Chebyshev Functions

The two functions and defined below are known as the Chebyshev functions.

The function is defined by

			(1)
			(2)
			(3)

(Hardy and Wright 1979, p. 340), where is the th prime, is the prime counting function, and is the primorial. This function has the limit

(4)

and the asymptotic behavior

(5)

(Bach and Shallit 1996; Hardy 1999, p. 28; Havil 2003, p. 184). The notation is also commonly used for this function (Hardy 1999, p. 27).

The related function is defined by

			(6)
			(7)

where is the Mangoldt function (Hardy and Wright 1979, p. 340; Edwards 2001, p. 51). Here, the sum runs over all primes and positive integers such that , and therefore potentially includes some primes multiple times. A simple and beautiful formula for is given by

(8)

i.e., the logarithm of the least common multiple of the numbers from 1 to (correcting Havil 2003, p. 184). The values of for , 2, ... are 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, ... (OEIS A003418; Selmer 1976). For example,

(9)

The function also has asymptotic behavior

(10)

(Hardy 1999, p. 27; Havil 2003, p. 184).

The two functions are related by

(11)

(Havil 2003, p. 184).

Chebyshev showed that , , and (Ingham 1995; Havil 2003, pp. 184-185).

According to Hardy (1999, p. 27), the functions and are in some ways more natural than the prime counting function since they deal with multiplication of primes instead of the counting of them.