Year: 2023
Type: article
Abstract: Assuming the Riemann hypothesis, we prove the latest explicit version of the prime number theorem for short intervals. Using this result, and assuming the generalised Riemann hypothesis for Dirichlet L-functions is true, we then establish explicit formulae for $\psi(x,\chi)$, $\theta(x,\chi)$ and an explicit version of the prime number theorem for primes in arithmetic progressions that hold for general moduli $q\geq 3$. Finally, we restrict our attention to $q\leq 10\,000$ and use an exact computation to refine these results.
Author: Ethan Simpson Lee
Institution: University of Bristol
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,867
OpenAlex ID: W4385719707