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On the reciprocity law for the twisted second moment of Dirichlet $ L$-functions


Author: Sandro Bettin
Journal: Trans. Amer. Math. Soc. 368 (2016), 6887-6914
MSC (2010): Primary 11M06; Secondary 11M41, 11A55
DOI: https://doi.org/10.1090/tran/6771
Published electronically: February 11, 2016
MathSciNet review: 3471080
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Abstract: We investigate the reciprocity law, studied by Conrey and Young, for the second moment of Dirichlet $ L$-functions twisted by $ \chi (a)$ modulo a prime $ q$. We show that the error term in this reciprocity law can be extended to a continuous function of $ a/q$ with respect to the real topology. Furthermore, we extend this reciprocity result, proving an exact formula also involving shifted moments.

We also give an expression for the twisted second moment involving the coefficients of the continued fraction expansion of $ a/q$, and, consequently, we improve upon a classical result of Selberg on the second moment of Dirichlet $ L$-functions with two twists.

Finally, we obtain a formula connecting the shifted second moment of the Dirichlet $ L$-functions with the Estermann function. In particular cases, this result can be used to obtain some simple explicit exact formulae for the moments.


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Additional Information

Sandro Bettin
Affiliation: Centre de Recherches Mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal, Quebec H3C 3J7, Canada
Address at time of publication: Dipartimento di Matematica, Università di Genova, via Dodecaneso 35, 16146 Genova, Italy
Email: bettin@crm.umontreal.ca, bettin@dima.unige.it

DOI: https://doi.org/10.1090/tran/6771
Received by editor(s): February 14, 2014
Received by editor(s) in revised form: May 16, 2014, and July 31, 2014
Published electronically: February 11, 2016
Article copyright: © Copyright 2016 American Mathematical Society