Simple critical zeros and distinct zeros of the Riemann zeta-function in short intervals

RESUMO

Recently, on the non-trivial zeros of the Riemann zeta function, it is discovered by Claude and verified by Alpöge and Furman that more than 67.25% of the zeros are simple and on the critical line, and more than 83.62% are distinct. Later, Lamzouri gave a different and more direct proof. In this article, we will use the method of Lamzouri to give lower bounds on the number of the non-trivial zeros of the Riemann zeta function in short intervals. To prove the main result, we establish Montgomery's theorem on the pair correlation of zeros of the zeta function in short intervals by following the approach of Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh, and then use Lamzouri's inequality on any finite multiset of complex numbers which is invariant under complex conjugation.