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Zeros of Riemann zeta-type functions


In this thesis we study two topics concerning the zeros of the zeta function and the zeros of related functions.

The first topic is a proof of a generalization of Newman’s conjecture. Newman’s original conjecture, which was proved by Rodgers and Tao in 2018, states that certain deformations of the Riemann xi function have zeros off the critical line. We will show that it is possible to formulate an analogue of Newman’s conjecture for any function in the extended Selberg class, and we then prove that these analogous conjectures are true in every case. Our proof is necessarily quite different from Rogers and Tao’s proof because their work requires information about the zeros of the zeta function which is not known in the general case. Our proof also has the benefit of being simpler and more direct.

The second topic is a result on the distribution of the argument of the Riemann zeta function on the critical line. In particular we will prove an unconditional lower bound on the tails of this distribution. This can equivalently be stated as a quantitative estimate for how often the number of nontrivial zeta zeros up to a given height differs greatly from the expected number of such zeros.

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