Chebyshev bias is the observed tendency for prime number counts in certain residue classes to lead the counts in other residue classes. For example, the number of primes with remainder 3 upon division by 4 often exceeds the number with remainder 1 (left figure). Similarly, the count with remainder 2 upon division by 3 often exceeds the count with remainder 1 (right figure). Such comparisons are called prime races (Wagon 1994).
Consider the list of the first 
primes 
(mod 4). This list contains equal numbers
of remainders 3 and 1 (mod 4) for 
, 3, 7, 13, 89, 2943, 2945, 2947, 2949, 2951, 2953, 50371,
... (OEIS A038691; Wagon 1994, pp. 2-3).
The values of 
for which the list is biased towards 1 are 2946, 50378, 50380, 50382, 50383, 50384,
50385, ... (OEIS A096628).
Defining
 |
where 
is the modular prime counting function
counting primes at most 
with remainder 
upon division by 
, the values of 
for which 
are 
, 3, 7, 13, 89, 2943, 2945, 2947, ... (OEIS A038691).
For 
,
consider the reduced list 
(mod 3), omitting 
by convention and 
because its remainder is
0. This list contains equal numbers of remainders 2
and 1 at the values 
,
6, 8, 12, 14, 22, 38, 48, 50, ... (OEIS A096629).
These ties correspond to 
, since the omitted prime
number 2 contributes to 
.
For the full list 
(mod 3), the first index with more remainders 1 than
2 is 
.
The first few such indices are 23338590792, 23338590794, 23338590795, 23338590796,
... (OEIS A096630). Bays and Hudson (1978,
p. 571) found the corresponding first prime number 
on Dec. 25, 1976.
The consecutive prime bias concerns a different statistic: ordered pairs of residue classes of successive primes, rather than the counts of individual primes in each residue class (Lemke Oliver and Soundararajan 2016, pp. E4446-E4447).
with 
."
Math. Comp. 32, 571-576, 1978.