Dirichlet Series
The zeta function
is the most famous member of a whole family. A Dirichlet series attaches an
analytic function to any arithmetic sequence — the general tool for studying number-theoretic
functions with the methods of analysis.
The general form
Given an arithmetic function a(n), its Dirichlet series is
D(s) = \sum_{n=1}^{\infty} \frac{a(n)}{n^{s}}.
Choosing a(n) = 1 recovers \zeta(s). Choosing
the Möbius function
a(n) = \mu(n) gives, remarkably,
1/\zeta(s) — the analytic shadow of
Möbius inversion.
Multiplication mirrors convolution
The product of two Dirichlet series re-sums by divisors:
\left(\sum \frac{a(n)}{n^s}\right)\!\left(\sum \frac{b(n)}{n^s}\right) = \sum_{n} \frac{(a * b)(n)}{n^{s}}, \quad (a*b)(n) = \sum_{d\mid n} a(d)\,b(n/d).
So multiplying Dirichlet series performs the divisor-convolution of their coefficients — and a
function being
multiplicative
is exactly what gives its series an
Euler product.
Analysis and arithmetic translate cleanly into one another.
Why build them
Dirichlet series convert questions about sequences into questions about functions — where poles,
zeros and growth rates can be brought to bear. Dirichlet's own creation, the
L-functions built from characters, attach a series to
each residue class and unlock primes in
arithmetic progressions.
They are the prototype of the vast theory of L-functions at the frontier
of modern number theory.