Chebyshev’s bias for composite numbers with restricted prime divisors

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Abstract
Let π ( x ; d , a ) \pi (x;d,a) denote the number of primes p ≤ x p\le x with p ≡ a ( mod ⁡ d ) p\equiv a(\operatorname {mod} d) . Chebyshev’s bias is the phenomenon for which “more often” π ( x ; d , n ) > π ( x ; d , r ) \pi (x;d,n)>\pi (x;d,r) , than the other way around, where n n is a quadratic nonresidue mod d d and r r is a quadratic residue mod d d . If π ( x ; d , n ) ≥ π ( x ; d , r ) \pi (x;d,n)\ge \pi (x;d,r) for every x x up to some large number, then one expects that N ( x ; d , n ) ≥ N ( x ; d , r ) N(x;d,n)\ge N(x;d,r) for every x x . Here N ( x ; d , a ) N(x;d,a) denotes the number of integers n ≤ x n\le x such that every prime divisor p p of n n satisfies p ≡ a ( m o d d ) p\equiv a(\mathrm {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 ) ≥ N ( x ; 4 , 1 ) N(x;4,3)\ge N(x;4,1) for every x 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.

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