Nanjing Univ. J. Math. Biquarterly 32(2015), no. 2, NEW SERIES FOR SOME SPECIAL VALUES OF L-FUNCTIONS

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1 Naning Univ. J. Math. Biquarterly 2205, no. 2, NEW SERIES FOR SOME SPECIAL VALUES OF L-FUNCTIONS ZHI-WEI SUN Astract. Dirichlet s L-functions are natural extensions of the Riemann zeta function. In this aer we first give a rief survey of Aéry-lie series for some secial values of the zeta function and certain L-functions. Then, we estalish two theorems on transformations of certain inds of congruences. Motivated y the results and ased on our comutation, we ose 48 new conectural series most of which involve harmonic numers for such secial values and related constants. For examle, we conecture that and ζ5, 2 5 ζ2, 2.. Introduction The Riemann zeta function given y ζs for Rs > n s n lays an imortant role in analytic numer theory. As Euler roved, m 2π2m 2ζ2m 2m! B 2m for m, 2,, Mathematics Suect Classification. Primary B65, M06; Secondary 05A0, A07, Y60, F05, 65B0. Keywords: Central inomial coefficient, congruence, Dirichlet s L-function, harmonic numer, Riemann s zeta function, series. Suorted y the National Natural Science Foundation grant 740 of China.

2 2 ZHI-WEI SUN cf. [8,. -2], where B 0, B, B 2,... are Bernoulli numers defined y n + B 0, and B 0 for n, 2,,.... In articular, 0 ζ2 π2 π4 π6, ζ4 and ζ It is well nown that π is transcendental cf. [8,. 7-76] and thus all those ζ2m m, 2,,... are irrational. In 978 R. Aéry cf. [2] and [2] successfully estalished the irrationality of ζ y using the series which converges at a geometric rate with ratio /4 since 2 4 π ζ. as +. In fact, the last identity was first deduced y A. A. Marov [8] in 890. There are also some fast converging series for ζ2 similar to., e.g., 2 2 π2 8, π2 8, π2.2 see R. Matsumoto [9]. More generally, x arcsin 2 x for x 2. see, e.g., [5]. It is also well nown that ζ π4.4 cf. [6,. 89]. Recall that the harmonic numers are given y H n : n 0,, 2,.... 0< n For m 2,, 4,..., we call those numers : H m n 0< n n 0,, 2,... m

3 NEW SERIES FOR SOME SPECIAL VALUES OF L-FUNCTIONS harmonic numers of order m. In contrast with., it is also nown that for every n 2,,... we have arcsin 2n x/2 2n! 4 n for x 2, and in articular arcsin 4 x 2 2 x < < 2 < < n < H See [5]. Putting x in.5 we get H n 2 x 2 for x 2..5 π Taing derivatives of oth sides of the identity in.5, we otain 4 x 4 x 2 arcsin 2 It is well nown that in articular, 0 arcsin x H π and x 2 for x < x2+ for x 2; By [5], for each n, 2,... we also have arcsin 2n+ x/2 2 x 2+ 2n +! rovided x 2; in articular, arcsin x 2 2 x Note that 0 0 < 0 < H2 2 0 < 2 < < n< 2 π..7 4 n i 2 i for x 2. H2 4.

4 4 ZHI-WEI SUN It is also nown that 0 cf. [9] and < π π cf. I. J. Zucer [0, 2.2] and [0]. M. Koecher [5] and D. Leshchiner [6, 4a] indeendently roved that for each n 2,,... we have ζ2n + n in articular, + 2 0<m<n ζ5 2 m 2m < < < n m < 0< < < n < 0<<n 2 0<<n m 2 ; H2..0 Leshchiner [6] deduced this via a sohisticated analytic method as well as an elegant cominatorial aroach. Insired y this result, J. Borwein and D. Bradley [4] used the PSLQ algorithm for finding integer relations to discover that ζ H An extension of this was roved y G. Almvist and A. Granville [] in 999. Leshchiner [6, 4] also roved that for any n, 2,,... we have n 4 n ζ2n <m<n m 2 + 2m < < n < < n m 0<<n 2 2 0<<n m 2 2 ;

