Number Theory SeminarExport this event to calendar

Thursday, March 10, 2016 — 1:30 PM EST

Allysa Lumley, York University

“A Zero Density Result for the Riemann Zeta Function”

Let N(σ,T) denote the number of nontrivial zeros of the Riemann zeta function with real part greater than σ and imaginary part between 0 and T. We provide explicit upper bounds for N(σ,T) commonly referred to as a zero density result. In 1940, Ingham showed the following asymptotic result

3(1σ) 5 N(σ,T)=O(T 2σ log T).

Ramar ́e recently proved an explicit version of this estimate:

3
N(σ,T) 4.9(3T)8(1σ) log52σ(T)+51.5log2 T,

forσ0.52andT 2000.
We discuss a generalization of the method used in these two results which yields an explicit bound

of a similar shape while also improving the constants. Furthermore, we present the effect of these improvements on explicit estimates for the prime counting function ψ(x). This is joint work with Habiba Kadiri and Nathan Ng.

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