Counting zeros of the Riemann zeta function

Abstract

In this article, we show that $$\left|N(T)-\frac{T}{2\pi }\log \left(\frac{T}{2\pi e}\right)\right|\le 0.1038\log T+0.2573\log \log T+9.3675$$ where $N(T)$ denotes the number of non-trivial zeros $\rho$, with $0<\Im (\rho )\le T$, of the Riemann zeta function. This improves the previous result of Trudgian for sufficiently large $T$. The improvement comes from the use of various subconvexity bounds and ideas from the work of Bennett $et$ $al.$ on counting zeros of Dirichlet $L$-functions.

Accepted by J. Number Theory