On primitive Dirichlet characters and the Riemann hypothesis

Abstract

For every positive integer n, let X n ′ be the set of primitive Dirichlet characters modulo n. We show that if the Riemann hypothesis is true, then the inequality | X 2 n k ′ | ⩽ C 2 e − γ φ ( 2 n k ) / log log ( 2 n k ) holds for all k ⩾ 1 , where n k is the product of the first k primes, γ is the Euler–Mascheroni constant, C 2 is the twin prime constant, and φ ( n ) is the Euler function. On the other hand, if the Riemann hypothesis is false, then there are infinitely many k for which the same inequality holds and infinitely many k for which it fails to hold.