Counting zeros of the Riemann zeta function

Abstract In this article, we show that | N ( T ) − T 2 π log ⁡ ( T 2 π e ) | ≤ 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.