On the reciprocity law for the twisted second moment of Dirichlet 𝐿-functions

Bettin, Sandro

doi:10.1090/tran/6771

On the reciprocity law for the twisted second moment of Dirichlet $L$-functions
HTML articles powered by AMS MathViewer

by Sandro Bettin PDF

Trans. Amer. Math. Soc. 368 (2016), 6887-6914 Request permission

Abstract:

We investigate the reciprocity law, studied by Conrey and Young, for the second moment of Dirichlet $L$-functions twisted by $\chi (a)$ modulo a prime $q$. We show that the error term in this reciprocity law can be extended to a continuous function of $a/q$ with respect to the real topology. Furthermore, we extend this reciprocity result, proving an exact formula also involving shifted moments.

We also give an expression for the twisted second moment involving the coefficients of the continued fraction expansion of $a/q$, and, consequently, we improve upon a classical result of Selberg on the second moment of Dirichlet $L$-functions with two twists.

Finally, we obtain a formula connecting the shifted second moment of the Dirichlet $L$-functions with the Estermann function. In particular cases, this result can be used to obtain some simple explicit exact formulae for the moments.

References

Tom M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976. MR 0434929
R. Balasubramanian, J. B. Conrey, and D. R. Heath-Brown, Asymptotic mean square of the product of the Riemann zeta-function and a Dirichlet polynomial, J. Reine Angew. Math. 357 (1985), 161–181. MR 783539
Abdelmejid Bayad and Abdelaziz Raouj, Mean values of $L$-functions and Dedekind sums, J. Number Theory 132 (2012), no. 8, 1645–1652. MR 2922335, DOI 10.1016/j.jnt.2012.01.014
Sandro Bettin and John Brian Conrey, A reciprocity formula for a cotangent sum, Int. Math. Res. Not. IMRN 24 (2013), 5709–5726. MR 3144177, DOI 10.1093/imrn/rns211
Sandro Bettin and Brian Conrey, Period functions and cotangent sums, Algebra Number Theory 7 (2013), no. 1, 215–242. MR 3037895, DOI 10.2140/ant.2013.7.215
H. M. Bui, A note on the second moment of automorphic $L$-functions, Mathematika 56 (2010), no. 1, 35–44. MR 2604981, DOI 10.1112/S002557930900028X
J. B. Conrey, The mean-square of Dirichlet L-functions, arxiv math.NT/0708.2699.
J. Brian Conrey and Amit Ghosh, Remarks on the generalized Lindelöf hypothesis, Funct. Approx. Comment. Math. 36 (2006), 71–78. MR 2296639, DOI 10.7169/facm/1229616442
J.-M. Deshouillers and H. Iwaniec, Kloosterman sums and Fourier coefficients of cusp forms, Invent. Math. 70 (1982/83), no. 2, 219–288. MR 684172, DOI 10.1007/BF01390728
Goran Djanković, On some exact reciprocity formulas for twisted second moments of Dirichlet $L$-functions, Bull. Korean Math. Soc. 49 (2012), no. 5, 1007–1014. MR 3012968, DOI 10.4134/BKMS.2012.49.5.1007
G. H. Hardy and J. E. Littlewood, Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes, Acta Math. 41 (1916), no. 1, 119–196. MR 1555148, DOI 10.1007/BF02422942
Dean Hickerson, Continued fractions and density results for Dedekind sums, J. Reine Angew. Math. 290 (1977), 113–116. MR 439725, DOI 10.1515/crll.1977.290.113
Henryk Iwaniec, On mean values for Dirichlet’s polynomials and the Riemann zeta function, J. London Math. Soc. (2) 22 (1980), no. 1, 39–45. MR 579806, DOI 10.1112/jlms/s2-22.1.39
Henryk Iwaniec and Peter Sarnak, The non-vanishing of central values of automorphic $L$-functions and Landau-Siegel zeros. part A, Israel J. Math. 120 (2000), no. part A, 155–177. MR 1815374, DOI 10.1007/s11856-000-1275-9
Jerzy Kaczorowski and Alberto Perelli, On the structure of the Selberg class. V. $1<d<5/3$, Invent. Math. 150 (2002), no. 3, 485–516. MR 1946551, DOI 10.1007/s00222-002-0236-9
Jerzy Kaczorowski and Alberto Perelli, A uniform version of Stirling’s formula. part 1, Funct. Approx. Comment. Math. 45 (2011), no. part 1, 89–96. MR 2865415, DOI 10.7169/facm/1317045234
Norman Levinson, More than one third of zeros of Riemann’s zeta-function are on $\sigma =1/2$, Advances in Math. 13 (1974), 383–436. MR 564081, DOI 10.1016/0001-8708(74)90074-7
Huaning Liu and Wenpeng Zhang, On the mean value of $L(m,\chi )L(n,\overline \chi )$ at positive integers $m,n\geq 1$, Acta Arith. 122 (2006), no. 1, 51–56. MR 2217323, DOI 10.4064/aa122-1-5
Stéphane R. Louboutin, A twisted quadratic moment for Dirichlet $L$-functions, Proc. Amer. Math. Soc. 142 (2014), no. 5, 1539–1544. MR 3168461, DOI 10.1090/S0002-9939-2014-11721-7
C. J. Mozzochi, Linking numbers of modular geodesics, Israel J. Math. 195 (2013), no. 1, 71–95. MR 3101243, DOI 10.1007/s11856-012-0161-6
Y. N. Petridis and M. S. Risager, Modular symbols have a normal distribution, Geom. Funct. Anal. 14 (2004), no. 5, 1013–1043. MR 2105951, DOI 10.1007/s00039-004-0481-8
Atle Selberg, Contributions to the theory of Dirichlet’s $L$-functions, Skr. Norske Vid.-Akad. Oslo I 1946 (1946), no. 3, 62. MR 22872
K. Soundararajan, Nonvanishing of quadratic Dirichlet $L$-functions at $s=\frac 12$, Ann. of Math. (2) 152 (2000), no. 2, 447–488. MR 1804529, DOI 10.2307/2661390
K. Soundararajan, Extreme values of zeta and $L$-functions, Math. Ann. 342 (2008), no. 2, 467–486. MR 2425151, DOI 10.1007/s00208-008-0243-2
Ilan Vardi, Dedekind sums have a limiting distribution, Internat. Math. Res. Notices 1 (1993), 1–12. MR 1201746, DOI 10.1155/S1073792893000017
Matthew P. Young, The reciprocity law for the twisted second moment of Dirichlet $L$-functions, Forum Math. 23 (2011), no. 6, 1323–1337. MR 2855052, DOI 10.1515/FORM.2011.053
Matthew P. Young, The fourth moment of Dirichlet $L$-functions, Ann. of Math. (2) 173 (2011), no. 1, 1–50. MR 2753598, DOI 10.4007/annals.2011.173.1.1