arXiv:2108.01540v1 [math.NT] 3 Aug 2021 functions h dnia qain fDirichlet example, of For equations [1-10]. identical formulae asymptotic the or identities some got have where otnain ayshlr aesuidtema au fDirichlet of value mean the studied have scholars Many continuation. .aog1913cm2p2okezaggalcm3 zzboyzyl@163 3. [email protected] [email protected] 044). iihe character Dirichlet 1.Introduction Keywords. colo ahmtc n ttsis otwsenPolyte Northwestern Statistics, and Mathematics of School Let hswr sspotdb aua cec ai eerhProject Research Basic Science Natural by supported is work This s q sacmlxnme with number complex a is Let L egv pcfi dniisfor identities specific give we ae,w hwsm pca ucinsre,adgv oenwiden new some give Dirichlet and the series, function for special tities some show we paper, ntenwiette fDirichlet of identities new the On ( L ≥ colo otaeEgneig ia ioogUniversity Jiaotong Xi’an Engineering, Software of School ,χ s, ( ,χ s, q ea nee and integer an be 3 Dirichlet eoeteDirichlet the denote ) ≥ sdfie by defined is ) χ ea integer, an be 3 ( χ − χ X mod )= 1) respectively, q − L 1 fntos dniy as us ucinseries. function sums; Gauss identity; -functions; | L ogMa Rong (1 109 epesRpbi fChina of Republic People’s 710049, 102 epesRpbi fChina of Republic People’s 710072, χ , L L ) fntosivligGussm.Specially, sums. Gauss involving -functions -functions L | χ Re 2 ( χ uogZhang Yulong ,χ s, = L eaDrcltcaatrmodulo character Dirichlet a be Abstract ( L eaDrcltcaatrmodulo character Dirichlet a be fntoswhen -functions L s π fntoscrepnigto corresponding -functions ) 1 (2 = ) 2 > 12 ϕ χ , ( 1 .I a eetne oall to extended be can It 1. q n X ). ige Zhang Jinglei ∞ ) =1 Y p | χ q n (

n s 1 ) − , 3 s p 1 2 1 = hia nvriy ia,Shaanxi Xi’an, University, chnical ! − , 2 fSaniProvince(2021JM- Shaanxi of ihteodadeven and odd the with 2 π 2 4 ϕ q ia,Shaanxi Xi’an, 2 2 ( .com q χ ) nthis In . la 1 a got has [1] Alkan (1) ; L s q q Dirichlet . fntosand -functions sn analytic using and , - L - 4 4 2 π ϕ(q) 1 π ϕ(q) 1 L(2, χ) = 1 4 + 2 1 2 , (2) χ mod q | | 180 p|q − p ! 18q p|q − p ! χ(−X1) = 1 Y Y where ϕ(q) is the Euler’s totient function. In addition, in contrast with these results, he also has given approximate formulas which are relatively easy to be derived by Abel’s summation and the Polya-Vinogradov inequality, that is ϕ(q) L(1, χ) 2 = + O(√q log q), (3) χ mod q | | 2 χ(−X1) = −1 where ϕ(q) is defined the same as before and the signal O only depends on constants.

He also has remarked that if χ and r have the same parity, then it is always possible to determine the average value of L(r, χ) 2 when χ ranges over all odd or even character | | modula q.

To proceed, we shall introduce the G(z, χ) [3] corresponding to χ modulo q and Bernoulli numbers [3] which are defined as

q−1 2πikz G(z, χ) := χ(k)e q ; kX=0 n n Bn = Bk, k! kX=0 n n! where z is an integer, k = k!(n−k)! is the binomial coefficient and B0 = 1.   In the proof of formulae (1) and (2), Alkan [2] has mainly constructed an identity between the Gauss sum and Dirichlet L-functions which is satisfied for the special conditions, that is,

2[ s ] s−k ( 1)s+1qs! q 2 s j − L(s, χ)= G(j, χ) Bk , (4) is2s−1πs k! q ! jX=1 kX=0 where s 1 and χ have the same parity. ≥ Of course, Zhang [14] also has obtained formulae (1) and (2) with the help of the identity between the Dirichlet L-functions and the generalized Dedekind sum

2 2n (n!) d 2 S(h, n, q) = n−1 2n 2n−1 χ(h) L(n, χ) 4 π q ϕ(d) χ mod q | | d|q n X χ(−1)X = (1) (n!)2 ∞ 1+( 1)n 2 − , (5) −4nπ2n rn ! rX=1 where the generalized Dedekind sum is defined by q a ah S(h, n, q)= B¯ ( )B¯ ( ), n q n q aX=1 2 which B¯n(x) is defined as following B (x [x]), if x is not an integer, B¯ (x)= n (6) n 0, − if x is an integer. 

