Von Mangoldt function

In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt.

Definition

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

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

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

The von Mangoldt function satisfies the identity^[1]

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

that is, the sum is taken over all integers d which divide n. This is proved by the fundamental theorem of arithmetic, since the terms that are not powers of primes are equal to 0.

For instance, let n=12. Recall the prime factorization of 12, 12=2²·3, which will turn up in the example.

Take the summation over all distinct positive divisors d of n:

$\sum _{d\,\mid \,12}\Lambda (d)=\Lambda (1)+\Lambda (2)+\Lambda (3)+\Lambda (4)+\Lambda (6)+\Lambda (12)$

$=\Lambda (1)+\Lambda (2)+\Lambda (3)+\Lambda (2^{2})+\Lambda (2\times 3)+\Lambda (2^{2}\times 3)$

$=0+\log 2+\log 3+\log 2+0+0\,\!$

$=\log(2\times 3\times 2)=\log 12.\,\!$

This provides an example of how the summation of the von Mangoldt function equals log (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.

Dirichlet series

The von Mangoldt function plays an important role in the theory of Dirichlet series, and in particular, the Riemann zeta function. 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 logarithmic derivative is then

${\frac {\zeta ^{\prime }(s)}{\zeta (s)}}=-\sum _{n=1}^{\infty }{\frac {\Lambda (n)}{n^{s}}}.$

These are special cases of a more general relation on Dirichlet series.^[2] If one has

$F(s)=\sum _{n=1}^{\infty }{\frac {f(n)}{n^{s}}}$

for a completely multiplicative function $f(n)$, and the series converges for $\Re (s)>\sigma _{0}$, then

${\frac {F^{\prime }(s)}{F(s)}}=-\sum _{n=1}^{\infty }{\frac {f(n)\Lambda (n)}{n^{s}}}$

converges for $\Re (s)>\sigma _{0}$.

Mellin transform

The Mellin transform of the Chebyshev function can be found by applying Perron's formula:

${\frac {\zeta ^{\prime }(s)}{\zeta (s)}}=-s\int _{1}^{\infty }{\frac {\psi (x)}{x^{s+1}}}\,dx$

which holds for $\Re (s)>1$.

Exponential series

An exponential series involving the von Mangoldt function, summed up to the first $10^{9}$ terms

Hardy and Littlewood examine the series^[3]

$F(y)=\sum _{n=2}^{\infty }\left(\Lambda (n)-1\right)e^{-ny}$

in the limit $y\to 0^{+}$. Assuming the Riemann hypothesis, they demonstrate that

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

Curiously, they also show that this function is oscillatory as well, with diverging oscillations. In particular, there exists a value $K>0$ such that

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

infinitely often. The graphic to the right indicates that this behaviour is not at first numerically obvious: the oscillations are not clearly seen until the series is summed in excess of 100 million terms, and are only readily visible when $y<10^{-5}$.

Riesz mean

The Riesz mean of the von Mangoldt function is given by

$\sum _{n\leq \lambda }\left(1-{\frac {n}{\lambda }}\right)^{\delta }\Lambda (n)=-{\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }{\frac {\Gamma (1+\delta )\Gamma (s)}{\Gamma (1+\delta +s)}}{\frac {\zeta ^{\prime }(s)}{\zeta (s)}}\lambda ^{s}ds$

$={\frac {\lambda }{1+\delta }}+\sum _{\rho }{\frac {\Gamma (1+\delta )\Gamma (\rho )}{\Gamma (1+\delta +\rho )}}+\sum _{n}c_{n}\lambda ^{-n}.$

Here, $\lambda$ and $\delta$ are numbers characterizing the Riesz mean. One must take $c>1$. The sum over $\rho$ is the sum over the zeroes of the Riemann zeta function, and

$\sum _{n}c_{n}\lambda ^{-n}\,$

can be shown to be a convergent series for $\lambda >1$.

Expansion of terms

The terms of the von Mangoldt function can be expanded into series which have numerators that form a symmetric matrix starting:

$T(n,k)={\begin{bmatrix}1&1&1&1&1&1\\1&-1&1&-1&1&-1\\1&1&-2&1&1&-2\\1&-1&1&-1&1&-1\\1&1&1&1&-4&1\\1&-1&-2&-1&1&2\end{bmatrix}}$

This matrix is defined by the recurrence:

$T(n,1)=1,\;T(1,k)=1,\;n\geq k:-\sum \limits _{i=1}^{k-1}T(n-i,k),\;n<k:-\sum \limits _{i=1}^{n-1}T(k-i,n)$

The von Mangoldt function can then for $n>1$ be calculated as ^[4]:

$\Lambda (n)=\sum \limits _{k=1}^{\infty }{\frac {T(n,k)}{k}}$

See also

• Prime-counting function

Notes

• ^ Allan Gut, Some remarks on the Riemann zeta distribution (2005)
• ^ Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001
• ^ Hardy, G. H.; Littlewood, J. E. (1916). "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes" (PDF). Acta Mathematica. 41: 119–196. doi:10.1007/BF02422942. {{cite journal}}: Unknown parameter |lastauthoramp= ignored (|name-list-style= suggested) (help)
• ^ Mats Granvik, Do these series converge to the Mangoldt function (2011)

References

• S.A. Stepanov (2001) [1994], "Mangoldt function", Encyclopedia of Mathematics, EMS Press