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Transactions of the American Mathematical Society

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Explicit formulae for averages of Goldbach representations


Authors: Jörg Brüdern, Jerzy Kaczorowski and Alberto Perelli
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 11P32, 11N05
DOI: https://doi.org/10.1090/tran/7799
Published electronically: March 20, 2019
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Abstract: We prove an explicit formula, analogous to the classical explicit formula for $ \psi (x)$, for the Cesàro-Riesz mean of any order $ k>0$ of the number of representations of $ n$ as a sum of two primes. Our approach is based on a double Mellin transform and the analytic continuation of certain functions arising therein.


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Additional Information

Jörg Brüdern
Affiliation: Mathematisches Institut, Bunsenstrasse 3-5, 37073 Göttingen, Germany
Email: bruedern@uni-math.gwdg.de

Jerzy Kaczorowski
Affiliation: Faculty of Mathematics and Computer Science, Adam Mickiewicz University, 61-614 Poznań, Poland; and Institute of Mathematics of the Polish Academy of Sciences, 00-956 Warsaw, Poland
Email: kjerzy@amu.edu.pl

Alberto Perelli
Affiliation: Dipartimento di Matematica, Università di Genova, via Dodecaneso 35, 16146 Genova, Italy
Email: perelli@dima.unige.it

DOI: https://doi.org/10.1090/tran/7799
Keywords: Goldbach problem, explicit formulae
Received by editor(s): January 22, 2018
Received by editor(s) in revised form: November 12, 2018
Published electronically: March 20, 2019
Additional Notes: This research was partially supported by a grant from Deutsche Forschungsgemeinschaft, by the grant PRIN-2015 “Number Theory and Arithmetic Geometry”, and by grant 2017/25/B/ST1/00208 “Analytic methods in number theory” from the National Science Centre, Poland. The third author is a member of the GNAMPA group of INdAM
Article copyright: © Copyright 2019 American Mathematical Society