Abstract
We obtain an asymptotic formula for the smoothly weighted first moment of quadratic Dirichlet L-functions at central values, with explicit main terms and an error term that is “square-root” of the main term.
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M.W. Alderson and M.O. Rubinstein, Conjectures and experiments concerning the moments of L(1/2, χd), Exp. Math., 21(3):307–328, 2012, https://doi.org/10.1080/10586458.2012.687238.
S. Bochner, A theorem on analytic continuation of functions in several variables, Ann. of Math. (2), 39(1):14–19, 1938, https://doi.org/10.2307/1968709.
M. Čech, The Ratios conjecture for real Dirichlet characters and multiple Dirichlet series, preprint, arXiv:2110.04409,
P. Gao and L. Zhao, First moment of central values of quadratic Dirichlet L-functions, preprint, arXiv:2303.11588.
D. Goldfeld and J. Hoffstein, Eisenstein series of \(\frac{1}{2}\) -integral weight and the mean value of real Dirichlet L-series, Invent. Math., 80(2):185–208, 1985, https://doi.org/10.1007/BF01388603.
D.R. Heath-Brown, A mean value estimate for real character sums, Acta Arith., 72(3):235–275, 1995, https://doi.org/10.4064/aa-72-3-235-275.
H. Iwaniec and E. Kowalski, Analytic number theory, Colloq. Publ., Am. Math. Soc., Vol. 53, AMS, Providence, RI, 2004, https://doi.org/10.1090/coll/053.
M. Jutila, On the mean value of L(\(\frac{1}{2}\) 2, χ) for real characters, Analysis, 1(2):149–161, 1981, https://doi.org/10.1524/anly.1981.1.2.149.
H.L. Montgomery and R.C. Vaughan, Multiplicative number theory. I. Classical theory, Camb. Stud. Adv. Math., Vol. 97, Cambridge Univ. Press, Cambridge, 2007.
10. K. Soundararajan, Nonvanishing of quadratic Dirichlet L-functions at s =
A.I. Vinogradov and L.A. Takhtadzhyan, Analogues of the Vinogradov-Gauss formula on the critical line, Zap. Nauchn. Sem. Leningrad. Otd. Mat. Inst. Steklova, 109:41–82, 180–181, 182–183, 1981.
M.P. Young, The first moment of quadratic Dirichlet L-functions, Acta Arith., 138(1):73–99, 2009, https://doi.org/10.4064/aa138-1-4.
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Wen, T. The first moment of quadratic Dirichlet L-functions at central values. Lith Math J 64, 210–226 (2024). https://doi.org/10.1007/s10986-024-09628-0
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DOI: https://doi.org/10.1007/s10986-024-09628-0