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BOUNDS FOR MOMENTS OF QUADRATIC DIRICHLET CHARACTER SUMS

Part of: Zeta and $L$-functions: analytic theory Exponential sums and character sums

Published online by Cambridge University Press:  06 May 2024

PENG GAO Open the ORCID record for PENG GAO [Opens in a new window]  and
LIANGYI ZHAO Open the ORCID record for LIANGYI ZHAO [Opens in a new window]
PENG GAO
Affiliation:
School of Mathematical Sciences, Beihang University, Beijing 100191, PR China e-mail: penggao@buaa.edu.cn
LIANGYI ZHAO*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia
*
e-mail: l.zhao@unsw.edu.au
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Abstract

We establish upper bounds for moments of smoothed quadratic Dirichlet character sums under the generalized Riemann hypothesis, confirming a conjecture of M. Jutila [‘On sums of real characters’, Tr. Mat. Inst. Steklova 132 (1973), 247–250].

Keywords

quadratic Dirichlet charactermoments

MSC classification

Primary: 11L40: Estimates on character sums
Secondary: 11M06: $zeta (s)$ and $L(s, chi)$

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 111 , Issue 1 , February 2025 , pp. 43 - 47
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The first author is supported in part by NSFC grant no. 11871082 and the second author by the FRG Grant PS71536 at the University of New South Wales.

References

Armon, M. V., ‘Averages of real character sums’, J. Number Theory 77(2) (1999), 209–226.CrossRefGoogle Scholar
Čech, M., ‘Moments of real Dirichlet $L$ -functions and multiple Dirichlet series’, Preprint, 2024, arXiv:2402.07473.Google Scholar
Gao, P., ‘Sharp upper bounds for moments of quadratic Dirichlet $L$ -functions’, Preprint, 2021, arXiv:2101.08483.Google Scholar
Harper, A. J., ‘Sharp conditional bounds for moments of the Riemann zeta function’, Preprint, 2013, arXiv:1305.4618.Google Scholar
Iwaniec, H. and Kowalski, E., Analytic Number Theory, American Mathematical Society Colloquium Publications, 53 (American Mathematical Society, Providence, RI, 2004).Google Scholar
Jutila, M., ‘On character sums and class numbers’, J. Number Theory 5 (1973), 203–214.CrossRefGoogle Scholar
Jutila, M., ‘On sums of real characters’, Tr. Mat. Inst. Steklova 132 (1973), 247–250.Google Scholar
Montgomery, H. L. and Vaughan, R. C., ‘Mean values of character sums’, Canad. J. Math. 31(3) (1979), 476–487.CrossRefGoogle Scholar
Shen, Q., ‘The fourth moment of quadratic Dirichlet $L$ -functions’, Math. Z. 298(1–2) (2021), 713–745.CrossRefGoogle Scholar
Soundararajan, K., ‘Nonvanishing of quadratic Dirichlet $L$ -functions at $s=\frac{1}{2}$ ’, Ann. of Math. (2) 152(2) (2000), 447–488.CrossRefGoogle Scholar
Szabó, B., ‘High moments of theta functions and character sums’, Mathematika 70(2) (2024), Article no. e12242, 37 pages.CrossRefGoogle Scholar
Virtanen, H., ‘The mean fourth power of real character sums’, Acta Arith. 103(3) (2002), 249–257.CrossRefGoogle Scholar