Primes in arithmetic progressions

Abstract

Strengthening work of Rosser, Schoenfeld, and McCurley, we establish explicit Chebyshev-type estimates in the prime number theorem for arithmetic progressions, for all moduli <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k less-than-or-equal-to 72"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mn>72</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k \le 72</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and other small moduli.