# A logarithmic improvement in the Bombieri-Vinogradov theorem

## TL;DR

This paper enhances the Bombieri-Vinogradov theorem by reducing the logarithmic factor from (log x)^(5/2) to (log x)^2 using advanced sieve techniques and Vaughan's identity.

## Contribution

It introduces a weighted Vaughan's identity and leverages sieve estimates to achieve a significant logarithmic improvement in the theorem.

## Key findings

- Reduced the logarithmic factor in the theorem from (log x)^(5/2) to (log x)^2
- Provided both effective and non-effective versions of the improved result
- Applied advanced sieve methods and weighted identities for the enhancement

## Abstract

In this paper we improve the best known to date result of Dress-Iwaniec-Tenenbaum, getting (log x)^2 instead of (log x)^(5/2). We use a weighted form of Vaughan's identity, allowing a smooth truncation inside the procedure, and an estimate due to Barban-Vehov and Graham related to Selberg's sieve. We give effective and non-effective versions of the result.