Highly biased prime number races

Abstract

Chebyshev observed in a letter to Fuss that there tends to be more primes of the form $4n+3$ than of the form $4n+1$. The general phenomenon, which is referred to as Chebyshev’s bias, is that primes tend to be biased in their distribution among the different residue classes $modq$. It is known that this phenomenon has a strong relation with the low-lying zeros of the associated $L$-functions, that is, if these $‘L‘$-functions have zeros close to the real line, then it will result in a lower bias. According to this principle one might believe that the most biased prime number race we will ever find is the $Li\left(x\right)$ versus $\pi \left(x\right)$ race, since the Riemann zeta function is the $‘L‘$-function of rank one having the highest first zero. This race has density $0.99999973\dots$, and we study the question of whether this is the highest possible density. We will show that it is not the case; in fact, there exist prime number races whose density can be arbitrarily close to $1$. An example of a race whose density exceeds the above number is the race between quadratic residues and nonresidues modulo $4849845$, for which the density is $0.999999928\dots$. We also give fairly general criteria to decide whether a prime number race is highly biased or not. Our main result depends on the generalized Riemann hypothesis and a hypothesis on the multiplicity of the zeros of a certain Dedekind zeta function. We also derive more precise results under a linear independence hypothesis.