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Title:
Remarks on Landau–Siegel zeros
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Abstract:
One of the central problems in comparative prime number theory involves understanding primes in
arithmetic progressions. The distribution of primes in arithmetic progressions are sensitive to real zeros near $s = 1$ of L-functions associated to primitive real Dirichlet characters. The Generalized Riemann Hypothesis implies that such L-functions have no zeros near $s = 1$. In 1935, Siegel proved the strongest known upper bound for the largest such real zero, but his result is vastly inferior to what is known unconditionally for other L-functions. We exponentially improve Siegel’s bound under a mild hypothesis that permits real zeros to lie close to $s = 1$. Our hypothesis can be verified for almost all primitive real characters. Our work extends to other families of L-functions. This is joint work with Jesse Thorner and Alexandru Zaharescu.
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