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 :

.

Properties[edit]

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)), 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 we define a sequence

then

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

The prime zeta function is related with the Artin's constant by

where Ln is the nth Lucas number.[1]

Specific values are:

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

Analysis[edit]

Integral[edit]

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:

s approximate value OEIS
1 OEISA137245
2 OEISA221711
3
4

Derivative[edit]

The first derivative is

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

s approximate value OEIS
2 OEISA136271
3
4
5

Generalizations[edit]

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 a 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.

References[edit]

  • 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:0811.4739free to read. 
  • 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:1008.2547free to read. 

External links[edit]