Prime zeta function

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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 :


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 s where ns is a pole (only ns = 1 when n is a squarefree number greater than or equal to 1), or zero of the Riemann zeta function ζ(.). The line is a natural boundary as the singularities cluster near all points of this line.

If one defines a sequence


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

The prime zeta function is related to Artin's constant by

where Ln is the nth Lucas number.[1]

Specific values are:

s approximate value P(s) OEIS
1 [2]
2 OEISA085548
3 OEISA085541
4 OEISA085964
5 OEISA085965
9 OEISA085969



The integral over the prime zeta function is usually anchored at infinity, because the pole at prohibits defining 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:

s approximate value OEIS
1 OEISA137245
2 OEISA221711


The first derivative is

The interesting values are again those where the sums converge slowly:

s approximate value OEIS
2 OEISA136271
3 OEISA303493
4 OEISA303494
5 OEISA303495


Almost-prime zeta functions[edit]

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 are a product of not necessarily distinct primes) define a sort of intermediate sums:

where is the total number of prime factors.

k s approximate value OEIS
2 2 OEISA117543
2 3
3 2 OEISA131653
3 3

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[edit]

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.


  1. ^ Weisstein, Eric W. "Artin's Constant". MathWorld.
  2. ^ See divergence of the sum of the reciprocals of the primes.
  • 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.
  • Li, Ji (2008). "Prime graphs and exponential composition of species". J. Comb. 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.

External links[edit]