The Bombieri-Vinogradov theorem with explicit constants

The Bombiery-Vinogradov theorem is an important result concerning the distribution of prime numbers
in arithmetic progressions, averaged over a range of moduli. 
Generally speaking, it addresses the error term in the Prime Number Theorem for arithmetic progressions,
averaged over the moduli q up to Q. It is sometimes called Generalized Riemann Hypothesis on average,
since without averaging the similar result would be about of the strength of GRH.
We improve by a log x factor the strongest effective result to date (due to Akbary and Hambrook). The
key ingredient is an effective version of Vaughan's identity and a Polya-Vinogradov inequality. In order
to reduce the logarithmic factor in Vaughan's inequality (and hence in the Bombieri-Vinogradov theorem)
we apply an explicit variant of an identity connected to the Mobius function, which was derived by
Helfgott in his work on the ternary Goldbach problem. We will also discuss why further reductions by
log factors are more challenging to achieve.