A reciprocity relation for the twisted second moment of the Riemann Zeta function

Khan, Rizwanur

doi:10.1090/proc/17003

A reciprocity relation for the twisted second moment of the Riemann Zeta function
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by Rizwanur Khan;

Proc. Amer. Math. Soc. 153 (2025), 81-90

DOI: https://doi.org/10.1090/proc/17003

Published electronically: October 29, 2024

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Abstract:

We prove a reciprocity relation for the twisted second moment of the Riemann Zeta function. This provides an analogue to a formula of Conrey for Dirichlet $L$-functions.

References

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
Sandro Bettin, On the reciprocity law for the twisted second moment of Dirichlet $L$-functions, Trans. Amer. Math. Soc. 368 (2016), no. 10, 6887–6914. MR 3471080, DOI 10.1090/tran/6771
Sandro Bettin, Vorrapan Chandee, and Maksym Radziwiłł, The mean square of the product of the Riemann zeta-function with Dirichlet polynomials, J. Reine Angew. Math. 729 (2017), 51–79. MR 3680370, DOI 10.1515/crelle-2014-0133
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
Valentin Blomer and Rizwanur Khan, Twisted moments of $L$-functions and spectral reciprocity, Duke Math. J. 168 (2019), no. 6, 1109–1177. MR 3934595, DOI 10.1215/00127094-2018-0060
J. B. Conrey, The mean-square of Dirichlet $L$-functions, preprint arXiv:0708.2699, 2007.
J. B. Conrey, More than two fifths of the zeros of the Riemann zeta function are on the critical line, J. Reine Angew. Math. 399 (1989), 1–26. MR 1004130, DOI 10.1515/crll.1989.399.1
J. B. Conrey, A. Ghosh, and S. M. Gonek, Large gaps between zeros of the zeta-function, Mathematika 33 (1986), no. 2, 212–238 (1987). MR 882495, DOI 10.1112/S0025579300011219
Goran Djanković, The reciprocity law for the twisted second moment of Dirichlet $L$-functions over rational function fields, Bull. Aust. Math. Soc. 98 (2018), no. 3, 383–388. MR 3877269, DOI 10.1017/S0004972718000874
I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 8th ed., Elsevier/Academic Press, Amsterdam, 2015. Translated from the Russian; Translation edited and with a preface by Daniel Zwillinger and Victor Moll; Revised from the seventh edition [MR2360010]. MR 3307944
P. Humphries and R. Khan, $L^p$-norm bounds for automorphic forms via spectral reciprocity, preprint arXiv:2208.05613.
Yoichi Motohashi, Spectral theory of the Riemann zeta-function, Cambridge Tracts in Mathematics, vol. 127, Cambridge University Press, Cambridge, 1997. MR 1489236, DOI 10.1017/CBO9780511983399
Asbjørn Christian Nordentoft, A note on additive twists, reciprocity laws and quantum modular forms, Ramanujan J. 56 (2021), no. 1, 151–162. MR 4311681, DOI 10.1007/s11139-020-00270-1
K. Soundararajan, Mean-values of the Riemann zeta-function, Mathematika 42 (1995), no. 1, 158–174. MR 1346680, DOI 10.1112/S0025579300011438
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