Abstract
Let χ be a nonprincipal Dirichlet character modulo a prime number p ≽ 3 and let \({\mathfrak{a}_{\cal X}}: = {1 \over 2}\left( {1{ - _{\cal X}}\left( { - 1} \right)} \right)\). Define the mean value
We give an identity for \({{\cal M}_p}\left( { - s,{\cal X}} \right)\,\,\) which, in particular, shows that
for fixed \(0 < \sigma < {1 \over 2}\) and \(\left| {t: = \,{\mathfrak{J}_s}} \right| = o\left( {{p^{\left( {1 - 2\sigma } \right)/\left( {3 + 2\sigma } \right)}}} \right)\).
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References
H. Davenport: Multiplicative Number Theory. Graduate Texts in Mathematics 74. Springer, New York, 2000.
E. Elma: On a problem related to discrete mean values of Dirichlet L-functions. J. Number Theory 217 (2020), 36–43.
A. Ivić: The Riemann Zeta-Function: Theory and Applications. Dover Publications, Mineola, 2003.
S. Kanemitsu, J. Ma, W. Zhang: On the discrete mean value of the product of two Dirichlet L-functions. Abh. Math. Semin. Univ. Hamb. 79 (2009), 149–164.
H. Liu, W. Zhang: On the mean value of \(L\left( {m{,_{\cal X}}} \right)L\left( {n{,_{\overline {\cal X} }}} \right)\) at positive integers m, n ≽ 1. Acta Arith. 122 (2006), 51–56.
S. Louboutin: Quelques formules exactes pour des moyennes de fonctions L de Dirichlet. Can. Math. Bull. 36 (1993), 190–196.
S. Louboutin: The mean value of ∣L(k, χ)∣2 at positive rational integers k ≽ 1. Colloq. Math. 90 (2001), 69–76.
K. Matsumoto: Recent developments in the mean square theory of the Riemann zeta and other zeta-functions. Number Theory. Trends in Mathematics. Birkhäuser, Basel, 2000, pp. 241–286.
H. L. Montgomery, R. C. Vaughan: Multiplicative Number Theory. I. Classical Theory. Cambridge Studies in Advanced Mathematics 97. Cambridge University Press, Cambridge, 2007.
Y. Motohashi: A note on the mean value of the zeta and L-functions. I. Proc. Japan Acad., Ser. A 61 (1985), 222–224.
E. C. Titchmarsh: The Theory of the Riemann Zeta-Function. Oxford Science Publications. Clarendon Press, Oxford, 1986.
Z. Xu, W. Zhang: Some identities involving the Dirichlet L-function. Acta Arith. 130 (2007), 157–166.
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Elma, E. On discrete mean values of Dirichlet L-functions. Czech Math J 71, 1035–1048 (2021). https://doi.org/10.21136/CMJ.2021.0189-20
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DOI: https://doi.org/10.21136/CMJ.2021.0189-20