Prime Zeta Values

Summary: §281–§282. Euler’s numerical table of $S(n)={\sum }_{p}1/p^n$ — the prime zeta function — to 12 decimal places for every even $n$ from 2 to 36, obtained by inverting the logarithmic relation between $\log \zeta (n)$ and the prime-power sums (§278).

Sources: chapter15

Last updated: 2026-05-11

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The table (§282)

For the series of reciprocals of $n$-th powers of all primes,

$S(n):={\sum }_{p\text{ prime}}\frac{1}{p^n}=\frac{1}{2^n}+\frac{1}{3^n}+\frac{1}{5^n}+\frac{1}{7^n}+\frac{1}{11^n}+\frac{1}{13^n}+\cdots ,$

Euler tabulates (source: chapter15, §282):

$n$	$S(n)$
2	$0.452247420041222$
4	$0.076993139764252$
6	$0.017070086850639$
8	$0.004061405366515$
10	$0.000993603573633$
12	$0.000246026470033$
14	$0.000061244396725$
16	$0.000015282026219$
18	$0.000003817278702$
20	$0.000000953961123$
22	$0.000000238450446$
24	$0.000000059608184$
26	$0.000000014901555$
28	$0.000000003725333$
30	$0.000000000931323$
32	$0.000000000232830$
34	$0.000000000058207$
36	$0.000000000014551$

Euler’s observation: “The remaining sums decrease by about one fourth at each step” (source: chapter15, §283). This is consistent with the dominant term $1/2^n$, which exactly quarters at each $n\to n+2$, plus smaller corrections from $1/3^n,1/5^n,\dots$ which become negligible.

The method (§281)

Euler does not sum these series directly — at $n=2$ that would require summing $1/p^2$ over all primes, which is hopeless without knowing the primes in closed form. Instead he inverts the §278 identity

$\log \zeta (n)=S(n)+\frac{1}{2}S(2n)+\frac{1}{3}S(3n)+\frac{1}{4}S(4n)+\cdots$

For large $n$ the right side is dominated by $S(n)$ (and the next term $S(2n)/2$ is already exponentially smaller). Knowing $\zeta (n)$ in closed form for even $n$ from Chapter 10 (e.g. $\zeta (2)={\pi }^2/6$, $\zeta (4)={\pi }^4/90$, $\zeta (6)={\pi }^6/945$, etc.), Euler solves recursively for $S(n)$:

$S(n)=\log \zeta (n)-\frac{1}{2}S(2n)-\frac{1}{3}S(3n)-\cdots$

Since $S(2n),S(3n),\dots$ also have closed-form-$\zeta$-determined values via the same recurrence (and they decay geometrically), the computation bootstraps from the largest $n$ downward: $S(36),S(34),\dots$ are computed first (the higher $n$ make all but the leading $\sum 1/2^{kn}$ negligible), then $S(34)$ uses $S(68),S(102),\dots$ which Euler approximates well, and so on, working down to $S(2)$.

A second sieve-based method (§281, parallel)

Euler also gives a related identity for odd-index-only prime sums

$\tilde{S}(n):=\frac{1}{3^n}+\frac{1}{5^n}+\frac{1}{7^n}+\frac{1}{11^n}+\cdots (\text{primes only, minus 2}),$

derived by removing the $1/2^n$ contribution. Manipulating

$S=(M-1)(1-\frac{1}{2^n})(1-\frac{1}{3^n})+\frac{1}{6^n}-\frac{1}{25^n}-\frac{1}{35^n}-\cdots$

(source: chapter15, §281) lets one recover $S(n)$ from the closed-form $M=\zeta (n)$ minus a rapidly-convergent correction series in composite squarefree indices.

This second form is the one Euler emphasises is convenient “provided only that $n$ is reasonably large” — the residual terms $1/9^n,1/15^n,1/21^n,\dots$ (composites of small primes) decay quickly when $n\ge 8$ or so.

Why no closed form?

The values $S(n)=\sum 1/p^n$ are not rational multiples of ${\pi }^n$ even at even $n$. They are believed to be transcendental but no closed form in terms of classical constants is known.

The closed form for $\log \zeta (n)$ (e.g. $\log ({\pi }^2/6)$ at $n=2$) decomposes via the §278 identity into the linear combination $S(n)+S(2n)/2+S(3n)/3+\cdots$ — and only the linear combination is closed-form. Individual prime-power sums $S(n)$, $S(2n)$, etc. each carry irreducible prime-distribution information.

This contrast — closed-form for $\zeta (2k)$, no closed form for $\sum 1/p^{2k}$ — is the analytic counterpart of: integers are easy (we know all of them), primes are hard (their distribution involves the zeros of $\zeta$).

Numerical accuracy

Euler’s 12-digit values match modern computations to all displayed digits. For example, $S(2)=\mathrm{0.452247420041065...}$ — Euler’s $0.452247420041222$ is correct to 14 digits with an end-of-table rounding wobble of about $2\cdot 10^{-13}$ in the last two displayed digits, consistent with the geometric tail truncation.

The very small magnitudes at $n=30$–$36$ ($\sim 10^{-9}$ to $10^{-11}$) show that Euler was carrying around 15+ digits of precision throughout the §278 recurrence — a remarkable hand-computation feat.

Related pages

• chapter-15-on-series-which-arise-from-products
• divergence-of-prime-reciprocals
• euler-product-formula
• zeta-at-even-integers
• basel-problem
• logarithmic-series
• squarefree-and-mobius-series