Counting zeros of the Riemann zeta function

Abstract

In this article, we show that N (T ) − T 2π log T 2πe ≤ 0.1038 log T + 0.2573 log log T + 9.3675 where N (T ) denotes the number of non-trivial zeros ρ, with 0 < Im(ρ) ≤ 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.