PrimeZetaP

The prime zeta function $P(s) = \sum_p p^{-s}$, the sum over primes. Computed via the Möbius/ζ identity $P(s) = \sum_{k \ge 1} \mu(k)/k \cdot \ln\zeta(ks)$ rather than sieving primes directly — the identity converges geometrically, sieving does not.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguagePrimeZetaP✓↗

PrimeZetaP(s)$P(s)$, for $\operatorname{Re}(s) > 0$.

Typed signatures

Type	Package	Applies
(number) -> number	analytic	always

Domain: Special functions

Details

• Reuses this package's own Zeta (already extended to complex arguments) rather than a fresh prime-summation kernel; $\ln\zeta(ks) \to 0$ geometrically as $k$ grows, so the sum settles in a few dozen terms at double precision.
• For $\operatorname{Re}(s) \le 1$ the identity is an analytic continuation, and each $\ln\zeta(x)$, $x = ks$, takes Wolfram's branch: the principal logarithm when $\operatorname{Re}(x) \ge 1/2$, and $x\ln 2 + (x-1)\ln\pi + \operatorname{Log}\sin(\pi x/2) + \operatorname{LogGamma}(1-x) + \operatorname{Log}\zeta(1-x)$ (the functional equation, each piece principal) when $\operatorname{Re}(x) < 1/2$. For real $s$ both are the principal logarithm, so $\operatorname{Im} P(s) = \pi \sum_{ks < 1} \mu(k)/k$. For complex $s$ the second differs from the first by a multiple of $2\pi i$ that grows with $\operatorname{Im}(s)$, and the value is not continuous across the real axis.
• $s = 1/k$ for a squarefree $k$ is ComplexInfinity, as in Wolfram; $1/4$ is finite because $\mu(4) = 0$.
• It declines (stays unevaluated) for $\operatorname{Re}(s) \le 0$, at a zero of $\zeta(ks)$ and where $\zeta(ks)$ lies on the logarithm's cut, for $\operatorname{Re}(s)$ below about $0.005$, and for $|\operatorname{Im}(ks)| > 350$ while $\operatorname{Re}(ks) < 1/2$.
• Checked against wolframscript's N[PrimeZetaP[s], 30] over a grid of real and complex $s$ in the strip, matching to 30 digits on the bignum path (where it answers) and to about $10^{-12}$ at doubles.
• N(x, d) past a double's digits runs the same sum on the BigDecimal $\zeta$ kernel, for real and complex $s$, to $d$ digits; it declines at an approximated pole, and where the terms (about $3.4 d/\operatorname{Re}(s)$) would be too many.

Examples

3 more kept as data, checked against the oracles

Basic #

$\sum_p 1/p^2$

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In$N(\mathrm{PrimeZetaP}(2))$Out$0.452\,247\,420\,041\,065\,5$

Known value · Wolfram 15.0.0 PrimeZetaP

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In$N(\mathrm{PrimeZetaP}(3))$Out$0.174\,762\,639\,299\,443\,55$

Known value · Wolfram 15.0.0 PrimeZetaP

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In$N(\mathrm{PrimeZetaP}(2), 50)$Out$0.452\,247\,420\,041\,065\,498\,506\,543\,364\,832\,247\,934\,173\,231\,343\,239\,89$

Scope #

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In$\mathrm{PrimeZetaP}(2.5)$Out$0.273\,680\,737\,993\,234$

Known value · Wolfram 15.0.0 PrimeZetaP

$\operatorname{Re}(s) < 1$: each $\ln\zeta(ks)$ with $ks < 1$ is negative, so contributes $\pm i\pi\mu(k)/k$, and $\operatorname{Im} P = \pi(1 - 1/2) = \pi/2$

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In$N(\mathrm{PrimeZetaP}(\frac{4}{11}), 50)$Out$-1.435\,100\,337\,745\,163\,433\,710\,852\,727\,776\,389\,558\,234\,430\,910\,384\,8+1.570\,796\,326\,794\,896\,619\,231\,321\,691\,639\,751\,442\,098\,584\,699\,687\,6i$

$P(3/10)$: $\operatorname{Im} P = \pi(1 - 1/2 - 1/3) = \pi/6$

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In$N(\mathrm{PrimeZetaP}(\frac{3}{10}))$Out$-1.328\,034\,068\,791\,346\,8+0.523\,598\,775\,598\,298\,9i$

Known value · Wolfram 15.0.0 PrimeZetaP

$\ln\zeta(2s)$ is infinite at $s = 1/2$ and $\mu(2) = -1$

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In$\mathrm{PrimeZetaP}(\frac{1}{2})$Out$\tilde∞$

complex $s$ with $\operatorname{Re}(s) < 1/2$: Wolfram continues $\ln\zeta(s)$ through the functional equation, which puts $\operatorname{Im} P$ at $5.42$, $2\pi$ off the principal $-0.86$

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In$N(\mathrm{PrimeZetaP}(\frac{3}{10}+2i))$Out$-0.469\,945\,293\,401\,886\,95+5.420\,612\,621\,861\,013i$

Known value · Wolfram 15.0.0 PrimeZetaP

Applications #

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In$\mathrm{ReplaceAll}(γ+\mathrm{PrimeZetaP}(1+\exp(-n))-n, \bigl\lbrack\mathrm{Rule}(n, 20)\bigr\rbrack)$Out$-20+γ+\mathrm{PrimeZetaP}(1+e^{-20})$

About

Examples	12
Checked against	Wolfram Language 12 agree
Crosswalk	Wolfram Language

Implementation

primitive · kernela kernel — an algorithm over mutable state that would not be readable as a tree

nativeenginetypescriptpackages/symbols/analysis/analytic/src/prime-zeta.ts

See also: Zeta