Prime zeta function
The Euler product for the Riemann zeta function ζ(s) implies that
which by Möbius inversion gives
When s goes to 1, we have . This is used in the definition of Dirichlet density.
This gives the continuation of P(s) to , with an infinite number of logarithmic singularities at points where ns is a pole or zero of ζ(s). The line is a natural boundary as the singularities cluster near all points of this line.
If we define a sequence
(Exponentiation shows that this is equivalent to Lemma 2.7 by Li.)
The prime zeta function is related with the Artin's constant by
Specific values are:
|s||approximate value P(s)||OEIS|
The integral over the prime zeta function is usually anchored at infinity, because the pole at prohibits to define a nice lower bound at some finite integer without entering a discussion on branch cuts in the complex plane:
The noteworthy values are again those where the sums converge slowly:
The first derivative is
The interesting values are again those where the sums converge slowly:
Almost-prime Zeta Functions
As the Riemann Zeta Function is a sum of inverse powers over the integers and the Prime Zeta Function a sum of inverse powers of the prime numbers, the k-primes (the integers which a are a product of not necessarily distinct primes) define a sort of intermediate sums:
where is the total number of prime factors.
Each integer in the denominator of the Riemann Zeta Function may be classified by its value of the index , which decomposes the Riemann Zeta Function into an infinite sum of the :
Prime Modulo Zeta Functions
Constructing the sum not over all primes but only over primes which are in the same modulo class introduces further types of infinite series that are a reduction of the Dirichlet L-function.
- Merrifield, C. W. (1881). "The Sums of the Series of Reciprocals of the Prime Numbers and of Their Powers". Proceedings of the Royal Society 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.
- 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.