Von Mangoldt function

The von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt.

The von Mangoldt function, conventionally written as Λ(n), is defined as

$\Lambda (n)={\begin{cases}\ln p&{\mbox{if }}n=p^{k}{\mbox{ for some prime }}p{\mbox{ and integer }}k\geq 1\\0&{\mbox{otherwise.}}\end{cases}}$

It is an example of an important arithmetic function that is neither multiplicative nor additive.

The von Mangoldt function satisfies the identity [T. Apostol, Thrm: 2.10]

$\ln n=\sum _{d\,\mid \,n}\Lambda (d),\,$

that is, the sum is taken over all integers d which divide n. The summatory von Mangoldt function, ψ(x), also known as the Chebyshev function, is defined as

$\psi (x)=\sum _{n\leq x}\Lambda (n).$

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.

The Riemann zeta function may be expressed in terms of the von Mangoldt function by (see Allan Gut reference):

$\zeta (\sigma +it)=\exp \left\{\sum _{n=2}^{\infty }{\frac {\Lambda (n)}{\log(n)}}\,n^{-(\sigma +it)}\right\}$

for $\sigma >1$.

See also

• Prime counting function

References

• Allan Gut, Some remarks on the Riemann zeta distribution
• Tom Apostol, Introduction to analytic number theory, Springer-Verlag, New York, 1976