Chapter 15 — On Series Which Arise From Products

Summary: §264–§296. Euler develops the duality between an infinite product $\prod (1\pm {\alpha }_{i}z{)}^{\pm 1}$ and the series obtained by multiplying it out. When the ${\alpha }_i$ range over the primes (or reciprocals of prime powers), the resulting series have number-theoretic meaning: squarefree numbers, all natural numbers, Möbius-style sign patterns. The central result is the Euler product formula $\zeta (n)={\prod }_p(1-1/p^n{)}^{-1}$ (§274, §283), and its most famous consequence — the divergence of $\sum 1/p$ obtained by taking logarithms at $n=1$ (§279). The chapter closes with a vast catalogue of series for $\pi /4$, $\pi /2$, $\pi /(2\sqrt{2})$, $\pi /(3\sqrt{3})$ etc. expressed as signed sums or products over primes classified mod 4, mod 6, mod 8 (§285–§296).

Sources: chapter15

Last updated: 2026-05-11

---

Movement 1 — Products as symmetric-coefficient series (§264–§269)

For any (finite or infinite) product of linear factors

$(1+\alpha z)(1+\beta z)(1+\gamma z)(1+\delta z)\cdots =1+Az+B{z}^2+C{z}^3+\cdots ,$

the coefficient $A$ is the sum of the ${\alpha }_i$, $B$ the sum of products taken two at a time, $C$ three at a time, etc. — the elementary symmetric polynomials in the ${\alpha }_i$ (source: chapter15, §264). The two specialisations $z=1$ and $z=-1$ collapse the polynomial coefficients into single signed series (§265–§266).

Letting the ${\alpha }_i$ range over the primes $2,3,5,7,11,\dots$ gives a series listing the squarefree natural numbers; with ${\alpha }_i=1/p^n$ the series is ${\sum }_{k}1/k^n$ over squarefree $k$, and with negative factors $(1-1/p^n)$ the signs become Möbius-style, depending on the parity of the number of prime factors. See squarefree-and-mobius-series.

Movement 2 — Inverse products and the Euler product formula (§270–§277)

The reciprocal product

$\frac{1}{(1-\alpha z)(1-\beta z)(1-\gamma z)\cdots }=1+Az+B{z}^2+\cdots$

now allows each ${\alpha }_i$ to repeat: $B$ is the sum of products of two factors not necessarily distinct, and so on (§270). At $z=1$ this is the sum over all products of the ${\alpha }_i$ with repetition (§271).

With ${\alpha }_i=1/p$ ranging over the primes, unique factorization of integers gives the harmonic series (§273):

$\frac{1}{(1-\frac{1}{2})(1-\frac{1}{3})(1-\frac{1}{5})(1-\frac{1}{7})\cdots }=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots .$

With ${\alpha }_i=1/p^n$ for any $n$ this is the Euler product formula (§274):

${\prod }_{p\text{ prime}}\frac{1}{1-1/p^n}=1+\frac{1}{2^n}+\frac{1}{3^n}+\frac{1}{4^n}+\cdots .$

Combined with the §269 product, this yields the reciprocal relation $PQ=1$ (§275): the product $\prod (1-1/p^n)$ and the Möbius series sum to $1/\zeta (n)$. See euler-product-formula.

Movement 3 — Logarithms: divergence of $\sum 1/p$ (§278–§282)

Taking the natural log of the Euler product and applying the §118 series to each factor yields

$\log M={\sum }_{k=1}^{\infty }\frac{1}{k}\left(\frac{1}{2^{kn}}+\frac{1}{3^{kn}}+\frac{1}{5^{kn}}+\frac{1}{7^{kn}}+\cdots \right),$

a double sum that transposes $\log \zeta (n)$ into a series in prime power sums (§278). At $n=1$ the left side is $\log \log \infty$ but every inner sum except the first is finite, forcing $\sum 1/p=\infty$ (§279). See divergence-of-prime-reciprocals.

At even $n$ the same identity expresses $\log ({\pi }^{2k}/\text{rational})$ as a convergent double sum, and Euler tabulates $S(n)={\sum }_{p}1/p^n$ to 12 decimal places for $n=2,4,6,\dots ,36$ (§281–§282). See prime-zeta-values.

Movement 4 — Sieve derivation of the Euler product (§283–§284)

Independently of Movement 2, §283 derives the Euler product by an Eratosthenes-style sieve on the series itself. Starting from $A=\sum 1/k^n$, one subtracts $A/2^n$ to remove even-index terms, then $B/3^n$ to remove multiples of 3, then $C/5^n$, and so on; at the end only the term 1 survives, giving

$A(1-\frac{1}{2^n})(1-\frac{1}{3^n})(1-\frac{1}{5^n})\cdots =1.$

The same technique applied in §284 to the alternating odd-denominator series $A=1-1/3^n+1/5^n-\cdots$ (which equals $\pi /4$ at $n=1$, by Leibniz) sieves out by primes with signs determined by class mod 4: primes of the form $4m-1$ contribute $(1+1/p^n)$, primes of the form $4m+1$ contribute $(1-1/p^n)$.

Movement 5 — Prime-signed series for $\pi$ (§285–§296)

The §284 product expression for $\pi /4$, together with the Basel-problem product expression for ${\pi }^2/6$, opens a long catalogue:

• §285–§286: $\pi /4={\prod }_{p}p/(p\pm 1)$, $\pi /2={\prod }_{p}p/p^{\ast }$, ratios giving prime-only Wallis-style products, and the comparison with Wallis.
• §287: cubic analogue ${\pi }^3/32={\prod }_p{p}^3/(p^3\pm 1)$.
• §288–§289: multiplying by individual factors converts a product over odd primes into one over all primes, yielding $\pi /6=1-1/2-1/3+\cdots$, $\pi /2=1+1/2-1/3-\cdots$, etc., with composite signs given multiplicatively.
• §290–§291: similar manipulations give series summing to $3\pi /2$, $0$, and infinity; in general, finite changes to the prime-sign pattern produce 0 if cofinitely many primes are negative and $\infty$ if cofinitely many are positive.
• §292–§294: applying the sieve to the §176 character series with mod 6 classification yields $\pi /(3\sqrt{3})=\prod p^n/(p^n\pm 1)$ with sign by class mod 6.
• §295: applying the sieve to the §179 series gives $\pi /(2\sqrt{2})$ products and series with signs by class mod 8.
• §296: Euler signals that the catalogue is unbounded — the same machinery applied to any of the §171–§180 character series gives a corresponding prime-classified product.

See prime-sign-series-for-pi for the full taxonomy and worked examples.

Significance

This chapter is the analytic-number-theory pivot of the Introductio. It introduces:

• The Euler product formula $\zeta (s)=\prod (1-p^{-s}{)}^{-1}$, the foundational identity that lets one read primes off the zeta function.
• The divergence of $\sum 1/p$, the first proof that primes are “denser than squares” — a sharpening of Euclid’s theorem that there are infinitely many primes.
• A wealth of prime-classified series that are modern Dirichlet $L$-functions in disguise: the §284 product is $L(1,{\chi }_4)=\pi /4$ in its Euler-product form; the §294 product is $L(1,{\chi }_6)$; §295 is $L(1,{\chi }_8)$.

The next chapter (Chapter 16) turns from products to additive representations: partitions of integers.

Related pages

• euler-product-formula
• squarefree-and-mobius-series
• divergence-of-prime-reciprocals
• prime-zeta-values
• prime-sign-series-for-pi
• basel-problem
• zeta-at-even-integers
• arctangent-series
• circular-arc-series
• logarithmic-series
• wallis-product