On the error term in the explicit formula of Riemann--von Mangoldt

Abstract

We provide an explicit O(x/T ) error term for the Riemann-von Mangoldt formula by making results of Wolke (1983) and Ramaré (2016) explicit. We also include applications to primes between consecutive powers, the error term in the prime number theorem, an inequality of Ramanujan, and a result due to Cramér on gaps between primes.

Density results for the zeros of zeta applied to the error term in the prime number theorem

“…(1) As in [26], we use the partial numerical verification of the Riemann Hypothesis H 0 = 3 • 10 12 from [25]. As in [4] we use R = 5.5666305 for the zero-free region following [19, Theorem 1 and Section 6.1]. That is if ̺ = β + iγ is a non trivial zero of the Riemann zeta function, then…”

“…After accounting for this correction, the constant 0.63 changes to 0.77. This also has the unfortunate effect of rendering for instance the results of Trudgian [32], Platt-Trudgian [24,26], and Cully-Hugill-Johnston [4] concerning error terms in the prime number theorem unreliable. In the present work, despite a worse subconvexity bound for ζ, we are able to recover, and actually improve on these previous results for E ψ (x).…”

Some explicit estimates for the error term in the prime number theorem

“…Next we give a recent estimate on the error term in the Riemann-von Mangoldt formula due to Cully-Hugill and the first author [CHJ21]. Here we convert their result into a specific form that is useful for our application.…”

“…In this paper we make refinements to Platt and Trudgian's method, in turn giving improvements to their estimates that hold for a wider range of x. Most notably, we modify a technique recently employed by Broadbent et al [Bro+21,Section A.2] and make use of a new explicit error term for the Riemann-von Mangoldt formula [CHJ21]. We also give improvements to other preliminary results in [PT21b] by incorporating a recent verification of the Riemann hypothesis up to height 3 • 10 12 [PT21a].…”

Estimates for $L$-functions in the critical strip under GRH with effective applications

“…Although applications of conditional and effective estimates for L-functions in the critical strip to various number-theoretic problems exist, see [CHJ21,p. 20] for instance, results in this direction are quite obscure.…”