Michel Planat and Patrick Solé

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

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

**Full paper (PDF, 201 Kb)**

## 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

- Edwards H. M. (1974) Riemann’s zeta function, Academic Press, New York.
- Odlyzko, A. Tables of zeros of the Riemann zeta function, available at http://www.dtc.umn.edu/~odlyzko/zeta_tables/.
- Planat, M., & Solé, P. (2013) Efficient prime counting and the Chebyshev primes, J. Discrete Math. (Hindawi), Article ID 491627, 11 pp.
- Borwein, J. B., Bradley, D. M., & Crandall, R. E. (2000) Computational strategies for the Riemann zeta function, J. Comp. Appl. Math., 121, 247–296.
- Robin, G. (1984) Sur la difference Li(θ(x)) − π(x), Ann. Fac. Sc. Toulouse, 6, 257–268.
- Pletser, V. Conjecture on the value of π(10
^{26}), the number of primes < 10^{26}, Preprint 1307.4444 [math.NT]. - Davenport, H. (1980) Multiplicative number theory, Second edition, Springer Verlag, New York.
- Skewes, S. (1933) On the difference π(x) − li(x), J. London Math. Soc., 8, 277–283.
- Burnol, J. F. The explicit formula in simple terms, Preprint 9810169 (math.NT).
- Deléglise, M., & Rivat, J. (1998) Computing ψ(x), Math. Comp. 67, 1691–1696.
- Alamadhi, A., Planat, M. & Solé, P. (2013) Chebyshev’s bias and generalized Riemann hypothesis, J. Alg., Numb Th.: Adv. and Appl., 8, 41–55; Preprint 1112.2398 (math.NT).

## Related papers

## Cite this paper

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