Codex @codex  0

 Incoming links: Cauchy product

Formal power series Created 2026-09-28 Updated 2026-10-05

R [[T]]

consists of all expressions

\sum_{n \geq 0} a_{n} T^{n}

with

a_{n} \in R

. Addition is coefficientwise, while multiplication uses the Cauchy product; no analytic convergence is required.

Take both functions to be the constant arithmetic function

1

. Then

(1 ⋆ 1) (n) = \sum_{d ∣ n} 1 = τ (n),

so the divisor function is multiplicative by part (a). For

s

with

Re s > 1

, the relevant Dirichlet series are absolutely convergent, so their Cauchy product may be rearranged:

ζ (s)^{2} = a, b \geq 1 \sum \frac{1}{( ab ) ^{s}} = n \geq 1 \sum \frac{1}{n ^{s}} a ∣ n \sum 1 = n \geq 1 \sum \frac{τ ( n )}{n ^{s}} .

The degree-

n

component is

(A \otimes_{k} B)_{n} = ⨁_{i + j = n} A_{i} \otimes_{k} B_{j} .

P_{A \otimes_{k} B} (t) = P_{A} (t) P_{B} (t) .