Zeta at Even Integers

Summary: §168: the systematic table of $\zeta (2k)={\sum }_{n=1}^{\infty }1/n^{2k}$ as a rational multiple of ${\pi }^{2k}$, computed by newtons-identities from the sine product. Euler tabulates through $\zeta (26)$ and notes the “extraordinary usefulness” of the irregular rational sequence that appears.

Sources: chapter10, chapter15

Last updated: 2026-05-11

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The table

For each positive integer $k$,

${\sum }_{n=1}^{\infty }\frac{1}{n^{2k}}=\frac{2^{2k-2}}{(2k+1)!}{c}_k{\pi }^{2k},$

where $c_k$ is a positive rational. Euler tabulates the first thirteen values (source: chapter10, §168):

$2k$	$\sum 1/n^{2k}$	$c_k$
2	${\pi }^2/6$	$1$
4	${\pi }^4/90$	$1/3$
6	${\pi }^6/945$	$1/3$
8	${\pi }^8/9450$	$3/5$
10	${\pi }^{10}/93555$	$5/3$
12	$\frac{691{\pi }^{12}}{638512875}$	$691/105$
14	$\frac{2{\pi }^{14}}{18243225}$	$35/1$
16	$\frac{3617{\pi }^{16}}{325641566250}$	$3617/15$
18	$\frac{43867{\pi }^{18}}{38979295480125}$	$43867/21$
20	$\frac{1222277{\pi }^{20}}{\dots }$	$1222277/55$
22	$\frac{854513{\pi }^{22}}{\dots }$	$854513/3$
24	$\frac{1181820455{\pi }^{24}}{\dots }$	$1181820455/273$
26	$\frac{76977927{\pi }^{26}}{\dots }$	$76977927/1$

Euler’s comment on the $c_k$ sequence: “We could continue with more of these, but we have gone far enough to see a sequence which at first seems quite irregular, $1,1/3,1/3,3/5,5/3,691/105,35/1,\dots$, but it is of extraordinary usefulness in several places” (source: chapter10, §168).

Sums of reciprocal odd squares

The same machinery applied to the cosh product (§169) gives

${\sum }_{k=0}^{\infty }\frac{1}{(2k+1{)}^{2n}}.$

Setup: from the §157 product

$\frac{e^x+e^{-x}}{2}=1+\frac{x^2}{1\cdot 2}+\frac{x^4}{1\cdot 2\cdot 3\cdot 4}+\cdots ={\prod }_{k=0}^{\infty }\left(1+\frac{4x^2}{(2k+1{)}^2{\pi }^2}\right),$

substitute $x^2={\pi }^{2}z/4$, and read off

$A=\frac{{\pi }^2}{1\cdot 2\cdot 4}=\frac{{\pi }^2}{8},B=\frac{{\pi }^4}{1\cdot 2\cdot 3\cdot 4\cdot 4^2},C=\frac{{\pi }^6}{1\cdot 2\cdot \dots \cdot 6\cdot 4^3},\dots$

The roots are ${\alpha }_k=1/(2k+1{)}^2$ for $k=0,1,2,\dots$. By Newton’s recurrence:

$P={\sum }_{k=0}^{\infty }\frac{1}{(2k+1{)}^2}=\frac{{\pi }^2}{8},$

$Q={\sum }_{k=0}^{\infty }\frac{1}{(2k+1{)}^4}=\frac{{\pi }^4}{96},R={\sum }_{k=0}^{\infty }\frac{1}{(2k+1{)}^6}=\frac{{\pi }^6}{960},$

and so on. (The full table requires §170’s even/odd splits to interleave with the even-zeta values; see odd-and-alternating-zeta-decomposition.)

The qualitative pattern

Euler observes in §168: any infinite series of the form $1+1/2^n+1/3^n+\cdots$ with $n$ even is a rational multiple of ${\pi }^n$. The same is conjectured true (and remains true today) for every even $n$, and the rational coefficient grows exuberantly: $c_{13}=76977927$ has eight digits.

