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

We provide an explicit $O(x\log x/T)$ 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 and an inequality of Ramanujan.

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