Odd and Alternating Zeta Decomposition

Summary: §170: from the full sum $M=1+1/2^n+1/3^n+\cdots$, the four basic variants — even-only, odd-only, alternating — fall out by elementary algebra. The mechanism that converts any closed form for $\zeta (n)$ into closed forms for $\sum 1/(2k{)}^n$, $\sum 1/(2k+1{)}^n$, and $\sum (-1{)}^{k+1}/k^n$.

Sources: chapter10

Last updated: 2026-04-30

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The decomposition

Let

$M=1+\frac{1}{2^n}+\frac{1}{3^n}+\frac{1}{4^n}+\frac{1}{5^n}+\cdots =\zeta (n).$

Multiplying through by $1/2^n$:

$\frac{M}{2^n}=\frac{1}{2^n}+\frac{1}{4^n}+\frac{1}{6^n}+\frac{1}{8^n}+\cdots \text{(even terms only)}.$

Subtracting:

$M-\frac{M}{2^n}=\frac{2^n-1}{2^n}M=1+\frac{1}{3^n}+\frac{1}{5^n}+\frac{1}{7^n}+\cdots \text{(odd terms only)}.$

Subtracting twice over:

$M-\frac{2M}{2^n}=\frac{2^{n-1}-1}{2^{n-1}}M=1-\frac{1}{2^n}+\frac{1}{3^n}-\frac{1}{4^n}+\frac{1}{5^n}-\cdots \text{(alternating)}.$

(source: chapter10, §170). Euler’s comment: “If $n$ is an even number and the sum is $A{\pi }^n$, then $A$ will be a rational number” — i.e. the rationality result of zeta-at-even-integers propagates through all four variants.

Worked tabulation

For $n=2$, with $M={\pi }^2/6$ (Basel):

series	sum
all $1/k^2$	${\pi }^2/6$
even $1/(2k{)}^2$	${\pi }^2/24$
odd $1/(2k+1{)}^2$	${\pi }^2/8$
alternating $\sum (-1{)}^{k+1}/k^2$	${\pi }^2/12$

The odd-terms-only result ${\pi }^2/8$ recovers the value computed independently in §169 from the cosh product — an internal consistency check.

For $n=4$, with $M={\pi }^4/90$:

series	sum
all $1/k^4$	${\pi }^4/90$
even $1/(2k{)}^4$	${\pi }^4/1440$
odd $1/(2k+1{)}^4$	${\pi }^4/96$
alternating $\sum (-1{)}^{k+1}/k^4$	$7{\pi }^4/720$

Why this matters

The decomposition is purely formal: it does not depend on the closed form of $M$. Whatever methods evaluate $M$ — Newton’s identities on the sine product, coefficient comparison, or anything else — the same closed form propagates to the even, odd, and alternating restrictions by linear algebra.

A consequence Euler exploits later: every zeta-style series can be split into “divisible by $p$” and “not divisible by $p$” parts using the same trick with $1/p^n$ in place of $1/2^n$. §177 splits by $1/3$, e.g.

${\sum }_{k\text{not divisible by }3}\frac{1}{k^2}=\frac{{\pi }^2}{6}-{\sum }_{k\ge 1}\frac{1}{(3k{)}^2}=\frac{{\pi }^2}{6}-\frac{{\pi }^2}{54}=\frac{8{\pi }^2}{54}=\frac{4{\pi }^2}{27}.$

This is the seed of an important tool: writing the zeta function as an Euler product over primes.

Modern footnote: Dirichlet $L$-functions and the Euler product

The four decompositions are the simplest case of a general principle. For any Dirichlet character $\chi$ mod $m$, the series ${\sum }_{n\ge 1}\chi (n)/n^s$ defines a Dirichlet $L$-function. The $\chi =$ trivial character recovers $\zeta (s)$; the principal character mod $2$ recovers Euler’s odd-only sum; the non-trivial character mod $4$ recovers the Leibniz-style alternating series. The decomposition

$\zeta (s)={\prod }_p\frac{1}{1-p^{-s}}$

(the Euler product) is the systematic version of the §170 “subtract the multiples of $p$” trick. Euler’s manipulations in §170 and §177 are early instances of the prime-factorization viewpoint that became foundational under Dirichlet (1837) and Riemann (1859).

Related pages

• basel-problem
• zeta-at-even-integers
• newtons-identities
• circular-arc-series
• sine-infinite-product
• cosine-infinite-product
• chapter-10-on-the-use-of-the-discovered-factors-to-sum-infinite-series