5 0 NEW SERIES FOR SOME SPECIAL VALUES OF L-FUNCTIONS 5 where the last sum is regarded as when m n ; in articular, < π4 24. Let χ e a Dirichlet character modulo a ositive integer m. Dirichlet L-function associated with the character χ is given y χ Ls, χ : for Rs >. s The Dirichlet eta function is defined y 4 βs L s, for Rs > 0, 2 + s where denotes the Kronecer symol. As Euler oserved, 0 β2n + n E 2n 4 n+ 2n! π2n+ for all n 0,, 2,... The cf..6 of [8,. 2], where E 0, E, E 2,... are Euler numers defined y n E 0, and E n 0 for n, 2,,.... In articular, 2 + π 4, π 2, 2 + 5π S. Ramanuan found that for 0 < x π/2 we have 2 sin 2+ x sin2x x log 2 sin x cf. B. C. Berndt and P. T. Joshi [,. 89], consequently π 4 2 log 2 + G 2 and K 0

6 6 ZHI-WEI SUN cf. [,. 9-40], where G : β2 is the Catalan constant, and K : L 2, with the Legendre symol. In contrast with. and.4, in 985 I. J. Zucer [0] roved the following remarale identities: where πk 4 ζ, 2 π2 5 log 2 + πg ζ, π2 log π ζ + 64 L, L : L 4, Z.-W. Sun [22] conectured that K and πk 26 9 ζ, π2 9 πl + ζ 9 ζ5, K; 2 the first formula here was confirmed in 202 y Kh. Hessami Pilehrood and T. Hessami Pilehrood [], and the second one was later roved y J. Guillera and M. Rogers [9]. L. van Hamme [20] investigated corresonding -adic congruences for certain hyergeometric series involving the Gamma function or π Γ/2 2. This stimulated later studies of -adic congruences for some well-nown series. For examle, R. Tauraso [29] roved that. and

7 NEW SERIES FOR SOME SPECIAL VALUES OF L-FUNCTIONS 7 the first identity in.2 have the following -adic analogs: 2 2 H mod and H mod, 2.2 where > 5 is a rime. It is nown that H 2 B / mod for any rime >. Z.-W. Sun [22, 25] showed that / E and /2 2B 2 mod for any rime > 5. By [22, Lemma 2.], if is an odd rime then 2 2 2/ 2 mod 2. for all,...,. The aer [29] contains a similar technique. Motivated y this and.4, Z.-W. Sun [22] conectured that 4 H B 5 mod, H 2 27 H mod 4,.5 where > 7 is a rime. This remains oen ut their mod versions have een confirmed y Kh. Hessami Pilehrood and T. Hessami Pilehrood [2]. Similarly, motivated y.0,. and., we conecture that /2 2 for every rime >, and that and /2 2 2 H 2 4 0B mod.6 5H B 7 mod.7 5H B 7 mod.8

8 8 ZHI-WEI SUN for any rime > 7. Note that if is an odd rime and m is a ositive integer with m then for every,..., we have H m Hm m m H m mod. 0<< Let > e a rime. Motivated y the first identity in.7, Z.-W. Sun [2] showed that 2 2 / mod 2, +/ E mod 2..9 In view of.8 and the last equality in.9, Z.-W. Sun [2] conectured that if > 5 then /2 0 +/ H mod, B mod 2,.2 / H B 5 mod..22 The congruences.20 and.2 were confirmed in [] and [25] resectively. Here we ose a curious conectural series for K as well as its related -adic congruences. Conecture.. i We have and ii For any rime >, we have B K..2 2 mod mod 2, where B n x denotes the Bernoulli olynomial of degree n.

9 NEW SERIES FOR SOME SPECIAL VALUES OF L-FUNCTIONS 9 Remar.. In [22, Con. 5.4] the author made the following conecture for any rime > : If mod and x 2 + y 2 with x, y Z and x mod, then 0 0 If 2 mod, then x 48 2x mod 2, 2 4 x mod mod 2. +/2 +/6 In view of.24, it is interesting to investigate what rimes > satisfy the congruence B 2 / 0 mod. The first such a rime is and the second one must e greater than By the standard heuristic arguments cf. [7, ], it seems that there should e infinitely many such rimes. In the next section we estalish two theorems on transformations of certain inds of congruences. Motivated y the results and ased on our comutation via the PSLQ algorithm, we ose in Sections -5 many conectural series involving harmonic numers or higher-order harmonic numers for some secial values of L-functions, most of which are mainly motivated y their -adic analogs we found first. All the conectural series in this aer converge at geometric rates and so they can e easily checed via some mathematical softwares lie Mathematica. 2. Transformations of certain inds of congruences Theorem 2.. Let e an odd rime and let a e an integer with a. Let, c Z, m, r Z + {, 2,,...} and ε {±}. Then 2 r ε a r + c ε m m 2 ε m m 2ε a r+ 2 a r+ 2 ε ε + c mod, m m 2.

