Explicit formulae for primes in arithmetic progressions, I

Abstract

We shall give an explicit formula for $\psi(x, q, a)$ with an error term of the form $C/\log^\alpha x$ under the condition that $q<\log^{\alpha_1} x$ is nonexceptional, for various values of $\alpha$ and $\alpha_1$. We shall also give an explicit formula for $\psi(x, q, a)$ with error terms $C/\log^A x$ working whether $q$ is exceptional or nonexceptional, but under the condition that $\frac{0.4923A}{\pi}q^{1/2}\log^2 q<\log x/\log\log x$. Moreover, we shall give an explicit form of Bombieri-Vinogradov theorem over non-exceptional moduli.