Michel Planat and Patrick Solé

Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132

Volume 21, 2015, Number 3, Pages 38—44

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## Details

### Authors and affiliations

Michel Planat

*Institut FEMTO-ST, CNRS,
15 B Avenue des Montboucons, F-25044 Besançon, France
*

Patrick Solé

*Telecom ParisTech, 46 rue Barrault, 75634 Paris Cedex 13, France.
Mathematics Department, King Abdulaziz University,
Jeddah, Saudi Arabia
*

### Abstract

Prime number theorem asserts that (at large ) the prime counting function is approximately the logarithmic integral . In the intermediate range, Riemann prime counting function deviates from by the asymptotically vanishing sum depending on the critical zeros of the Riemann zeta function . We find a fit [with three to four new exact digits compared to ] by making use of the Von Mangoldt explicit formula for the Chebyshev function . Another equivalent fit makes use of the Gram formula with the variable . Doing so, we evaluate in the range , with the help of the first Riemann zeros . A few remarks related to Riemann hypothesis (RH) are given in this context.

### Keywords

- Prime counting
- Chebyshev psi function
- Riemann hypothesis

### AMS Classification

- Primary 11N05
- Secondary 11A25, 11N37

### References

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## Cite this paper

Planat, M. & Solé, P. (2015). Improving Riemann prime counting. Notes on Number Theory and Discrete Mathematics, 21(3), 38-44.