# Function of von Mangoldt

In mathematics, the function of von Mangoldt is a arithmetic Fonction named in the honor of the German Mathématicien Hans von Mangoldt.

## Definition

The function of von Mangoldt, written in a conventional way $\ Lambda \left(N\right) \,$, is defined by

$\ Lambda \left(N\right) = \ begin \left\{boxes\right\} \ ln p & \ mbox \left\{if\right\} n=p^k \ mbox \left\{for certain prime numbers\right\} p \ mbox \left\{and an entirety\right\} K \ Ge 1, \ \ 0 & \ mbox \left\{differently.\right\} \ end \left\{boxes\right\}$

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

The function of von Mangoldt satisfies the identity

$\ ln N = \ sum_ \left\{D \, \ mid \, N\right\} \ Lambda \left(d\right), \,$

i.e., the sum is taken on all the Entier S D which divides N . The function sommatoire of von Mangoldt , $\ psi \, \left(X\right)$, so known like the function of Tchebychev, is defined like

$\ psi \left(X\right) = \ sum_ \left\{N \ X\right\} \ Lambda \left(N\right).$

von Mangoldt provided a rigorous proof of a Formule clarifies for $\ psi \left(X\right) \,$, implying a sum on the zero not-commonplace ones of the function zeta of Riemann . It was an important part of the first demonstration of the Théorème of the prime numbers.

## Series of Dirichlet

The function of von Mangoldt plays a big role in the theory of the series of Dirichlet, as well as the Fonction zeta of Riemann. In particular, one has

$\ log \ zeta \left(S\right) = \ sum_ \left\{n=2\right\} ^ \ infty \ frac \left\{\ Lambda \left(N\right)\right\}\left\{\ log \left(N\right)\right\}\, \ frac \left\{1\right\} \left\{n^s\right\}$

for $\ Re \left(S\right) > 1$. The Dérivée logarithmic curve is then

$\ frac \left\{\ zeta^ \ premium \left(S\right)\right\}\left\{\ zeta \left(S\right)\right\} = - \ sum_ \left\{n=1\right\} ^ \ infty \ frac \left\{\ Lambda \left(N\right)\right\}\left\{n^s\right\}.$

Those are particular cases of a more general relation of series of Dirichlet

$F \left(there\right) = \ sum_ \left\{n=2\right\} ^ \ infty \ left \left(\ Lambda \left(N\right) - 1 \ right\right) e^ \left\{- ny\right\}$

and showed that

$F \left(there\right) = \ mathcal \left\{O\right\} \ left \left(\ sqrt \left\{\ frac \left\{1\right\} \left\{there\right\}\right\} \ right\right).$

Curiously, they as showed as this function is oscillatory, with divergent oscillations. In particular, there exists a value $K>0$ such as

$F \left(there\right) < - \ frac \left\{K\right\} \left\{\ sqrt \left\{there\right\}\right\}$ and $F \left(there\right) > \ frac \left\{K\right\} \left\{\ sqrt \left\{there\right\}\right\}$

infinitely often. The graph on the line indicates that this behavior is not obvious on the first numbers: the oscillations are not seen clearly until the series is summoned by excess up to 100 million terms, and are only visible when $y<10^ \left\{- 5\right\}$.

## The report/ratio of Riesz

The Rapport of Riesz of the function of von Mangoldt is given by

$\ sum_ \left\{N \ the \ lambda\right\} \ left \left(1 \ frac \left\{N\right\} \left\{\ lambda\right\} \ right\right) ^ \ delta \ Lambda \left(N\right)$

$= - \ frac \left\{1\right\} \left\{2 \ pi I\right\} \ int_ \left\{Ci \ infty\right\} ^ \left\{c+i \ infty\right\}$

\ frac {\ Gamma (1+ \ delta) \ Gamma (S)}{\ Gamma (1+ \ delta+s)} \ frac {\ zeta^ \ premium (S)}{\ zeta (S)} \ lambda^s ds

$= \ frac \left\{\ lambda\right\} \left\{1+ \ delta\right\} +$

\ sum_ \ rho \ frac {\ Gamma (1+ \ delta) \ Gamma (\ rho)}{\ Gamma (1+ \ delta+ \ rho)} + \ sum_n c_n \ lambda^ {- N}.

Here, $\ lambda \,$ and $\ delta \,$ is numbers characterizing the report/ratio of Riesz. One must take $c>1 \,$. The sum on $\ rho \,$ is the sum on the zeros of the function zeta of Riemann, and $\ sum_n c_n \ lambda^ \left\{- N\right\} \,$ can be shown like a convergent series for $\ lambda > 1 \,$.

## References

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