On primitive Dirichlet characters and the Riemann hypothesis

Abstract

For any natural number $n$, let $X_n^{\prime }$ be the set of primitive Dirichlet characters modulo $n$. We show that if the Riemann hypothesis is true, then the inequality $|X_{2n_k}^{\prime }|\le C_2{e}^{-\gamma }\varphi (2n_k)/\log \log (2n_k)$ holds for all $k\ge 1$, where $n_k$ is the product of the first $k$ primes, $\gamma$ is the Euler-Mascheroni constant, $C_2$ is the twin prime constant, and $\varphi (n)$ 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.

7 pages