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GOLDBACH REPRESENTATIONS IN ARITHMETIC PROGRESSIONS AND ZEROS OF DIRICHLET L-FUNCTIONS

Part of: Additive number theory; partitions Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  24 August 2018

Gautami Bhowmik ,
Karin Halupczok ,
Kohji Matsumoto  and
Yuta Suzuki
Gautami Bhowmik
Affiliation:
Laboratoire Paul Painlevé, Labex-Cempi, Université Lille 1, 59655 Villeneuve d’Ascq Cedex, France email bhowmik@math.univ-lille1.fr
Karin Halupczok
Affiliation:
Mathematisch-Naturwissenschaftliche Fakultät, Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1, 40225 Düsseldorf, Germany email karin.halupczok@uni-duesseldorf.de
Kohji Matsumoto
Affiliation:
Graduate School of Mathematics, Nagoya University, Furocho, Chikusa-ku, Nagoya 464-8602, Japan email kohjimat@math.nagoya-u.ac.jp
Yuta Suzuki
Affiliation:
Graduate School of Mathematics, Nagoya University, Furocho, Chikusa-ku, Nagoya 464-8602, Japan email m14021y@math.nagoya-u.ac.jp
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Abstract

Assuming a conjecture on distinct zeros of Dirichlet $L$-functions we get asymptotic results on the average number of representations of an integer as the sum of two primes in arithmetic progression. On the other hand the existence of good error terms gives information on the location of zeros of $L$-functions. Similar results are obtained for an integer in a congruence class expressed as the sum of two primes.

MSC classification

Primary: 11P32: Goldbach-type theorems; other additive questions involving primes 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses
Secondary: 11M41: Other Dirichlet series and zeta functions
Type
Research Article
Information
Mathematika , Volume 65 , Issue 1 , 2019 , pp. 57 - 97
Copyright
Copyright © University College London 2018 

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Footnotes

The fourth author is supported by Grant-in-Aid for JSPS Research Fellow (Grant Number: JP16J00906) and had the partial aid of CEMPI for his stay at Lille.

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