Chebyshev Functions

The Chebyshev functions are the two functions and defined below for real number arguments .

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, the product of the primes at most . Empty sums and products have values 0 and 1, respectively. This function has the limit

(4)

and the asymptotic behavior

(5)

as through the positive integers (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. Both functions are right-continuous step functions and vanish for . At a prime , has a jump of ; at each prime power , has a jump of . The displayed sums include the full jump at their upper endpoint. A simple and beautiful formula for , for , is given by

(8)

i.e., the natural logarithm of the least common multiple of the integers from 1 to , where is the floor function (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).

For , the two functions are related by

(11)

(Havil 2003, p. 184).

The assertions , , and are equivalent forms of the prime number theorem (Ingham 1995; NIST DLMF, §25.16(i)). Chebyshev (1852, pp. 377-379) obtained two-sided bounds for and .

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.