Estimates of 𝜓,𝜃 for large values of 𝑥 without the Riemann hypothesis

Dusart, Pierre

doi:10.1090/mcom/3005

Estimates of $\psi ,\theta$ for large values of $x$ without the Riemann hypothesis
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Math. Comp. 85 (2016), 875-888 Request permission

Abstract:

The enlargement of known zero-free regions has enabled us to find better effective estimates for classical number-theoretic functions linked to the distribution of prime numbers. In particular we draw the quintessence of the method of Rosser and Schoenfeld on the upper bounds for the usual Chebyshev prime and prime power counting functions to find an upper bound function directly linked to a zero-free region.

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