The qualitative conclusion: $\zeta (2k)/{\pi }^{2k}$ is rational for every positive integer $k$.

Modern footnote: Bernoulli numbers

Euler’s $c_k$ are essentially the Bernoulli numbers. The modern formula is

$\zeta (2k)=\frac{(-1{)}^{k+1}(2\pi {)}^{2k}}{2(2k)!}{B}_{2k},$

where $B_{2k}$ are the Bernoulli numbers $B_2=1/6$, $B_4=-1/30$, $B_6=1/42$, $B_8=-1/30$, $B_{10}=5/66$, $B_{12}=-691/2730$, … Comparing with Euler’s parametrization $c_k=2(2k+1)|B_{2k}|$:

$\zeta (2k)=\frac{2^{2k-2}{c}_k}{(2k+1)!}{\pi }^{2k}=\frac{2^{2k-2}\cdot 2(2k+1)|B_{2k}|}{(2k+1)!}{\pi }^{2k}=\frac{2^{2k-1}|B_{2k}|}{(2k)!}{\pi }^{2k}.$

Verification: $c_1=2\cdot 3\cdot 1/6=1$, $c_2=2\cdot 5\cdot 1/30=1/3$, $c_3=2\cdot 7\cdot 1/42=1/3$, $c_4=2\cdot 9\cdot 1/30=3/5$, $c_6=2\cdot 13\cdot 691/2730=691/105$. The “irregularity” Euler noticed is the genuine irregularity of the Bernoulli sequence — a sequence whose number-theoretic density (via Kummer’s congruences and the Herbrand–Ribet theorem) is one of the central topics of $p$-adic analytic number theory.

The Bernoulli numbers were known to Jakob Bernoulli (in connection with sums of $k$-th powers of consecutive integers) before Euler’s Introductio, but the link between them and the zeta values is Euler’s discovery — recorded here, computed term-by-term in §168 without a closed form. Euler returned to the subject decades later and produced the modern formula; the ${\pi }^{2k}$ scaling is already evident in the table above.

What is not solved here

The series $\zeta (2k+1)$ at odd indices — $\sum 1/n^3$, $\sum 1/n^5$, $\dots$ — are conspicuously absent from the table. They cannot be obtained by Euler’s method: the sine product gives even-power sums via the Newton recurrence, never odd-power sums of $1/k$ (the odd powers of ${\alpha }_k=1/k^2$ are $1/k^{2(\text{odd})}=1/k^{\text{even}}$, so the recurrence still produces even-zeta values). Apéry (1978) proved $\zeta (3)$ irrational; in 2026 it remains unknown whether $\zeta (3)/{\pi }^3$ is rational, and similarly for $\zeta (5),\zeta (7),\dots$.

Re-derivation via primes (Chapter 15)

The closed-form $\zeta (2k)$ values are also the inputs for Chapter 15’s Euler product formula $\zeta (2k)={\prod }_p(1-1/p^{2k}{)}^{-1}$ and the §281 inversion that computes ${\sum }_{p}1/p^{2k}$ numerically by bootstrapping from $\log \zeta (2k)$. The §285 prime-only Wallis-style product for ${\pi }^2/6$ — see prime-sign-series-for-pi — is the simplest non-trivial consequence.

Related pages

• basel-problem
• newtons-identities
• odd-and-alternating-zeta-decomposition
• sine-infinite-product
• cosine-infinite-product
• exponential-infinite-product
• pi
• log-pi-via-products
• log-sine-via-products
• chapter-10-on-the-use-of-the-discovered-factors-to-sum-infinite-series
• chapter-11-on-other-infinite-expressions-for-arcs-and-sines
• chapter-15-on-series-which-arise-from-products
• euler-product-formula
• prime-zeta-values
• prime-sign-series-for-pi