Goldbach representations and exceptional zeros of Dirichlet L-functions

Ade Irma Suriajaya (Kyushu University)

15-Jun-2021, 08:00-08:30 (3 years ago)

Abstract: G. H. Hardy and J. E. Littlewood in 1922 studied the number of representations of a positive number as a sum of prime numbers. They conjectured that all large even numbers can be written as a sum of two odd primes and also conjectured an asymptotic formula for the number of representations. This conjecture gives a quantitative statement of the well-known Goldbach's conjecture. J. Fei in 2016 used a weaker form of this Hardy-Littlewood Goldbach's Conjecture and showed that we could almost eliminate the possible existence of the Landau-Siegel zeros of Dirichlet L-functions associated with characters modulo q congruent to 3 mod 4. To be more precise, Fei showed that we can narrow the interval which may contain a possible exceptional zero of the corresponding Dirichlet L-function. G. Bhowmik and K. Halupczok in a recent preprint extended Fei's result to all odd characters with a slightly weaker conjecture. Independently, C. Jia in his recent preprint used a slightly different form of weak Hardy-Littlewood Goldbach's Conjecture to obtain results similar to Bhowmik and Halupczok's. We extended the weak Hardy-Littlewood Goldbach's Conjecture as close as possible to the original Hardy-Littlewood Goldbach's Conjecture and improved the arguments to extend Fei, Bhowmik and Halupczok, and Jia's results to all Dirichlet L-functions associated with real quadratic characters. This is a joint work with Daniel A. Goldston.

Following Goldston's talk at an AIM seminar early last month, J. Friedlander and H. Iwaniec further improved our result and succeeded in showing that the weak Hardy-Littlewood Goldbach's Conjecture we used indeed implies that there are no Landau-Siegel zeros. As in Friedlander and Iwaniec's approach, using an improved estimate on the prime number theorem for primes in arithmetic progressions, we are able to further improve our arguments to obtain Friedlander and Iwaniec's result. In this talk, I would like to explain the slightly different conjectures and approaches used in this study and introduce relevant results.

number theory

Audience: researchers in the topic


Japan Europe Number Theory Exchange Seminar

Series comments: The purpose of the Japan Europe Number Theory Exchange Seminar is to give a opportunity for researchers in Japan and Europe to exchange their research projects by giving short talks (30 min). The target audience are researchers of any level in the area of number theory.

Start for Fall 2021: 26th October

Organizers: Henrik Bachmann*, Nils Matthes
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