On the Shanks-Rényi race problem mod 5 (Q1343635)

If as usual \(\pi (x,q, a)\) denotes the number of primes \(\leq x\) and \(\equiv a\pmod q\), then the race problem asks whether, for any permutation \(a_ 1, a_ 2, \dots, a_{\varphi(q)}\) of reduced residue classes \(\pmod q\), there are infinitely many values of \(m\) for which \[ \pi(m, q, a_ 1)> \pi(m, q, a_ 2)> \dots> \pi(m,q, a_{\varphi (q)}). \] The strong Race Hypothesis asserts that for each permutation, the set of \(m\) for which the above inequalities hold, has positive lower density. The problem can also be formulated replacing \(\pi (x,q, a)\) by \(\psi(x,q, a)\), the usual summatory function over \(\Lambda (n)\). With this replacement, the author proves the strong Race Hypothesis \(\pmod 5\) assuming the Generalized Riemann Hypothesis \(\pmod 5\).