Dirichlet series transform multiplicative arithmetic into complex analysis. Their Euler products encode prime factorization, while analytic behavior near singularities controls average order and prime distribution.
Dirichlet series
Definition. The Dirichlet series of an arithmetic function \(a(n)\) is \[ D_a(s)=\sum_{n=1}^{\infty}\frac{a(n)}{n^s}, \qquad s=\sigma+it. \] Absolute convergence in one half-plane implies uniform convergence on every smaller closed half-plane and hence holomorphy there.
Proposition. If the relevant series converge absolutely, then \[ D_{a*b}(s)=D_a(s)D_b(s). \]
Proof. Absolute convergence permits rearrangement: \[ \sum_{m,n\ge1}\frac{a(m)b(n)}{(mn)^s} =\sum_{k\ge1}\frac1{k^s}\sum_{mn=k}a(m)b(n) =\sum_{k\ge1}\frac{(a*b)(k)}{k^s}. \]
Euler products
Theorem. If \(a\) is multiplicative and its Dirichlet series converges absolutely, then \[ D_a(s)=\prod_p\left(\sum_{k=0}^{\infty} \frac{a(p^k)}{p^{ks}}\right). \] If \(a\) is completely multiplicative, this becomes \[ D_a(s)=\prod_p(1-a(p)p^{-s})^{-1}. \]
Proof. Expand a finite product over \(p\le P\). Unique factorization gives exactly the terms \(a(n)n^{-s}\) whose prime divisors are at most \(P\). Absolute convergence allows passage to the limit \(P\to\infty\).
The Riemann zeta function
For \(\sigma>1\), \[ \zeta(s)=\sum_{n\ge1}n^{-s} =\prod_p(1-p^{-s})^{-1}. \] Since no Euler factor vanishes and the logarithm converges absolutely, \(\zeta(s)\ne0\) in \(\sigma>1\). Also, \[ \frac1{\zeta(s)}=\sum_{n\ge1}\frac{\mu(n)}{n^s},\qquad -\frac{\zeta'(s)}{\zeta(s)} =\sum_{n\ge1}\frac{\Lambda(n)}{n^s}, \] where \(\Lambda(p^k)=\log p\) and is zero otherwise.
Logarithmic derivative. \[ \log\zeta(s)=\sum_p\sum_{k\ge1}\frac1{k p^{ks}}. \] Differentiating termwise gives \[ -\frac{\zeta'}{\zeta}(s) =\sum_p\sum_{k\ge1}\frac{\log p}{p^{ks}} =\sum_n\frac{\Lambda(n)}{n^s}. \]
Partial summation
Proposition (Abel summation). If \(A(x)=\sum_{n\le x}a_n\) and \(f\in C^1\), then \[ \sum_{n\le x}a_nf(n) =A(x)f(x)-\int_1^xA(t)f'(t)\,\mathrm{d}t. \]
This is discrete integration by parts. Taking \(f(t)=t^{-s}\) links growth of \(A(x)\) with analytic continuation or convergence of its Dirichlet series. Tauberian theorems provide converses under positivity or boundary regularity assumptions.
Example. Since \(\sigma_k=\operatorname{id}^k*1\), \[ \sum_{n\ge1}\frac{\sigma_k(n)}{n^s} =\zeta(s)\zeta(s-k) \] for \(\operatorname{Re}s>k+1\). A convolution identity becomes a product of two classical analytic functions.