Prime zeta function

In mathematics, the Prime zeta function is an analogue of the Riemann zeta function, studied by Glaisher (1891). It is defined as the following infinite series, which converges for $\Re (s)>1$:

$P(s)=\sum _{p\,\in \mathrm {\,primes} }{\frac {1}{p^{s}}}$.

The Euler product for the Riemann zeta function ζ(s) implies that

$\log \zeta (s)=\sum _{n>0}{\frac {P(ns)}{n}}$

which by Möbius inversion gives

$P(s)=\sum _{n>0}\mu (n){\frac {\log \zeta (ns)}{n}}$

When s goes to 1, we have $P(s)\sim \log \zeta (s)\sim \log \left({\frac {1}{s-1}}\right)$. This is used in the definition of Dirichlet density.

This gives the continuation of P(s) to $\Re (s)>0$, with an infinite number of logarithmic singularities at points where ns is a pole or zero of ζ(s). The line $\Re (s)=0$ is a natural boundary as the singularities cluster near all points of this line.

If we define a sequence

$a_{n}=\prod _{p^{k}\mid n}{\frac {1}{k}}=\prod _{p^{k}\mid \mid n}{\frac {1}{k!}}$

then

$P(s)=\log \sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}.$

(Exponentiation shows that this is equivalent to Lemma 2.7 by Li.)

Specific values

$P(1)={\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{5}}+{\tfrac {1}{7}}+{\tfrac {1}{11}}+\cdots =\infty .$

$P(2)=0{,}45224{\text{ }}74200{\text{ }}41065{\text{ }}49850\ldots$ (sequence A085548 in the OEIS)

$P(3)=0{,}17476{\text{ }}26392{\text{ }}99443{\text{ }}53642\ldots$ (sequence A085541 in the OEIS)

$P(4)=0{,}07699{\text{ }}31397{\text{ }}64246{\text{ }}84494\ldots$ (sequence A085964 in the OEIS)

References

• Merrifield, C. W. (1881). "The Sums of the Series of Reciprocals of the Prime Numbers and of Their Powers". Proc. Roy. Soc. London. 33: 4–10. doi:10.1098/rspl.1881.0063. JSTOR 113877.
• Fröberg, Carl-Erik (1968). "On the prime zeta function". Nordisk Tidskr. Informationsbehandling (BIT). 8 (3): 187–202. doi:10.1007/BF01933420. MR 0236123.
• Glaisher, J. W. L. (1891). "On the Sums of Inverse Powers of the Prime Numbers". Quart. J. Math. 25: 347–362.
• Mathar, Richard J. (2008). "Twenty digits of some integrals of the prime zeta function". arXiv:0811.4739. Bibcode:2008arXiv0811.4739M. {{cite arXiv}}: Unknown parameter |bibcode= ignored (help)
• Li, Ji (2008). "Prime graphs and exponential composition of species". J. Combin. Theory A. 115: 1374–1401. doi:10.1016/j.jcta.2008.02.008. MR 2455584.
• Mathar, Richard J. (2010). "Table of Dirichlet L-series and prime zeta modulo functions for small moduli". arXiv:1008.2547. Bibcode:2010arXiv1008.2547M. {{cite arXiv}}: Unknown parameter |bibcode= ignored (help)

External links

• Prime Zeta Function, in Wolfram Mathworld