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.

Every integer greater than two can be expressed as the sum of a prime and a square-free number. Expanding on recent work, we provide explicit and asymptotic results when divisibility conditions are imposed on the square-free number. For example, we show for odd k ≤ 10 5 and even k ≤ 2 • 10 5 that any even integer n ≥ 40 can be expressed as the sum of a prime and a squarefree number coprime to k. We also discuss applications to other Goldbach-like problems.

Using a recent verification of the Riemann hypothesis up to height $$3\cdot 10^{12}$$ 3 · 10 12 , we provide strong estimates on $$\pi (x)$$ π ( x ) and other prime counting functions for finite ranges of x. In particular, we get that $$|\pi (x)-li(x)|<\sqrt{x}\log x/8\pi $$ | π ( x ) - l i ( x ) | < x log x / 8 π for $$2657\le x\le 1.101\cdot 10^{26}$$ 2657 ≤ x ≤ 1.101 · 10 26 . We also provide weaker bounds that hold for a wider range of x, and an application to an inequality of Ramanujan.

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.