Although Alkan has obtained formulae (1) and (2) with the help of the connection in [2] between the Dirichlet L-function and Gauss sum, he can only give the connection for s and χ having the same parity. In fact, when s and χ had the different parity, Alkan has got the right of formula (4) is zero. Therefore following his method, he could not get the identities when s and χ have the different parity. Based on that, we would like to study the connection between Dirichlet L-functions and Gauss sums under the condition that s and χ have the different parity. By the method, we can also give a new connection between Dirichlet L-functions and Gauss sums under the condition s and χ having the same parity. I think it’s very interesting and significant because we can know more about the connections between values of Dirichlet L-functions on positive integers and Gauss sum and do more research about Dirichlet L-functons.

In this paper, we will generate identities between Dirichlet L-functions and Gauss sums, and get some new identities of Dirichlet L-functions. That is, we will prove the following:

Theorem 1 Let q 3 be an integer and χ be a Dirichlet character modulo q. If s> 1 ≥ is an integer, s and χ have the different parity, we have 2πnj 2πnj ( i)s( mod 2) q ∞ sin(s+1)( mod 2)( ) coss( mod 2)( ) L(s, χ)= − G(j, χ) q q , (7) q ns jX=1 nX=0 where G(j, χ) is the Gauss sum corresponding the Dirichlet character χ modulo q.

Theorem 2 Let q 3 be an integer and χ be a Dirichlet character modulo q. If s is ≥ an integer, s and χ have the same parity, we have ( i)s q xs sπxs−1 [s/2] L(s, χ)= − G(j, χ) + ζ(2k)(xs)(2k) , (8)   − qs! j=1 2 − 2 k=1 2πj X X x= q   where G(j, χ) is the Gauss sum corresponding the Dirichlet character χ modulo q, ζ(2k) is the Riemann zeta function at value 2k, and [x] denotes the biggest integer less than x.

Specially, if s = 2 and χ is the even or odd Dirichlet character modula q respectively, we have

Corollary 1 Let q 3 be an integer and χ be an even and odd Dirichlet character ≥ modulo q respectively, we have π2 q π2 q π2 L(2, χ) = j2G(j, χ) jG(j, χ)+ , χ( 1)=1; (9) q3 − q2 6q − jX=1 jX=1 3 q 2πj 1 q x L(2, χ) = G(j, χ) log 2 sin dx, χ( 1) = 1, (10) q 0 2 − − jX=1 Z   where G(j, χ) is the Gauss sum corresponding the Dirichlet character χ modulo q.

Note. Theoretically speaking, we can give all the identities for L(r, χ) for every integer r 1 for the corresponding even and odd Dirichlet character χ respectively ≥ except L(1, χ) for the even character because of the condition of Theorem 2 . How to get the identity of L(1, χ) for an even character is still an open problem. 2. Some Lemmas

To prove the theorems, we give the following lemmas.

Lemma 1 If a function sequence G (x) satisfies the differential equation { n }n=0,1,2···

d G (x)=(n + 1)G (x), dx n+1 n with G0(x) is a nonzero integer. Then the function sequence will be determined by a constant sequence a in the form of { n} n n n−k Gn(x)= akx , k! kX=0 where an is the constant term of polynomial Gn(x).

Proof: Obviously Gn(x) is a polynomial of degree n, assume that

n n n−k Gn(x)= akx . k! kX=0

Multiplying both sides by n + 1 and integrating, then we have

n n xn−k+1 (n + 1)Gn(x)dx = (n + 1) ak k! n k +1 Z kX=0 − n (n + 1)! = a xn−k+1 + a k!(n k + 1)! k n+1 kX=0 − n+1 n +1 n+1−k = akx k ! kX=0 = Gn+1(x). (11)

Taking the derivative of formula (11), then we can get d (n + 1)G (x)= G (x). n dx n+1 This has proved Lemma 1.

4 Lemma 2 If x (0, 2π), we have ∈ ∞ sin nx π x = − ; (12) n 2 nX=1 ∞ cos nx x = log 2 sin . (13) n − 2 nX=1  

Proof: As we know, for a complex variable z, when z < 1, we have the Taylor series | | expansion of log(1 z) as − ∞ zn log(1 z)= . − − n nX=1 We can also extend it to all z 1 except z = 1 by using analytic continuation. | | ≤ Therefore, let z = eix, it is allowed for x (0, 2π) ∈ ∞ cos nx ∞ sin nx log(1 eix)= + i . (14) − − n n nX=1 nX=1 − ix x π i Substitute 1 e = 2(1 cos x)e 2 into equation (14), we get − − q x π log(1 eix) = log 2(1 cos x) − i − − − − − 2 q x x π = log 2 sin − i −  2  − 2 ∞ cos nx ∞ sin nx = + i n n nX=1 nX=1

Consider real part and imaginary part respectively, we immediately have

∞ sin nx π x = − ; n 2 nX=1 ∞ cos nx x = log 2 sin . n − 2 nX=1  

That proves Lemma 2. 2. Proofs of Theorems

With the help of these lemmas, it is feasible to prove theorems and the corollary. Firstly we will prove Theorem 1.

