Von Mangoldt function

Definition

The von Mangoldt function, denoted by Λ(n), is defined as

Λ ( n ) = { log ⁡ p if  n = p k  for some prime  p  and integer  k ≥ 1 , 0 otherwise.

The values of Λ(n) for the first nine positive numbers are

0 , log ⁡ 2 , log ⁡ 3 , log ⁡ 2 , log ⁡ 5 , 0 , log ⁡ 7 , log ⁡ 2 , log ⁡ 3 ,

which is related to (sequence A014963 in the OEIS).

The summatory von Mangoldt function, ψ(x), also known as the Chebyshev function, is defined as

ψ ( x ) = ∑ n ≤ x Λ ( n ) .

Von Mangoldt provided a rigorous proof of an explicit formula for ψ(x) involving a sum over the non-trivial zeros of the Riemann zeta function. This was an important part of the first proof of the prime number theorem.

Properties

The von Mangoldt function satisfies the identity

log ⁡ ( n ) = ∑ d ∣ n Λ ( d ) .

The sum is taken over all integers d that divide n. This is proved by the fundamental theorem of arithmetic, since the terms that are not powers of primes are equal to 0. For example, consider the case n = 12 = 2² × 3. Then

∑ d ∣ 12 Λ ( d ) = Λ ( 1 ) + Λ ( 2 ) + Λ ( 3 ) + Λ ( 4 ) + Λ ( 6 ) + Λ ( 12 ) = Λ ( 1 ) + Λ ( 2 ) + Λ ( 3 ) + Λ ( 2 2 ) + Λ ( 2 × 3 ) + Λ ( 2 2 × 3 ) = 0 + log ⁡ ( 2 ) + log ⁡ ( 3 ) + log ⁡ ( 2 ) + 0 + 0 = log ⁡ ( 2 × 3 × 2 ) = log ⁡ ( 12 ) .

By Möbius inversion, we have

Λ ( n ) = − ∑ d ∣ n μ ( d ) log ⁡ ( d )   .

Dirichlet series

The von Mangoldt function plays an important role in the theory of Dirichlet series, and in particular, the Riemann zeta function. In particular, one has

log ⁡ ζ ( s ) = ∑ n = 2 ∞ Λ ( n ) log ⁡ ( n ) 1 n s , Re ( s ) > 1.

The logarithmic derivative is then

ζ ′ ( s ) ζ ( s ) = − ∑ n = 1 ∞ Λ ( n ) n s .

These are special cases of a more general relation on Dirichlet series. If one has

F ( s ) = ∑ n = 1 ∞ f ( n ) n s

for a completely multiplicative function  f (n), and the series converges for Re(s) > σ₀, then

F ′ ( s ) F ( s ) = − ∑ n = 1 ∞ f ( n ) Λ ( n ) n s

converges for Re(s) > σ₀.

Chebyshev function

The second Chebyshev function ψ(x) is the summatory function of the von Mangoldt function:

ψ ( x ) = ∑ p k ≤ x log ⁡ p = ∑ n ≤ x Λ ( n )   .

The Mellin transform of the Chebyshev function can be found by applying Perron's formula:

ζ ′ ( s ) ζ ( s ) = − s ∫ 1 ∞ ψ ( x ) x s + 1 d x

which holds for Re(s) > 1.

Exponential series

Hardy and Littlewood examined the series

F ( y ) = ∑ n = 2 ∞ ( Λ ( n ) − 1 ) e − n y

in the limit y → 0⁺. Assuming the Riemann hypothesis, they demonstrate that

F ( y ) = O ( 1 y ) and F ( y ) = Ω ± ( 1 y )

In particular this function is oscillatory with diverging oscillations: there exists a value K > 0 such that both inequalities

F ( y ) < − K y ,  and  F ( z ) > K z

hold infinitely often in any neighbourhood of 0. The graphic to the right indicates that this behaviour is not at first numerically obvious: the oscillations are not clearly seen until the series is summed in excess of 100 million terms, and are only readily visible when y < 10⁻⁵.

Riesz mean

The Riesz mean of the von Mangoldt function is given by

∑ n ≤ λ ( 1 − n λ ) δ Λ ( n ) = − 1 2 π i ∫ c − i ∞ c + i ∞ Γ ( 1 + δ ) Γ ( s ) Γ ( 1 + δ + s ) ζ ′ ( s ) ζ ( s ) λ s d s = λ 1 + δ + ∑ ρ Γ ( 1 + δ ) Γ ( ρ ) Γ ( 1 + δ + ρ ) + ∑ n c n λ − n .

Here, λ and δ are numbers characterizing the Riesz mean. One must take c > 1. The sum over ρ is the sum over the zeroes of the Riemann zeta function, and

∑ n c n λ − n

can be shown to be a convergent series for λ > 1.

Approximation by Riemann zeta zeros

The real part of the sum over the zeta zeros:

− ∑ i = 1 ∞ n ρ ( i ) , where ρ(i) is the i-th zeta zero, peaks at primes, as can be seen in the adjoining graph, and can also be verified through numerical computation. It does not sum up to the Von Mangoldt function.

The Fourier transform of the von Mangoldt function gives a spectrum with spikes at ordinates equal to imaginary part of the Riemann zeta function zeros. This is sometimes called a duality.