An effective version of the Bombieri-Vinogradov theorem

In the talk, we deal with a new effective version of the Bombieri-Vinogradov theorem. This theorem improves the previous result belonging to F. Dress, H. Iwaniec and G. Tenenbaum [1]. Namely, we prove the following
Theorem. Suppose that $x\geqslant 4$, $1\leqslant Q_{1}\leqslant Q\leqslant x^{\,1/2}$ and let $l(q)$ denotes the smallest prime divisor of $q$. Then
$$ \sum\limits_{\substack{q\leqslant Q \\ l(q)>Q_{1}}}\max_{2\leqslant y\leqslant x}\max_{(a,q)=1}\biggl|\psi(y;q,a)\,-\,\frac{\psi(y)}{\varphi(q)}\biggr|\,\ll\, \bigl(xQ_{1}^{-1}\,+\,Qx^{\,1/2}\,+\,x^{\,95/96}\log{x}\bigr)(\log{x})^{3}. $$
(Here we get the factor $(\log{x})^{3}$ instead $(\log{x})^{7/2}$ from [1]). In the proof, we use a weighted form of Vaughan’s identity, allowing a smooth truncation inside the procedure and a technique of Graham [2] related to Selberg’s sieve.
[1] F. Dress, H. Iwaniec, G. Tenenbaum, Sur une somme liée à la fonction de Möbius. J. Reine Angew. Math. 340 (1983). P. 53 – 58.
[2] S. Graham, An asymptotic estimate related to Selberg’s sieve. J. Number Theory. 10:1 (1978). P. 83 – 94.