Proof of Theorem 1. In order to prove Theorem 1, we need to discuss two equations between Dirichlet L-functions and Gauss sums under the different condition χ( 1) = 1 and χ( 1) = 1 respectively. − − −

5 When χ( 1) = 1 and s> 1, we have − q ∞ cos 2πnj G(j, χ) q ns jX=1 nX=1 πnji − πnji q q ∞ 2 2 2πmji e q + e q = χ(m)e q 2ns jX=1 mX=1 nX=1 ∞ q q 1 2π(m+n)ji 2π(m−n)ji = χ(m) e q + e q 2n nX=1 mX=1 jX=1   ∞ q = [χ( n)+ χ(n)] 2ns − nX=1 ∞ χ(n) = q ns nX=1 = qL(s, χ), which is equivalent to

2πnj 1 q ∞ sin( ) L(s, χ)= G(j, χ) q . (15) q ns jX=1 nX=1

When χ( 1) = 1, we have − − q ∞ sin( 2πnj ) i G(j, χ) q ns jX=1 nX=1 πnji − πnji q q ∞ 2 2 2πmji e q e q = χ(m)e q − 2ns jX=1 mX=1 nX=1 ∞ q q 1 2π(m+n)ji 2π(m−n)ji = χ(m) e q e q 2ns − nX=1 mX=1 jX=1   ∞ q = [χ( n) χ(n)] 2ns − − nX=1 ∞ χ(n) = q − ns nX=1 = qL(s, χ), − which is equivalent to

2πnj i q ∞ cos( ) L(s, χ)= G(j, χ) q . (16) −q ns jX=1 nX=1

According to the equations (15) and (16), when s and χ have the same parity, we have 2πnj 2πnj ( i)s( mod 2) q ∞ sins( mod 2)( ) cos(s+1)( mod 2)( ) L(s, χ)= − G(j, χ) q q . (17) q ns jX=1 nX=1 6 When s and χ have the different parity, s> 1, we have 2πj 2πj ( i)s( mod 2) q ∞ sin(s+1)( mod 2)( ) coss( mod 2)( ) L(s, χ)= − G(j, χ) q q . q ns jX=1 nX=0

This proves Theorem 1. Next we will prove Theorem 2 from the formula (17).

Proof of Theorem 2. Consider the part of trigonometric function separately ∞ sins( mod 2)( 2πnj ) cos(s+1)( mod 2)( 2πnj ) q q , ns nX=1 which is equivalent to the following function sequence we denote as Fs(x) at the 2πj { } point x = q ∞ sin nx ∞ cos nx ∞ sin nx ∞ cos nx ∞ sin nx , , , , (18) n n2 n3 n4 n5 ··· nX=1 nX=1 nX=1 nX=1 nX=1

Therefore we can rewrite the formula (17) as following ( i)s( mod 2) q 2πj L(s, χ)= − G(j, χ)Fs . (19) q q ! jX=1

For formula (18), we have recursion obviously, F (x)= π−x 1 2 (20) F (x)=( 1)nF ′ (x), ( n − n+1 where the leading term is obtained by Lemma 2. If we give another function sequence G (x) which is defined by { n } G (x)= n!F (x), when n 0, 1(mod4); n n (21) G (x)=−n!F (x), when n ≡2, 3(mod4).  n n ≡ Then we have G (x) is a function sequence satifying the requirement of Lemma 1. { n } Therefore, a sequence a exists to determine G (x) . What means, if we find the { n} { n } general term formula of a , we find that of F (x) . When 0

G0(x) = 1/2; x π G (x) = ; 1 2 − 2 x2 π2 G (x) = πx + ; 2 2 − 3 x3 3πx2 G (x) = + π2x + 0; 3 2 − 2 x4 4π2 G (x) = 2πx3 + π2x2 +0x + ; 4 2 − 15 x5 5πx4 10π2x3 4π2x G (x) = + + 0; 5 2 − 2 3 − 3 ······ 7 Then list the constant term of the polynomial G (x), we get the sequence a as n { n} following

a0 = 1/2; a = π/2; 1 − ∞ 1 a = (2k)! ; 2k n2k jX=1 a2k+1 = 0, (k =1, 2, 3 ) (22) ··· ∞ 1 where j=1 n2k is the value of Riemann ζ function on the even integers as P ( 1)k+1B (2π)2k ζ(2k)= − 2k . 2(2k)!

