where
(Hardy 1999, p. 28; Krantz 1999, p. 161; Edwards 2001, p. 50).

The summatory Mangoldt function, illustrated
above, is defined by

(8)

where
is the Mangoldt function, and is also known as the second Chebyshev
function (Edwards 2001, p. 51). is given by the so-called explicit
formula

(9)

for
and
not a prime or prime power
(Edwards 2001, pp. 49, 51, and 53), and the sum is over
all nontrivial zeros
of the Riemann zeta function , i.e., those in the critical
strip so
(Montgomery 2001), and interpreted as

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

(12)

as
(Dusart 1999).

Von Mangoldt proved his formula 30 years after Riemann's paper, which contained a related formula that inspired von Mangoldt's. Von Mangoldt's formula was then used
to prove the prime number theorem in the equivalent
form

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