10 0 ZHI-WEI SUN and also a 2 r /2 ε /2 m 0 / a r 2 + r 6/a 2 ε + c ε m m 2 + r 6/a 2m 0 ε ε c ε mod. m 2 + m 2.2 Proof. Let n /2. For {,..., }, clearly 2 if and only if > n. By., Thus 2 2 a r 2 2 a r mod for all,..., n. 2 ε + c ε m m a r+ 2 a r+ 2 ε + c ε m m a r+ 2 a r+ 2 r a 2 r+ 2 This roves 2.. ε + c ε m m ε + c ε m m ε + c ε m m ε m ε m m ε 0<s< ε s s + c ε m m s ε s ε + c mod. sm m

11 NEW SERIES FOR SOME SPECIAL VALUES OF L-FUNCTIONS For each 0,..., n, we have 2 2n 6 n mod 2. since 2 /2 n n n 2n /2 4 n 4 n 4 n 2n n 6 mod. n Therefore 2 ε + c ε m m a r 2 ε a r + c ε m m n 2n n n ε a n n r + c ε n m n m 0 n a a 2 ε 6 /2 r m 0 0 s< n a 2 2 r 2 + r 6/a 0 ε m εn 2 m 0 s< ε s 2s + c ε m 2 + m s0 ε n s n s + c ε n m /2 m n a 2 2 r 2 + r 6/a 0 ε ε ε s m 2m 2s + + c ε mod. m 2 + m This roves 2.2. In view of the aove, we have comleted the roof of Theorem 2..

12 2 ZHI-WEI SUN Remar 2.. Let e an odd rime, and let m Z + and ε {±}. It is well nown that H m 0 mod when m. Note also that ε m /2 ε + ε m m /2 + m ε and hence / m 0 mod if 2 m. ε mod m Corollary 2.. Let > 5 e a rime. For any, c Z, m Z + and ε {±}, we have 2 ε + c ε m m and also m 2ε m 2 m+ /2 0 ε Proof. In view of.2 and.9, 2 2 ε ε + c mod, m m ε + c ε m m ε c ε m 2 + m / mod mod. 2.5 Thus, y alying Theorem 2. with a r we immediately otain 2.4 and 2.5. Corollary 2.2. Let > 5 e a rime. For any, c Z, m Z + and ε {±}, we have 2 ε 2 + c ε m m m 2ε 2 ε ε + c mod, m m 2.6

13 NEW SERIES FOR SOME SPECIAL VALUES OF L-FUNCTIONS and also m 2 m+2 /2 0 ε ε + c ε m m ε c ε m 2 + m mod. 2.7 Proof. In view of.2 and.20, 2 / mod Alying Theorem 2. with a and r 2, we immediately otain 2.6 and 2.7. Corollary 2.. Let > e a rime. For any, c Z, m Z + and ε {±}, we have 2 ε + c ε 2 m m B ε m and also m 2ε m 2 m+ /2 0 ε ε ε + c m m mod, 2.8 ε + c ε 2 /2 m m B ε m ε 2 + m + c ε 2 + m mod. 2.9 Proof. Let n /2. By 2., n 2 n 2n n 2 + 2n mod. 8

14 4 ZHI-WEI SUN As conectured y the author and roved in [2], and So / H 2 2 H B mod 2 B mod. B 2 mod, 2.0 which confirms the mod version of.22. Alying Theorem 2. with a and r, we immediately otain 2.8 and 2.9. Theorem 2.2. Let e an odd rime and let m e an integer with mm 4. Let α e a ositive integer, and let a 0, a,..., a α e -adic integers. Define a 0 a for 0,,..., α. Then we have the congruences 0 α 0 2 mm 4 4 m a α and α 2 mm 4 m 4 m a α α α 0 2 m a mod 2. 2 m 4 2 m a mod, 0 where α denotes the Jacoi symol. 2.2 Proof. Let n α /2. By Lucas theorem see, e.g., [4], for each n +,..., a we have 2 a + 2 a 2 a 0 mod. 0 a + 0 Also, for any, {0,..., n} with + n, we have n /2 α /2 i /2 i 0 i< α mod. 2i i<

