In mathematics , the Mangoldt function , named after the German mathematician
Hans von Mangoldt , is a number-theoretic function that is usually referred to as.
Definitions and basic properties
The Mangoldt function is defined as
It is neither an additive nor a multiplicative function .
exp (Λ (n))
can be specified explicitly as
where denotes the least common multiple .
The first values of the sequence are
- 1, 2, 3, 2, 5, 1, 7, 2, 3, 1, 11, 1, 13, 1, 1, 2, 17, 1, 19, 1, 1, 1, ... (sequence A014963 in OEIS )
Summed up Mangoldt function
The summed Mangoldt function,
is also known as the Chebyshev function . It plays a role in proving the prime number theorem .
Divisional sums
where denotes the Möbius function .
Dirichlet series
The Mangoldt function plays an important role in the theory of the Dirichlet series .
It applies
The logarithmic derivation of this provides a connection between the Riemann function and the Mangoldt function:
More generally, it is even true: is multiplicative and its Dirichlet series
converges for certain , then applies
credentials