Chebyshev's bias for composite numbers with restricted prime divisors

Let $\pi(x;d,a)$ denote the number of primes $p\le x$with $p\equiv a(\operatorname{mod} d)$. Chebyshev's bias is the phenomenon for which ``more often'' $\pi(x;d,n)>\pi(x;d,r)$, than the other way around, where $n$ is a quadratic nonresidue mod $d$ and $r$ is a quadratic residue mod $d$. If $\pi(x;d,n)\ge \pi(x;d,r)$ for every $x$ up to some large number, then one expects that $N(x;d,n)\ge N(x;d,r)$ for every $x$. Here $N(x;d,a)$ denotes the number of integers $n\le x$such that every prime divisor $p$ of $n$ satisfies $p\equiv a(\operatorname{mod}d)$. In this paper we develop some tools to deal with this type of problem and apply them to show that, for example, $N(x;4,3)\ge N(x;4,1)$ for every $x$.

In the process we express the so-called second order Landau-Ramanujan constant as an infinite series and show that the same type of formula holds for a much larger class of constants.