15 NEW SERIES FOR SOME SPECIAL VALUES OF L-FUNCTIONS 5 As /2 2 / 4 for 0,, 2,..., we have 2 a 4 m 0 0 /2 4 a 4 m and hence α /2 /2 a 2 n 4 4 a m 4 i 2 4 m a 0 0 i0 2 m 4 a 2 m 4 a 0 2 /2 n n i i0 m 4 a 2 m m a m 4 0 mm 4 α α 4 m m 4 m m 4 0 This roves 2.. Similarly, 2 a 4 m 0 0 /2 /2 a 0 2 n 4 4 a + i m 4 0 i0 2 4 i m 4 4 i m 4 n n m 4 α r0 r m 2 m a mod. 4 m 4 /2 i 4 i m 4

16 6 ZHI-WEI SUN and hence α m a a m 4 2 a m 4 0 n n + i i i0 + 4 m 4 n 2 a m 4 2 m 4 a m m m 4 n 2 a m 4 2 a m 4 0 m 4 m n 2 a m 4 mm 4 α α 0 n 0<i n n 4 i m 4 n i 4 m n 4 m 4 m 4 n m 4 n m 2 m a m i mod. So 2.2 also holds. This concludes the roof. Remar 2.2. Theorem 2.2 was motivated y the identity.8 in [26]. Corollary 2.4. For 0,, 2,... define f x : x and g x : x. Let e an odd rime and let α e any ositive integer. Then, for any integer m 0, 4 mod, we have α 0 2 g x mm + 4 m + 4 α α 0 2 m f x mod 2.

17 and α m 0 NEW SERIES FOR SOME SPECIAL VALUES OF L-FUNCTIONS 7 2 g x m mm + 4 α 2 m+4+2 m f x mod. α Proof. By [28, Theorem 2.2], g x f x for all 0,, 2, Thus, y alying Theorem 2.2 we otain that α 2 g x m m 4 α 4 m 2 α m f x mod 0 and that α 2 m 4 m g x 0 m m 4 α 2 m 4 2 α m f x mod. 0 So, oth 2. and 2.4 hold. Remar 2.. The author [24, 2.6] roved that 2 f x x mod 2 for any rime > Conectural formulas involving ordinary harmonic numers Conecture.. i We have H 2 + 2/ 2 ζ, 2. H 2 + 2H 2 5 ζ, 2.2 H 2 + 7H πk. 2. 0

18 8 ZHI-WEI SUN ii Let > e a rime. Then /2 H 2 + 2/ 2 B 2 mod, H 2 + 2/ 2 4 H B 5 mod, H 2 + 2H H B 5 mod, H 2 + 7H B 2 mod. Remar.. A comination of. and.2 yields H / 2 ζ 2 for which Mathematica 9 could yield a roof after running the FullSimlify command half an hour. Comining.-. we find exact values of, 2 H 2 2 and H Recently G.-S. Mao and Z.-W. Sun [7] showed that for any rime > we have 2 H 2 H B 2 B 2 mod, mod.

19 NEW SERIES FOR SOME SPECIAL VALUES OF L-FUNCTIONS 9 Conecture.2. i We have And L v H /2 2 7 ζ,.4 4 2H 2 H ζ,.5 4 6H 2 H + 8 2πG,.6 2H 2 7H + 2 π2 2 6H 2 8H + 5 log 2,.7 26 ζ,.8 6H 2 0H πk, H + π2 log ζ + 4π2 log φ, ζ + π2 log 5 5 φ 6,.2 50 where φ is the famous golden ratio 5 + /2, the Lucas numers L 0, L, L 2,... are given y L 0 2, L, and L n+ L n + L n for all n, 2,,..., and v 0, v, v 2,..., are defined y v 0 2, v 5, and v n+ 5v n v n for all n, 2,,....

