Available in electronic format
Available in print format		 Mathematics of Computation
Journal of the American Mathematical Society
ISSN 1088-6842(e) ISSN 0025-5718(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Estimates of $\theta(x;k,l)$ for large values of $x$

Author(s): Pierre Dusart.
Journal: Math. Comp. 71 (2002), 1137-1168.
MSC (2000): Primary 11N13, 11N56; Secondary 11Y35, 11Y40
Posted: November 21, 2001
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: We extend a result of Ramaré and Rumely, 1996, about the Chebyshev function $\theta$ in arithmetic progressions. We find a map $\varepsilon(x)$ such that $\mid\theta(x;k,l)-x/\varphi(k)\mid<x\varepsilon(x)$ and $\varepsilon(x)=O\left(\frac{1}{\ln^a x}\right)\quad{(\forall a>0)}$, whereas $\varepsilon(x)$ is a constant. Now we are able to show that, for $x\geqslant1531$,

\begin{displaymath}\mid\theta(x;3,l)-x/2\mid<0.262\frac{x}{\ln x}\end{displaymath}

and, for $x\geqslant151$,

\begin{displaymath}\pi(x;3,l)>\frac{x}{2\ln x}.\end{displaymath}

References:

1.
KEVIN S. MCCURLEY, ``Explicit zero-free regions for Dirichlet $L$-functions'', J. Number Theory, 19, (1984) pp. 7-32. MR 85k:11041

2.
KEVIN S. MCCURLEY, ``Explicit estimates for $\theta(x;3,l)$ and $\psi(x,3,l)$'', 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 $\theta(x)$ and $\psi(x)$'', Math. Comp., 29, Number 129 (January 1975) pp. 243-269. MR 56:15581a

7.
LOWELL SCHOENFELD, ``Sharper bounds for the Chebyshev functions $\theta(x)$ and $\psi(x)$. 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

Similar Articles:

Retrieve articles in Mathematics of Computation with MSC (2000): 11N13, 11N56, 11Y35, 11Y40

Retrieve articles in all Journals with MSC (2000): 11N13, 11N56, 11Y35, 11Y40

Additional Information:

Pierre Dusart
Affiliation: Département de Math., LACO, 123 avenue Albert Thomas, 87060 Limoges cedex, France
Email: dusart@unilim.fr

DOI: 10.1090/S0025-5718-01-01351-5
PII: S 0025-5718(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
Posted: November 21, 2001
Copyright of article: Copyright 2001, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google