Let 
denote the number of primes 
which are congruent to 
modulo 
(i.e., the modular
prime counting function). Then one might expect that
 |
(Berndt 1994).
Although this is true for small numbers, Hardy and Littlewood showed that 
changes sign infinitely often. The effect was first
noted by Chebyshev in 1853, and is sometimes called the Chebyshev phenomenon. It
was subsequently studied by Shanks (1959), Hudson (1980), and Bays and Hudson (1977,
1978, 1979). The effect was also noted by Ramanujan, who incorrectly claimed that
(Berndt 1994).
The bias of the sign of 
is known as the Chebyshev
bias.
." Math. Comput. 32, 281-286, 1978.Bays,
C. and Hudson, R. H. "Numerical and Graphical Description of All Axis Crossing
Regions for the Moduli 4 and 8 which Occur Before 
." Internat. J. Math. Math. Sci. 2,
111-119, 1979.Berndt, B. C.