Averages of values of $L$-series
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Abstract:
We obtain an exact formula for the average of values of $L$-series over two independent odd characters. The average of any positive moment of values at $s=1$ is then expressed in terms of finite cotangent sums subject to congruence conditions. As consequences, bounds on such cotangent sums, limit points for the average of first moment of $L$-series at $s=1$ and the average size of positive moments of character sums related to the class number are deduced.References
- Emre Alkan, On Dirichlet $L$-functions with periodic coefficients and Eisenstein series, Monatsh. Math. 163 (2011), no. 3, 249–280. MR 2805873, DOI 10.1007/s00605-010-0211-2
- Emre Alkan, On the mean square average of special values of $L$-functions, J. Number Theory 131 (2011), no. 8, 1470–1485. MR 2793888, DOI 10.1016/j.jnt.2011.02.013
- Emre Alkan, Values of Dirichlet $L$-functions, Gauss sums and trigonometric sums, Ramanujan J. 26 (2011), no. 3, 375–398. MR 2860694, DOI 10.1007/s11139-010-9292-8
- Emre Alkan, Florin Stan, and Alexandru Zaharescu, Lehmer $k$-tuples, Proc. Amer. Math. Soc. 134 (2006), no. 10, 2807–2815. MR 2231602, DOI 10.1090/S0002-9939-06-08484-X
- Bruce C. Berndt and Alexandru Zaharescu, Finite trigonometric sums and class numbers, Math. Ann. 330 (2004), no. 3, 551–575. MR 2099193, DOI 10.1007/s00208-004-0559-5
- Harold Davenport, Multiplicative number theory, 3rd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000. Revised and with a preface by Hugh L. Montgomery. MR 1790423
- 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
- Ming Gao Qi, A class of mean square formulas for $L$-functions, J. Tsinghua Univ. 31 (1991), no. 3, 34–41 (Chinese, with English summary). MR 1168609
- Zhefeng Xu and Wenpeng Zhang, Some identities involving the Dirichlet $L$-function, Acta Arith. 130 (2007), no. 2, 157–166. MR 2357653, DOI 10.4064/aa130-2-5
- Wen Peng Zhang, A formula for quartic mean values of the $L$-function, Kexue Tongbao (Chinese) 34 (1989), no. 9, 647–650 (Chinese). MR 1020426
- Wen Peng Zhang, On the fourth power mean of Dirichlet $L$-functions, Lecture notes in contemporary mathematics, 1989, Sci. Press Beijing, Beijing, 1990, pp. 173–179. MR 1180537
- Wen Peng Zhang, On the mean value of the $L$-function, J. Math. Res. Exposition 10 (1990), no. 3, 355–360 (Chinese, with English summary). MR 1072441
- Wen Peng Zhang, A note on a class of mean square values of $L$-functions, J. Northwest Univ. 20 (1990), no. 3, 9–12 (Chinese, with English summary). MR 1077163
- Wen Peng Zhang, On the general Dedekind sums and one kind identities of Dirichlet $L$-functions, Acta Math. Sinica (Chinese Ser.) 44 (2001), no. 2, 269–272 (Chinese, with English and Chinese summaries). MR 1831528
Additional Information
- Emre Alkan
- Affiliation: Department of Mathematics, Koç University, Rumelifeneri Yolu, 34450, Sarıyer, Istanbul, Turkey
- Email: ealkan@ku.edu.tr
- Received by editor(s): August 9, 2011
- Received by editor(s) in revised form: August 15, 2011
- Published electronically: August 28, 2012
- Additional Notes: The author is supported by the Distinguished Young Scholar Award, Tüba-Gebip of Turkish Academy of Sciences
- Communicated by: Ken Ono
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 1161-1175
- MSC (2010): Primary 11M06, 11L05
- DOI: https://doi.org/10.1090/S0002-9939-2012-11506-0
- MathSciNet review: 3008864
Dedicated: Dedicated to the memory of Professor Cemal Koç