and its main purpose is to introduce various results about the Smarandache problems, such as Smarandache function and its asymptotic properties, series convergence, solutions about special equations. At the same time, we put forward to some new interesting problems either in order to research further. We hope this booklet will guide and inspire readers to these fields.