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

Abstract

We provide an explicit [Formula: see text] error term for the Riemann–von Mangoldt formula by making results of Wolke [On the explicit formula of Riemann–von Mangoldt, II, J. London Math. Soc. 2(3) (1983) 406–416] and Ramaré [Modified truncated Perron formulae, Ann. Math. Blaise Pascal 23(1) (2016) 109–128] explicit. We also include applications to primes between consecutive powers, the error term in the prime number theorem and an inequality of Ramanujan.