Lethbridge Number Theory and Combinatorics Seminar: Forrest Francis

Let $\phi(n)$ be Euler's totient function and let $q$ and $a$ be fixed coprime natural numbers. Denote by $S_{q,a}$ the set of natural numbers whose prime divisors are all congruent to $a$ modulo $q$. We can establish \[\limsup_{n \in S_{q,a}} \frac{n}{\phi(n) (\log(\phi(q)\log{n}))^{1/\phi(q)}}
= \frac{1}{C(q,a)},\]
where $C(q,a)$ is a constant associated with a theorem of Mertens. We may then wish to know whether there are infinitely many $n$ in $S_{q,a}$ for which \[ \frac{n}{\phi(n)(\log\phi(q)\log{n})^{1/\phi(q)}} > \frac{1}{C(q,a)}  \qquad (*)  \] is true. In the case $q=a=1$, Nicolas (1983) established that if the Riemann hypothesis is true, then ($*$) holds for all primorials (products of the form $\prod_{p \leq x} p$), but if the Riemann hypothesis is false then there are infinitely many primorials for which
($*$) is true and infinitely many primorials for which ($*$) is false.

In this talk we will show that, for some $q>1$, the work of Nicolas can be generalized by replacing the Riemann hypothesis with analogous conjectures for Dirichlet $L$-functions and replacing the primorials with products of the form \[\prod_{\substack{p \leq x \\ p \,\equiv \,a \,(\mathrm{mod}\,q)}} p.\]