## Abstract

We denote by

*ψ*(*x; q, a*) the sum of Λ(*n*)/*n*for all*n≤x*and congruent to*a*mod*q*and similarly by*ψ*(*x; q, a*) the sum of Λ(*n*) over the same set. We show that the error term in*ψ*(*x; q, a*) − (log*x*)/*ϕ*(*q*) −*C*(*q, a*) for a suitable constant*C*(*q, a*) can be controlled by that of*ψ*(*y; q, a*)−*y*/*ϕ*(*q*) for*y*of size*x*, up to a small error term. As a consequence, if a partial generalized Riemann hypothesis has been verified for the*L*-functions attached to characters modulo*q*up to height*H*, this error term is bounded by*O*(*e*^{−H/8}) when*x*≥*H*. Previous methods had at best*O*(1/*H*) instead. We further compute the asymptotics for the*L*-average of a quantity closely related to^{2}*C*(*q*,*a*).Original language | English |
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Pages (from-to) | 77-92 |

Number of pages | 15 |

Journal | Experimental Mathematics |

Volume | 26 |

Issue number | 1 |

Early online date | 8 Jul 2016 |

DOIs | |

Publication status | Published - 18 Jan 2017 |

## Keywords

- prime numbers in arithmetic progressions
- explicit formulae
- explicit estimates

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