prime zeta function

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

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

which by Möbius inversion gives

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

When s goes to 1, we have $P(s)\backslash sim\; \backslash log\backslash zeta(s)\backslash sim\backslash log\backslash left(\backslash frac\{1\}\{s-1\}\; \backslash right)$.

This is used in the definition of Dirichlet density.

This gives the continuation of P(s) to $\backslash Re(s)\; >\; 0$, 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 $\backslash Re(s)\; =\; 0$ is a natural boundary as the singularities cluster near all points of this line.

If one defines a sequence

:$a\_n=\backslash prod\_\{p^k\; \backslash mid\; n\}\; \backslash frac\{1\}\{k\}=\backslash prod\_\{p^k\; \backslash mid\; \backslash mid\; n\}\; \backslash frac\; 1\; \{k!\}$

then

:$P(s)=\backslash log\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{a\_n\}\{n^s\}.$

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

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

: $\backslash ln\; C\_\{\backslash mathrm\{Artin\}\}\; =\; -\; \backslash sum\_\{n=2\}^\{\backslash infty\}\; \backslash frac\{(L\_n-1)P(n)\}\{n\}$

where L_n is the nth Lucas number.{{MathWorld|ArtinsConstant|Artin's Constant}}

Specific values are:

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:

:$\backslash int\_s^\backslash infty\; P(t)\; \backslash ,\; dt\; =\; \backslash sum\_p\; \backslash frac\; 1\; \{p^s\backslash log\; p\}$

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

The first derivative is

: $P\text{'}(s)\; \backslash equiv\; \backslash frac\{d\}\{ds\}\; P(s)\; =\; -\; \backslash sum\_p\; \backslash frac\{\backslash log\; p\}\{p^s\}$

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

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:

: $P\_k(s)\backslash equiv\; \backslash sum\_\{n:\; \backslash Omega(n)=k\}\; \backslash frac\; 1\; \{n^s\}$

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

Each integer in the denominator of the Riemann zeta function $\backslash zeta$

may be classified by its value of the index $k$, which decomposes the Riemann zeta

function into an infinite sum of the $P\_k$:

:$\backslash zeta(s)\; =\; 1+\backslash sum\_\{k=1,2,\backslash ldots\}\; P\_k(s)$

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

:$P\_\{\backslash Omega\}(u,\; s)\; :=\; \backslash sum\_\{n\; \backslash geq\; 1\}\; \backslash frac\{u^\{\backslash Omega(n)\}\}\{n^s\}\; =\; \backslash prod\_\{p\; \backslash in\; \backslash mathbb\{P\}\}\; \backslash left(1-up^\{-s\}\backslash right)^\{-1\},$

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 $P\_k(s)\; =\; [u^k]\; P\_\{\backslash Omega\}(u,\; s)\; =\; h(x\_1,\; x\_2,\; x\_3,\; \backslash ldots)$ when the sequences correspond to $x\_j\; :=\; j^\{-s\}\; \backslash chi\_\{\backslash mathbb\{P\}\}(j)$ where $\backslash chi\_\{\backslash mathbb\{P\}\}$ denotes the characteristic function of the primes. Using Newton's identities, we have a general formula for these sums given by

:$P\_n(s)\; =\; \backslash sum\_\{\{k\_1+2k\_2+\backslash cdots+nk\_n=n\}\; \backslash atop\; \{k\_1,\backslash ldots,k\_n\; \backslash geq\; 0\}\}\; \backslash left[\backslash prod\_\{i=1\}^n\; \backslash frac\{P(is)^\{k\_i\}\}\{k\_i!\; \backslash cdot\; i^\{k\_i\}\}\backslash right]\; =\; -[z^n]\backslash log\backslash left(1\; -\; \backslash sum\_\{j\; \backslash geq\; 1\}\; \backslash frac\{P(js)\; z^j\}\{j\}\backslash right).$

Special cases include the following explicit expansions:

:

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.

• Divergence of the sum of the reciprocals of the primes

{{reflist}}

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• {{MathWorld|title=Prime Zeta Function|id=PrimeZetaFunction}}

Category:Zeta and L-functions