Circular-Arc Series

Summary: §171–§180: applying Newton’s identities to the §164 arc-form products $\cos (v/2)+\tan (g/2)\sin (v/2)$ and $\cos (v/2)+\cot (g/2)\sin (v/2)$ produces a vast family of closed-form series in which the denominators run over arithmetic progressions modulo $n$, with sign patterns coming from the elementary-symmetric structure. Special cases recover Leibniz’s $\pi /4$ series and many of its $\sqrt{2}$, $\sqrt{3}$ analogues.

Sources: chapter10

Last updated: 2026-05-11

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The master products

From §164,

$\cos \frac{v}{2}+\tan \frac{g}{2}\sin \frac{v}{2}=\left(1+\frac{v}{\pi -g}\right)\left(1-\frac{v}{\pi +g}\right)\left(1+\frac{v}{3\pi -g}\right)\left(1-\frac{v}{3\pi +g}\right)\left(1+\frac{v}{5\pi -g}\right)\cdots$

(source: chapter10, §171). Substituting $v=\pi x/n$, $g=m\pi /n$:

$\left(1+\frac{x}{n-m}\right)\left(1-\frac{x}{n+m}\right)\left(1+\frac{x}{3n-m}\right)\left(1-\frac{x}{3n+m}\right)\cdots =\cos \frac{\pi x}{2n}+\tan \frac{m\pi }{2n}\sin \frac{\pi x}{2n}.$

Expanding the right side as a power series in $x$:

$=1+\frac{\pi x}{2n}\tan \frac{m\pi }{2n}-\frac{{\pi }^2{x}^2}{2\cdot 4n^2}-\frac{{\pi }^3{x}^3}{2\cdot 4\cdot 6n^3}\tan \frac{m\pi }{2n}+\frac{{\pi }^4{x}^4}{2\cdot 4\cdot 6\cdot 8n^4}+\cdots ,$

so the elementary symmetric coefficients of the product are

$A=\frac{\pi }{2n}\tan \frac{m\pi }{2n},B=\frac{-{\pi }^2}{2\cdot 4n^2},C=\frac{-{\pi }^3}{2\cdot 4\cdot 6n^3}\tan \frac{m\pi }{2n},D=\frac{{\pi }^4}{2\cdot 4\cdot 6\cdot 8n^4},\dots$

(source: chapter10, §171). The roots — read off the linear factors — are

$\alpha =\frac{1}{n-m},\beta =-\frac{1}{n+m},\gamma =\frac{1}{3n-m},\delta =-\frac{1}{3n+m},ϵ=\frac{1}{5n-m},\dots$

(source: chapter10, §171), with the alternating sign pattern.

The power sums

Apply Newton’s recurrence to obtain $P,Q,R,S,T,V$. With $k=\tan (m\pi /2n)$ for brevity:

$P=\frac{1}{n-m}-\frac{1}{n+m}+\frac{1}{3n-m}-\frac{1}{3n+m}+\cdots =\frac{\pi k}{2n},$

$Q=\frac{1}{(n-m{)}^2}+\frac{1}{(n+m{)}^2}+\frac{1}{(3n-m{)}^2}+\frac{1}{(3n+m{)}^2}+\cdots =\frac{(k^2+1){\pi }^2}{4n^2},$

$R=\frac{1}{(n-m{)}^3}-\frac{1}{(n+m{)}^3}+\frac{1}{(3n-m{)}^3}-\cdots =\frac{(k^3+k){\pi }^3}{8n^3},$

$S=\frac{(3k^4+4k^2+1){\pi }^4}{48n^4},T=\frac{(3k^5+5k^3+2k){\pi }^5}{96n^5}$

(source: chapter10, §172). The pattern: even-power sums are positive (each term squared); odd-power sums are alternating, with the alternation matching the sign-pattern of the roots.

The cot-variant

The §173 partner

$\cos \frac{v}{2}+\cot \frac{g}{2}\sin \frac{v}{2}=\left(1+\frac{v}{g}\right)\left(1-\frac{v}{2\pi -g}\right)\left(1+\frac{v}{2\pi +g}\right)\left(1-\frac{v}{4\pi -g}\right)\cdots$

with the same substitution $v=\pi x/n$, $g=m\pi /n$, gives an analogous family with roots

$\alpha =\frac{1}{m},\beta =-\frac{1}{2n-m},\gamma =\frac{1}{2n+m},\delta =-\frac{1}{4n-m},ϵ=\frac{1}{4n+m},\dots$

and

$P=\frac{1}{m}-\frac{1}{2n-m}+\frac{1}{2n+m}-\frac{1}{4n-m}+\cdots =\frac{\pi }{2nk}$

(source: chapter10, §173–§174), and so on for $Q,R,S,T,V$.

