A Dirichlet series is an infinite series of the form
$$ \sum_{n=1}^{\infty} \frac{a_n}{n^{s}} = a_1 , 1^{-s}+ a_2 , 2^{-s}+ a_3 , 3^{-s}+ \cdots , $$
where $ (a_n)_{n\ge 1} $ is a sequence of complex numbers, $ s $ is a complex variable written as $ s=\sigma+it $ with real part $ \sigma$ and imaginary part $ t$, and $ n^{-s}=e^{-s\log n}$. The series generalizes the Riemann zeta function, which corresponds to the special case $a_n\equiv1$.
Convergence
For a given sequence $ (a_n) $, the Dirichlet series converges absolutely in a right half‑plane
$$ \sigma > \sigma_c, $$
where the abscissa of absolute convergence $ \sigma_c $ is defined by
$$ \sigma_c = \limsup_{n\to\infty}\frac{\log |a_1|+\dots+\log |a_n|}{\log n}. $$
It may also converge conditionally in a larger half‑plane $ \sigma > \sigma_a $ (the abscissa of convergence). The region of convergence is therefore a vertical strip in the complex plane.
Analytic properties
Within its half‑plane of convergence, a Dirichlet series defines a holomorphic function of $ s $. Under suitable conditions on the coefficients $ a_n $ (e.g., boundedness or moderate growth), the series can be analytically continued beyond its initial domain, often via functional equations or integral representations.
Important examples
- Riemann zeta function: $\displaystyle \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}$, convergent for $\sigma>1$ and analytically continued to $\mathbb{C}\setminus{1}$.
- Dirichlet L‑functions: For a Dirichlet character $\chi$ modulo $q$,
$$ L(s,\chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}. $$ These functions are central to analytic number theory and the study of prime distribution in arithmetic progressions. - Hurwitz zeta function: $\displaystyle \zeta(s,\alpha)=\sum_{n=0}^{\infty}\frac{1}{(n+\alpha)^{s}}$ with $0<\alpha\le1$.
Operations and transformations
- Multiplication: The product of two Dirichlet series corresponds to the Dirichlet convolution of their coefficient sequences: $$ \left(\sum_{n\ge1}\frac{a_n}{n^{s}}\right)!!\left(\sum_{n\ge1}\frac{b_n}{n^{s}}\right) =\sum_{n\ge1}\frac{(ab)_n}{n^{s}},\quad (ab)n=\sum{d\mid n}a_d,b_{n/d}. $$
- Differentiation: Formal differentiation with respect to $ s $ yields
$$ \frac{d}{ds}\left(\frac{1}{n^{s}}\right) = -\frac{\log n}{n^{s}} , $$ allowing the construction of series involving logarithmic factors. - Mellin transform: Dirichlet series arise naturally as Mellin transforms of arithmetic functions.
Applications
Dirichlet series are a fundamental tool in analytic number theory. They encode arithmetic information in their coefficients and enable the use of complex analysis to study prime numbers, divisor functions, and partition functions. The analytic continuation and functional equations of L‑functions derived from Dirichlet series underpin major results such as Dirichlet’s theorem on primes in arithmetic progressions and the proof of the prime number theorem.
In addition to number theory, Dirichlet series appear in:
- Probability theory, e.g., in the study of random multiplicative functions.
- Mathematical physics, particularly in spectral theory and quantum chaos, where Selberg zeta functions are expressed as Dirichlet series.
- Dynamical systems, through zeta functions counting periodic orbits.
Historical notes
The concept is named after German mathematician Peter Gustav Lejeune Dirichlet (1805–1859), who introduced the series now bearing his name in his 1837 work on the convergence of series of the form $\sum a_n/n^{s}$ and proved the first results concerning the distribution of primes in arithmetic progressions. Subsequent development by Bernhard Riemann, Edmund Landau, and many 20th‑century mathematicians expanded the theory to a broad class of L‑functions.