20 20 ZHI-WEI SUN ii For any rime > we have 2 2 2H 2 H + 2 2q 2 H mod, H 2 H + 8 6q 2 + 2E mod 2, 2H 2 7H + 2 2q q 2 2 mod 2, 2 2 2H 2 H B mod 2, 6H 2 8H + 5 9q 6H mod, 6H 2 0H + 7 9q B 2 2 H 2 4H 26 9 B mod 2, where q a for a 0 mod denotes the Fermat quotient a /. Remar.2. a Comining.5-.7 we find the exact values of 2 2 H 2 H 2 2, 2 2 and Similarly, comining.8-.0 we get the exact values of H H 2 2, 2 2 and The author [27, Remar 5.2] used Mathematica 7 to find that 4 H 4 H 2 2 7ζ and ζ. Alying. with x φ, φ we otain φ π 2 2 arcsin 2 2 9π mod 2,

21 NEW SERIES FOR SOME SPECIAL VALUES OF L-FUNCTIONS 2 and φ arcsin π 2 π Therefore, L φ 2 + φ 2 2 π with x 5 ± 5/2 yields that 2 2 2π /2 π 2 5 5/2 2 and So we have v π2. Conecture.. i We have 0H / 2 π4 0,. H 2 + 4H π4,.4 H 2 H + 2/ 4 ζ5, H 2 02H + 28/ π2 ζ,.6 97H 2 6H + 227/ πl. 8.7

22 22 ZHI-WEI SUN ii Let > e a rime. Then 2 H B 5 mod, 0H B 5 mod 2, 2 H H B 5 mod if > 5, 2 H 2 + 4H H B 5 mod, 2 H 2 H 22 5 B 5 mod 2, 2 H 2 H + 2/ 4 H B 5 mod, H 2 02H B 5 mod 2, 2 H 2 02H + 28/ 4 H B 5 mod. If > 5, then 2 97H 2 6H B 4 48 mod and 2 97H 2 6H + 227/ 4 97 H B 4 mod 2.

23 NEW SERIES FOR SOME SPECIAL VALUES OF L-FUNCTIONS 2 Conecture.4. i We have H H H ii Let e an odd rime. Then /2 0 If >, then /2 0 8G, ζ, β4 + 5 πζ H E mod H H B 5 mod. 4. Conectural formulas involving higher-order harmonic numers For any ositive integers and m, oviously m + Hm + m 0< /2 2 2 m 2 m H m /2. Conecture 4.. i We have 6H 2 /2 / π4. 4.

24 24 ZHI-WEI SUN Also, ii Let > e a rime. Then H ζ5 + 2ζ2ζ, ζ5 2ζ2ζ, ζ5 6ζ2ζ. 4.4 Also, /2 / H B 5 mod, H B 2 5 mod, 6H Conecture 4.2. i We have H H 2 45 B 5 mod B 5 mod, B 5 mod. + / ζ2, 4.5 H 2 /2 4 2 π6 6260, 4.6 H 4 / π

25 Also, NEW SERIES FOR SOME SPECIAL VALUES OF L-FUNCTIONS << π ζ2, π6 4 5 ζ2, ii Let > e a rime. Then If > 7, then π6 + 6ζ H + 2B 5 mod, H 2 / B 5 mod H H 4 When > 5, we have H H B 7 mod, 5 6 H B 7 mod, / H B 7 mod << 2 + B mod, 2B 2 5 mod, B 5 mod.

26 26 ZHI-WEI SUN Conecture 4.. We have H 5 + 4/ H + 8/ and ζ ζ ζ2ζ5 + ζζ4, ζ2ζ5 + ζζ4, ζ7 ζ2ζ ζζ Conectural formulas involving 0 ± /2 + m Conecture 5.. i We have πζ, G π2 0 + πζ L ii Let > e a rime. Then / Conecture 5.2. i We have < 0 < B 5 mod π2, π πG, 5.4 ζ, π , 5.6

27 0 NEW SERIES FOR SOME SPECIAL VALUES OF L-FUNCTIONS / < < ii Let > e a rime. Then 2 2 /2 0 /2 0 / < 5 B 5 mod 2, π , π , π H mod 2, B < 5 2 B 5 mod 2. Provided > 7, we have 2 and /2 0 / < H 2 mod 2 mod, B mod 2.