Special values

$m=1$, $n=2$ (§175): the Leibniz series and friends

Here $k=\tan (\pi /4)=1$, the §172 and §174 series coincide, and:

$\frac{\pi }{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots ,$

$\frac{{\pi }^2}{8}=1+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+\frac{1}{9^2}+\cdots ,$

$\frac{{\pi }^3}{32}=1-\frac{1}{3^3}+\frac{1}{5^3}-\frac{1}{7^3}+\cdots ,$

$\frac{{\pi }^4}{96}=1+\frac{1}{3^4}+\frac{1}{5^4}+\frac{1}{7^4}+\cdots ,\frac{{\pi }^6}{960}=1+\frac{1}{3^6}+\frac{1}{5^6}+\cdots$

(source: chapter10, §175). Euler observes: even-exponent series were already obtained in §169 (cosh product); the odd-exponent alternating series

$1-\frac{1}{3^{2n+1}}+\frac{1}{5^{2n+1}}-\frac{1}{7^{2n+1}}+\cdots$

are seen here for the first time. Each equals a rational multiple of ${\pi }^{2n+1}$.

$m=1$, $n=3$ (§176): $\sqrt{3}$ series

Here $k=\tan (\pi /6)=1/\sqrt{3}$:

$\frac{\pi }{6\sqrt{3}}=\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{10}+\frac{1}{14}-\frac{1}{16}+\cdots ,$

$\frac{{\pi }^2}{27}=\frac{1}{2^2}+\frac{1}{4^2}+\frac{1}{8^2}+\frac{1}{10^2}+\frac{1}{14^2}+\cdots ,$

$\frac{\pi }{3\sqrt{3}}=1-\frac{1}{2}+\frac{1}{4}-\frac{1}{5}+\frac{1}{7}-\frac{1}{8}+\cdots$

(source: chapter10, §176). The pattern of denominators: integers not divisible by 3, with alternating sign.

$m=1$, $n=4$ and $n=8$ (§179)

$k=\tan (\pi /8)=\sqrt{2}-1$, etc. After algebra:

$\frac{\pi }{2\sqrt{2}}=1+\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\frac{1}{9}+\frac{1}{11}-\frac{1}{13}-\cdots (n=4),$

$\frac{\pi }{4(2-\sqrt{2}{)}^{1/2}}=1+\frac{1}{7}-\frac{1}{9}-\frac{1}{15}+\frac{1}{17}+\cdots (n=8).$

(source: chapter10, §179). §180 extracts further combinations; the sign patterns become genuinely intricate. Euler comments that one “could let $n=16$ and $m=1,3,5,$ or $7$ which would show the sums of series in which the terms are $1,1/3,1/5,1/7,\dots$ and in which the various changes of positive and negative signs are different from those already seen” (source: chapter10, §180) — i.e. the technique generates an unlimited supply of “character sums” in modern terminology.

Modern footnote: Dirichlet $L$-values

The series above are exactly the values

$L(\chi ,n)={\sum }_{k\ge 1}\frac{\chi (k)}{k^n}$

for a Dirichlet character $\chi$ mod $4n$ (or a divisor thereof) at positive integer $s=n$. Specifically:

• $\pi /4=L({\chi }_4,1)$ for the non-trivial character mod $4$.
• $\pi /(2\sqrt{2})=L({\chi }_8,1)$ for the real quadratic character mod $8$.
• $\pi /(3\sqrt{3})=L(\chi ,1)$ for one of the characters mod $12$.

The general phenomenon — that $L(\chi ,n)\in \overline{\mathbb{Q}}\cdot {\pi }^n$ when $\chi$ is even and $n$ even (or $\chi$ odd and $n$ odd) — is the functional equation for Dirichlet L-functions. Euler is computing case-by-case examples of a uniform structural theorem first proved by Hurwitz a century later.

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