Explicit improvements of the Brun-Titchmarsh theorem for arbitrary intervals

We give an explicit version of Brun-Titchmarsh theorem applicable for arbitrary moduli and arbitrary intervals. For example, we show that $\pi (x+y;k,a)-\pi (x;k,a)<2y\mathrm{/}(\phi (k)(\log (y\mathrm{/}k)+0.8601))$ for any relatively prime positive integers $k$ and $a$ and any positive real numbers $x$ and $y$ such that $y>k$.