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.

19 pages, appending an explicit form of BV theorem

References in corpus (1)

• An explicit zero-free region for the Dirichlet L-functions

Cited by in corpus (5)

• Explicit Chen's theorem
• On the divisibility of odd perfect numbers, quasiperfect numbers and amicable numbers by a high power of a prime
• Medium-sized values for the Prime Number Theorem for primes in arithmetic progression
• An analog of perfect numbers involving the unitary totient function
• Explicit formulae for primes in arithmetic progressions, II