Leírás
In 1932, Ingham posed the question of what error term in the Prime Number Theorem (PNT) would follow from a general zero-free region for the Riemann zeta function. About 50 years later, this question was answered by Pintz, "up to an epsilon". In the past few years, the results of Pintz were extended to Beurling generalized number systems by Révész.
In this talk, I will discuss some recent work on this topic. First, we make precise the epsilon in the Pintz-Révész theorem, refining recent work of Johnston. Next, we show that this refinement is sharp by constructing, for every "reasonable" choice of function $\eta(t)$, a Beurling number system whose associated zeta function has infinitely many zeros on the contour $\sigma = 1 - \eta(t)$ and none to the right, and a matching error term in the PNT. The constructed examples exhibit several other interesting properties. If time permits, we will comment on zero-density estimates, Landau's local method for deducing zero-free regions from zeta-bounds, and Montgomery-style clustering results for zeros.
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