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

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 $x\geqslant 151$, \[ \pi (x;3,l)>\frac {x}{2\ln x}.\]