Estimating $\pi (x)$ and related functions under partial RH assumptions

Abstract

The aim of this paper is to give a direct interpretation of the validity of the Riemann hypothesis up to a certain height $T$ in terms of the prime-counting function $\pi (x)$. This is done by proving the well-known explicit Schoenfeld bound on the RH to hold as long as $4.92\sqrt{x/\log (x)}\le T$. Similar statements are proven for the Riemann prime-counting function and the Chebyshov functions $\psi (x)$ and $\vartheta (x)$. Apart from that, we also improve some of the existing bounds of Chebyshov type for the function $\psi (x)$.

16 pages, corrected version with a missing summand in Theorem 1 added (explicit bounds in tables 1 and 2 are still valid), corrigendum for journal version in preparation

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• The Riemann hypothesis is true up to $3\cdot 10^{12}$
• From explicit estimates for the primes to explicit estimates for the Möbius function II