Estimates of for large values of

Author:
Pierre Dusart

Journal:
Math. Comp. **71** (2002), 1137-1168

MSC (2000):
Primary 11N13, 11N56; Secondary 11Y35, 11Y40

DOI:
https://doi.org/10.1090/S0025-5718-01-01351-5

Published electronically:
November 21, 2001

MathSciNet review:
1898748

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We extend a result of Ramaré and Rumely, 1996, about the Chebyshev function in arithmetic progressions. We find a map such that and , whereas is a constant. Now we are able to show that, for ,

and, for ,

**1.**KEVIN S. MCCURLEY, ``Explicit zero-free regions for Dirichlet -functions'',*J. Number Theory*,**19**, (1984) pp. 7-32. MR**85k:11041****2.**KEVIN S. MCCURLEY, ``Explicit estimates for and '',*Math. Comp.*,**42**, Number 165 (January 1984) pp. 287-296. MR**85g:11085****3.**O. RAMARÉ and R. RUMELY, ``Primes in arithmetic progressions'',*Math. Comp.*,**65**, Number 213 (January 1996) pp. 397-425. MR**97a:11144****4.**J. BARKLEY ROSSER, ``Explicit bounds for some functions of prime numbers'',*Amer. J. Math.*,**63**, (1941) pp. 211-232. MR**2:150e****5.**J. BARKLEY ROSSER and L. SCHOENFELD, ``Approximate formulas for some functions of prime numbers'',*Illinois J. Math.***6**(1962) pp. 64-94. MR**25:1139****6.**J. BARKLEY ROSSER and L. SCHOENFELD, ``Sharper bounds for the Chebyshev functions and '',*Math. Comp.*,**29**, Number 129 (January 1975) pp. 243-269. MR**56:15581a****7.**LOWELL SCHOENFELD, ``Sharper bounds for the Chebyshev functions and . II'',*Math. Comp.*,**30**, Number 134 (April 1976) pp. 337-360. MR**56:15581b****8.**G. TENENBAUM, ``Introduction à la théorie analytique et probabiliste des nombres'', Institut Elie Cartan (1990) MR**97e:11005a****9.**J. VAN DE LUNE, H. J. J. TE RIELE and D. T. WINTER, ``On the zeros of the Riemann zeta function in the critical strip. IV''*Math. Comp.*,**46**, Number 174 (April 1986) pp. 667-681. MR**87e:11102**

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Additional Information

**Pierre Dusart**

Affiliation:
Département de Math., LACO, 123 avenue Albert Thomas, 87060 Limoges cedex, France

Email:
dusart@unilim.fr

DOI:
https://doi.org/10.1090/S0025-5718-01-01351-5

Keywords:
Bounds for basic functions,
arithmetic progression

Received by editor(s):
February 23, 1998

Received by editor(s) in revised form:
December 17, 1998, and August 21, 2000

Published electronically:
November 21, 2001

Article copyright:
© Copyright 2001
American Mathematical Society