Chebyshev observed in a letter to Fuss that there tends to be more primes of the form
than of
the form
.
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
. It is
known that this phenomenon has a strong relation with the low-lying zeros of the associated
-functions, that is,
if these
-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
versus
race, since the Riemann
zeta function is the
-function
of rank one having the highest first zero. This race has density
, 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
. An example of a
race whose density exceeds the above number is the race between quadratic residues and nonresidues
modulo
, for which
the density is
.
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.
Keywords
prime number races, primes in arithmetic progressions