Let [t] denote the round down of t and (xn)(t) denote the t-order derivative of xn, from Lemma 1 and formula (22), so we have

n n n−k Gn(x) = akx k! kX=0 xn nπxn−1 [n/2] n = + (2k)! ζ(2k)xn−2k 2 − 2 2k! kX=1 xn nπxn−1 [n/2] = + ζ(2k)(xn)(2k). (23) 2 − 2 kX=1

Plug the formula (23) into the formula (21), considering the formula (19) and we get

L(s, χ)= − 1 q xs sπxs 1 [s/2] s (2k) qs! j=1 G(j, χ) 2 2 + k=1 ζ(2k)(x ) 2πj , s 0(mod4); − − x= q ≡  h − i i qP xs sπxs 1 [Ps/2] s (2k)  G(j, χ) + ζ(2k)(x ) 2πj , s 1 (mod 4);  qs! j=1 2 2 k=1 x=  − q ≡   s s−1   1 Pq x sπx P[s/2] s (2k)  qs! j=1 G(j, χ) 2 2 + k=1 ζ(2k)(x ) 2πj , s 2 (mod 4); − x= q ≡ q h s s−1 [s/2] i  Pi G(j, χ) x sπx +P ζ(2k)(xs)(2k) , s 3 (mod 4).  qs! j=1 2 2 k=1 x= 2πj  − − q ≡  P h P i  In conclusion

( i)s q xs sπxs−1 [s/2] L(s, χ)= − G(j, χ) + ζ(2k)(xs)(2k) .   − qs! j=1 2 − 2 k=1 2πj X X x= q  

This proves Theorem 2.

8 Proof of Corollary 1 We can easily get formula (9) from Theorem 2 for s = 2. For formula (10), we remark that definitive integral appearing in the Corollary 1 is a improper integral. It converges because

π 4 x log(2 sin )dx = 4π log 2. Z0 2 −

Through the Lemma 2, when x (0, 2π) ∈ ∞ sin nx ∞ cos nx = + C n2 n nX=1 Z nX=1 x t = log 2(sin )dt Z0 − 2

2πj So when s = 2 in Theorem 2, taking x = q , we have

q 2πj 1 q x L(2, χ)= G(j, χ) log(2 sin )dx. q 0 2 jX=1 Z

The proof of Corollary 1 has already been completed.

References

[1] Alkan E. On the mean square average of special values of L-functions. Journal of Number Theory, 2011, 131(8): 1470-1485.

[2] Alkan E. Values of Dirichlet L-functions, Gauss sums and trigonometric sums. The Ramanujan Journal, 2011, 26(3): 375-398.

[3] Apostol T M. Introduction to . Springer Science, Business Media, 1998.

[4] Balasubramanian R. A note on Dirichlet’s L-functions. Acta Arithmetica, 1980, 38: 273-283.

[5] Heath-Brown D R. An asymptotic series for the mean value of DirichletL- functions. Commentarii Mathematici Helvetici, 1981, 56(1): 148-161.

[6] Heath-Brown D R. Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression. Proceedings of the London Mathematical Society, 1992, 3(2): 265-338.

[7] Louboutin S. Quelques formules exactes pour des moyennes de fonctions L de Dirichlet. Canadian Mathematical Bulletin, 1993, 36(2): 190-196.

9 [8] Louboutin S. The mean value of L(k, χ) 2 at positive rational integers k 1, Col- | | ≥ loquium Mathematicum. Instytut Matematyczny Polskiej Akademii Nauk, 2001, 90: 69-76.

[9] Ma R., Zhang Y., Grtzmann M. Some notes on identities for Dirichlet L-functions. Acta Mathematica Sinica, English Series, 2014, 30(5): 747-754.

[10] Ma R., Niu Y., Zhang Y., On asymptotic properties of the generalized Dirichlet L-functions. International Journal of Number Theory, 2019, 15(6): 1305-1321.

[11] Mordell L J. The reciprocity formula for Dedekind sums. American Journal of Mathematics, 1951, 73(3): 593-598.

[12] Walum H. An exact formula for an average of L-series. Illinois Journal of Mathe- matics, 1982, 26(1): 1-3.

[13] Zhang W. Lecture notes in contemporary mathematics, Beijing: Science Press China, 173-179, 1989.

[14] Zhang W. On the General Dedekind Sums and One Kind Identities of Dirichlet L-Functions. Acta Mathematica Sinica, 2001, 44(2): 269-272.

10