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Estimates of
Authors:Pierre Dusart.
Affiliation:Département de Math., LACO, 123 avenue Albert Thomas, 87060 Limoges cedex, France
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 $midtheta(x;k,l)-x/varphi(k)mid<xvarepsilon(x)$ and $varepsilon(x)=Oleft(frac{1}{ln^a x}right)quad{(forall a>0)}$, whereas $varepsilon(x)$ is a constant. Now we are able to show that, for $xgeqslant1531$,

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

and, for $xgeqslant151$,

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

Keywords:Bounds for basic functions   arithmetic progression
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