Mangoldt Summatory Function

The Mangoldt summatory function, illustrated above, is a summatory function defined by

(1)

where is the Mangoldt function. It is also called the second Chebyshev function (Edwards 2001, p. 51).

For not equal to a prime or prime power, it is given by the explicit formula

(2)

where the sum is over the nontrivial zeros of the Riemann zeta function , i.e., those in the critical strip so (Montgomery 2001), and is interpreted as

(3)

Vallée Poussin's version of the prime number theorem states that

(4)

for some (Davenport 1980, Vardi 1991). The prime number theorem is equivalent to the statement

(5)

as (Dusart 1999), or equivalently

(6)

Von Mangoldt rigorously derived the explicit formula in 1895, roughly three decades after the 1859 paper of Riemann that inspired it. The formula subsequently played a central role in proofs of the prime number theorem, whose equivalent formulation above is (Edwards 2001, pp. 49-54).

The Riemann hypothesis is equivalent to the estimate

(7)

(Davenport 1980, p. 114; Vardi 1991).

Vardi (1991, p. 155) also gives the formula

(8)

where is the floor function and is a factorial.