Mangoldt Function

The Mangoldt function is the function defined by

(1)

sometimes also called the lambda function. has the explicit representation

(2)

where denotes the least common multiple. The first few values of for , 2, ..., plotted above, are 1, 2, 3, 2, 5, 1, 7, 2, ... (OEIS A014963).

The Mangoldt function is implemented in the Wolfram Language as MangoldtLambda[n].

It satisfies the divisor sums

			(3)
			(4)
			(5)
			(6)

where is the Möbius function (Hardy and Wright 1979, p. 254).

The Mangoldt function is related to the Riemann zeta function by

(7)

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

(10)

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

(11)

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

(13)

The Riemann hypothesis is equivalent to

(14)

(Davenport 1980, p. 114; Vardi 1991).

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

(15)

where is the floor function and is a factorial.