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Mathematics of Computation

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

Author: Pierre Dusart
Journal: Math. Comp. 71 (2002), 1137-1168
MSC (2000): Primary 11N13, 11N56; Secondary 11Y35, 11Y40
Published electronically: November 21, 2001
MathSciNet review: 1898748
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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}

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Additional Information

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

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