28 28 ZHI-WEI SUN Conecture 5.. We have π2 ζ + 8 πβ4, π ζ πζ5, π ζ + π 2 β πζ5, π G π2 ζ, π ζ πζ References [] G. Almvist and A. Granville, Borwein and Bradley s Aéry-lie formulae for ζ4n +, Exeriment. Math , [2] R. Aéry, Irrationalité de ζ2 et ζ, Astérisque 6 979,. [] B. C. Berndt and P. T. Joshi, Chater 9 of Ramanuan s Second Noteoo: Infnite Series Identities, Tansformations, and Evaluations, Amer. Math. Soc., Providence, R.I., 98. [4] J. M. Borwein and D. M. Bradley, Searching symolically for Aéry-lie formulae for values of the Riemann zeta function, SIGSAM Bull. Algera. Symolic Maniulation, Assoc. Comut. Machinery 0 996, no. 2, 2 7. [5] J. M. Borwein and M. Chamerland, Integer owers of arcsin, Int. J. Math. Math. Sci , Article ID 98, 0. [6] L. Comtet, Advanced Cominatorics: The Art of Finite and Infinite Exansion, D. Reidel Pulishing, Dordrecht, Holland, 974.

29 NEW SERIES FOR SOME SPECIAL VALUES OF L-FUNCTIONS 29 [7] R. Crandall and C. Pomerance, Prime Numers: A Comutational Persective, Sringer, New Yor, 200. [8] P. Eymard and J.-P. Lafon, The numer π, Amer. Math. Soc., Providence, R.I., [9] J. Guillera and M. Rogers, Ramanuan series uside-down, J. Austral. Math. Soc , [0] Kh. Hessami Pilehrood and T. Hessami Pilehrood, Series acceleration formulas for eta values, Discrete Math. Theore. Comut. Sci , [] Kh. Hessami Pilehrood and T. Hessami Pilehrood, Bivariate identities for values of the Hurwitz zeta function and suercongruences, Electron. J. Comin , no. 2, Research Paer 5, 0. [2] Kh. Hessami Pilehrood and T. Hessami Pilehrood, Congruences arising from Aéry-tye series for zeta values, Adv. in Al. Math , [] Kh. Hessami Pilehrood, T. Hessami Pilehrood and R. Tauraso, Congruences concerning Jacoi olynomials and Aéry-lie formulae, Int. J. Numer Theory 8 202, [4] H. Hu and Z.-W. Sun, An extension of Lucas theorem, Proc. Amer. Math. Soc , [5] M. Koecher, Letter German, Math. Intelligence 2 979/980, no.2, [6] D. Leshchiner, Some new identities for ζ, J. Numer Theory 98, [7] G.-S. Mao and Z.-W. Sun, Two congruences involving harmonic numers with alications, Int. J. Numer Theory, in ress. Doi: 0.42/S [8] A. A. Maroff, Mémoiré sur la transformation de séries eu convergentes en séries tres convergentes, Mém. de l Acad. Im. Sci. de St. Pétersourg, 7 980, no. 9, 8. Availale at htt://www. math.mun.ca/ sergey/research/history/marov/marov890.html [9] R. Matsumoto, A collection of formulae for π, on-line version availale at htt:// small. [20] L. van Hamme, Some conectures concerning artial sums of generalized hyergeometric series, in: -adic Functional Analysis Nimegen, 996, , Lecture Notes in Pure and Al. Math., Vol. 92, Deer, 997. [2] A. van der Poorten, A roof that Euler missed...aéry s roof of the irrationality of ζ, Math. Intelligencer 978/79, [22] Z.-W. Sun, Suer congruences and Euler numers, Sci. China Math , [2] Z.-W. Sun, On congruences related to central inomial coefficients, J. Numer Theory 20, [24] Z.-W. Sun, Congruences for Franel numers, Adv. in Al. Math. 5 20, [25] Z.-W. Sun, -adic congruences motivated y series, J. Numer Theory 4 204, [26] Z.-W. Sun, Some new series for /π and related congruences, Naning Univ. J. Math. Biquarterly 204, [27] Z.-W. Sun, A new series for π and related congruences, Internat. J. Math , no. 8, ages.

30 0 ZHI-WEI SUN [28] Z.-W. Sun, Congruences involving g n x n n x, Ramanuan J., in ress. Doi: 0.007/s [29] R. Tauraso, More congruences for central inomial coefficients, J. Numer Theory 0 200, [0] I. J. Zucer, On the series 2 n and related sums, J. Numer Theory , Deartment of Mathematics, Naning University, Naning 2009, Peole s Reulic of China address: zwsun@nu.edu.cn