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.