Prime zeta function

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

${\displaystyle \int _{s}^{\infty }P(t)\,dt=\sum _{p}{\frac {1}{p^{s}\log p}}}$

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

s	approximate value ${\displaystyle \sum _{p}1/(p^{s}\log p)}$	OEIS
1	${\displaystyle 1.63661632\ldots }$	OEIS: A137245
2	${\displaystyle 0.50778218\ldots }$	OEIS: A221711
3	${\displaystyle 0.22120334\ldots }$
4	${\displaystyle 0.10266547\ldots }$

The first derivative is

${\displaystyle P'(s)\equiv {\frac {d}{ds}}P(s)=-\sum _{p}{\frac {\log p}{p^{s}}}}$

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

s	approximate value ${\displaystyle P'(s)}$	OEIS
2	${\displaystyle -0.493091109\ldots }$	OEIS: A136271
3	${\displaystyle -0.150757555\ldots }$	OEIS: A303493
4	${\displaystyle -0.060607633\ldots }$	OEIS: A303494
5	${\displaystyle -0.026838601\ldots }$	OEIS: A303495

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 ${\displaystyle k}$ not necessarily distinct primes) define a sort of intermediate sums:

${\displaystyle P_{k}(s)\equiv \sum _{n:\Omega (n)=k}{\frac {1}{n^{s}}}}$

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

k	s	approximate value ${\displaystyle P_{k}(s)}$	OEIS
2	2	${\displaystyle 0.14076043434\ldots }$	OEIS: A117543
2	3	${\displaystyle 0.02380603347\ldots }$
3	2	${\displaystyle 0.03851619298\ldots }$	OEIS: A131653
3	3	${\displaystyle 0.00304936208\ldots }$

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

${\displaystyle \zeta (s)=1+\sum _{k=1,2,\ldots }P_{k}(s)}$

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

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

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 ${\displaystyle P_{k}(s)=[u^{k}]P_{\Omega }(u,s)=h(x_{1},x_{2},x_{3},\ldots )}$ when the sequences correspond to ${\displaystyle x_{j}:=j^{-s}\chi _{\mathbb {P} }(j)}$ where ${\displaystyle \chi _{\mathbb {P} }}$ denotes the characteristic function of the primes. Using Newton's identities, we have a general formula for these sums given by

${\displaystyle P_{n}(s)=\sum _{{k_{1}+2k_{2}+\cdots +nk_{n}=n} \atop {k_{1},\ldots ,k_{n}\geq 0}}\left[\prod _{i=1}^{n}{\frac {P(is)^{k_{i}}}{k_{i}!\cdot i^{k_{i}}}}\right]=-[z^{n}]\log \left(1-\sum _{j\geq 1}{\frac {P(js)z^{j}}{j}}\right).}$

Special cases include the following explicit expansions:

${\displaystyle {\begin{aligned}P_{1}(s)&=P(s)\\P_{2}(s)&={\frac {1}{2}}\left(P(s)^{2}+P(2s)\right)\\P_{3}(s)&={\frac {1}{6}}\left(P(s)^{3}+3P(s)P(2s)+2P(3s)\right)\\P_{4}(s)&={\frac {1}{24}}\left(P(s)^{4}+6P(s)^{2}P(2s)+3P(2s)^{2}+8P(s)P(3s)+6P(4s)\right).\end{aligned}}}$

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.