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 (N) \,

, is defined by

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

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

The function of von Mangoldt satisfies the identity

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

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

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

von Mangoldt provided a rigorous proof of a Formule clarifies for $\ psi (X) \,$ , 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

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

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

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

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

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

and showed that

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

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

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

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 $y<10^ {- 5}$ .

The report/ratio of Riesz

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

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

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

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

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

\ 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 $\ sum_n c_n \ lambda^ {- N} \,$ can be shown like a convergent series for $\ lambda > 1 \,$ .

See too

Function of account of the prime numbers

References

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