Von Mangoldt function

In mathematics, the Mangoldt function, named after the German mathematician Hans von Mangoldt, a number theoretic function which is commonly referred to.

Definitions and basic properties

The Czech Mangold function is defined as

It is neither an additive function nor multiplicative function.

Exp ( Λ ( n ) )

Exp ( Λ ( n ) ) can be specified explicitly as

Being the least common multiples referred.

The first value of the sequence exp ( Λ (n)) are

Summed Mangoldt function

The summed Mangoldt function,

Is also referred to as Chebyshev function. It plays a role in the proof of the prime number theorem.

Divisor sums

The Möbius function called.

Dirichlet series

The Mangoldt function plays an important role in the theory of Dirichlet series.

It is

The logarithmic derivative which provides a connection between the Riemann function and the Mangoldt function:

More generally, even: Is multiplicative and their Dirichlet series

Converges for certain, then applies

Credentials

• Eric W. Weisstein: Mangoldt Function. In: MathWorld (English).
• SpringerLink

• Number theoretic function