On primitive Dirichlet characters and the Riemann hypothesis

For every positive integer n, let be the set of primitive Dirichlet characters modulo n. We show that if the Riemann hypothesis is true, then the inequality holds for all k⩾1, where nk is the product of the first k primes, γ is the Euler–Mascheroni constant, C2 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.