Estimates of $\theta (x;k,l)$ for large values of $x$
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- by Pierre Dusart;
- Math. Comp. 71 (2002), 1137-1168
- DOI: https://doi.org/10.1090/S0025-5718-01-01351-5
- Published electronically: November 21, 2001
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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 $x\geqslant 1531$, \[ \mid \theta (x;3,l)-x/2\mid <0.262\frac {x}{\ln x}\] and, for $x\geqslant 151$, \[ \pi (x;3,l)>\frac {x}{2\ln x}.\]References
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Bibliographic Information
- Pierre Dusart
- Affiliation: Département de Math., LACO, 123 avenue Albert Thomas, 87060 Limoges cedex, France
- Email: dusart@unilim.fr
- 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
- © Copyright 2001 American Mathematical Society
- 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
- MathSciNet review: 1898748