Prime Zeta Function

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

:$P(s)=\sum_ \frac=\frac+\frac+\frac+\frac+\frac+\cdots.$

Properties

The Euler product for the Riemann zeta function ''ζ''(''s'') implies that
: $\log\zeta(s)=\sum_ \frac n$
which by Möbius inversion gives

:$P(s)=\sum_ \mu(n)\frac n$

When ''s'' goes to 1, we have $P(s)\sim \log\zeta(s)\sim\log\left(\frac \right)$.
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 ''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 $\Re(s) = 0$ is a natural boundary as the singularities cluster near all points of this line.

If one defines a sequence
:$a_n=\prod_ \frac=\prod_ \frac 1$

then

:$P(s)=\log\sum_^\infty \frac.$

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

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

: $\ln C_ = - \sum_^ \frac$

where ''L''_''n'' is the ''n''th Lucas number.

Specific values are:

Analysis

Integral

The integral over the prime zeta function is usually anchored at infinity,
because the pole at $s=1$ prohibits defining a nice lower bound
at some finite integer without entering a discussion on branch cuts in the complex plane:

:$\int_s^\infty P(t) \, dt = \sum_p \frac 1$

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

Derivative

The first derivative is

: $P'(s) \equiv \frac P(s) = - \sum_p \frac$

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

Generalizations

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

: $P_k(s)\equiv \sum_ \frac 1$

where $\Omega$ is the total number of prime factors.

Each integer in the denominator of the Riemann zeta function $\zeta$
may be classified by its value of the index $k$, which decomposes the Riemann zeta
function into an infinite sum of the $P_k$:

:$\zeta(s) = 1+\sum_ P_k(s)$

Since we know that the Dirichlet series (in some formal parameter ''u'') satisfies

:$P_(u, s) := \sum_ \frac = \prod_ \left(1-up^\right)^,$

we can use formulas for the symmetric polynomial variants with a generating function of the right-hand-side type. Namely, we have the coefficient-wise identity that P_k(s) = ^kP_(u, s) = h(x_1, x_2, x_3, \ldots) when the sequences correspond to $x_j := j^ \chi_(j)$ where $\chi_$ denotes the characteristic function of the primes. Using Newton's identities, we have a general formula for these sums given by

:P_n(s) = \sum_ \left prod_^n \frac\right= - ^nlog\left(1 - \sum_ \frac\right).

Special cases include the following explicit expansions:

:$\beginP_1(s) & = P(s) \\ P_2(s) & = \frac\left(P(s)^2+P(2s)\right) \\ P_3(s) & = \frac\left(P(s)^3+3P(s)P(2s)+2P(3s)\right) \\ P_4(s) & = \frac\left(P(s)^4+6P(s)^2 P(2s)+3 P(2s)^2+8P(s)P(3s)+6P(4s)\right).\end$

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.

See also

* Divergence of the sum of the reciprocals of the primes

References

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External links

* {{MathWorld, title=Prime Zeta Function, id=PrimeZetaFunction

Zeta and L-functions