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Dirichlet Series

An arithmetic function $fn$ can be encoded into an infinite series of the form

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Infinite Series Attached to Arithmetic Functions

An arithmetic function f(n)f(n) can be encoded into an infinite series of the form

∑n=1∞f(n)ns, \sum_{n=1}^{\infty}\frac{f(n)}{n^s},

where ss is usually a real or complex variable.

Such a series is called a Dirichlet series.

Dirichlet series are fundamental because they transform arithmetic information into analytic information. Multiplicative structure becomes product structure, and divisor sums become ordinary multiplication of series.

General Form

A Dirichlet series has the form

F(s)=∑n=1∞anns, F(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s},

where ana_n is a sequence of complex numbers.

When

an=f(n), a_n=f(n),

for an arithmetic function ff, the series becomes

F(s)=∑n=1∞f(n)ns. F(s)=\sum_{n=1}^{\infty}\frac{f(n)}{n^s}.

The variable ss is typically written as

s=σ+it, s=\sigma+it,

where σ,t∈R\sigma,t\in\mathbb{R}.

Convergence depends mainly on the real part σ\sigma.

The Riemann Zeta Function

The most important Dirichlet series is the Riemann zeta function:

ζ(s)=∑n=1∞1ns. \zeta(s)=\sum_{n=1}^{\infty}\frac1{n^s}.

This series converges for

Re⁡(s)>1. \operatorname{Re}(s)>1.

The zeta function encodes the multiplicative structure of the integers and is deeply connected with prime numbers.

Euler discovered the product formula

ζ(s)=∏p11−p−s, \zeta(s) = \prod_p\frac1{1-p^{-s}},

where the product runs over all primes.

This identity follows from unique prime factorization.

Euler Products

Suppose ff is multiplicative. Then its Dirichlet series factors into an Euler product:

∑n=1∞f(n)ns=∏p(1+f(p)ps+f(p2)p2s+⋯ ), \sum_{n=1}^{\infty}\frac{f(n)}{n^s} = \prod_p \left( 1+\frac{f(p)}{p^s} +\frac{f(p^2)}{p^{2s}} +\cdots \right),

provided the series converges absolutely.

Each prime contributes one local factor.

The product formula works because every positive integer has a unique prime factorization.

Completely Multiplicative Case

If ff is completely multiplicative, then

f(pα)=f(p)α. f(p^\alpha)=f(p)^\alpha.

Therefore the local factor becomes a geometric series:

1+f(p)ps+f(p)2p2s+⋯=11−f(p)p−s. 1+\frac{f(p)}{p^s} +\frac{f(p)^2}{p^{2s}} +\cdots = \frac1{1-f(p)p^{-s}}.

Hence

∑n=1∞f(n)ns=∏p11−f(p)p−s. \sum_{n=1}^{\infty}\frac{f(n)}{n^s} = \prod_p \frac1{1-f(p)p^{-s}}.

This is one reason completely multiplicative functions are analytically convenient.

Dirichlet Convolution and Multiplication

Dirichlet series transform convolution into multiplication.

If

F(s)=∑n=1∞f(n)ns F(s)=\sum_{n=1}^{\infty}\frac{f(n)}{n^s}

and

G(s)=∑n=1∞g(n)ns, G(s)=\sum_{n=1}^{\infty}\frac{g(n)}{n^s},

then

F(s)G(s)=∑n=1∞(f∗g)(n)ns, F(s)G(s) = \sum_{n=1}^{\infty}\frac{(f*g)(n)}{n^s},

where

(f∗g)(n)=∑d∣nf(d)g(nd) (f*g)(n)=\sum_{d\mid n}f(d)g\left(\frac nd\right)

is the Dirichlet convolution.

This identity is analogous to the multiplication of ordinary power series.

Examples

Since

τ=1∗1, \tau=\mathbf{1}*\mathbf{1},

we obtain

∑n=1∞τ(n)ns=ζ(s)2. \sum_{n=1}^{\infty}\frac{\tau(n)}{n^s} = \zeta(s)^2.

Similarly,

σ=id⁡∗1, \sigma=\operatorname{id}*\mathbf{1},

so

∑n=1∞σ(n)ns=ζ(s)ζ(s−1). \sum_{n=1}^{\infty}\frac{\sigma(n)}{n^s} = \zeta(s)\zeta(s-1).

For Euler’s totient function,

φ=id⁡∗μ, \varphi=\operatorname{id}*\mu,

hence

∑n=1∞φ(n)ns=ζ(s−1)ζ(s). \sum_{n=1}^{\infty}\frac{\varphi(n)}{n^s} = \frac{\zeta(s-1)}{\zeta(s)}.

These identities connect arithmetic functions directly with analytic functions.

Möbius Function and the Reciprocal of ζ(s)\zeta(s)

Since

μ∗1=ε, \mu*\mathbf{1}=\varepsilon,

their Dirichlet series satisfy

(∑n=1∞μ(n)ns)ζ(s)=1. \left( \sum_{n=1}^{\infty}\frac{\mu(n)}{n^s} \right) \zeta(s) = 1.

Therefore

∑n=1∞μ(n)ns=1ζ(s). \sum_{n=1}^{\infty}\frac{\mu(n)}{n^s} = \frac1{\zeta(s)}.

This identity is fundamental in analytic number theory because zeros of ζ(s)\zeta(s) affect cancellation in sums involving μ(n)\mu(n).

Abscissa of Convergence

A Dirichlet series usually converges only in a half-plane

Re⁡(s)>σ0. \operatorname{Re}(s)>\sigma_0.

The boundary value σ0\sigma_0 is called the abscissa of convergence.

For example,

ζ(s)=∑n=1∞1ns \zeta(s) = \sum_{n=1}^{\infty}\frac1{n^s}

converges when

Re⁡(s)>1. \operatorname{Re}(s)>1.

This follows from comparison with the integral

∫1∞x−σ dx. \int_1^\infty x^{-\sigma}\,dx.

The location of the convergence boundary often reflects arithmetic growth properties of the coefficients.

Analytic Continuation

Many important Dirichlet series extend beyond their initial region of convergence.

For example, the zeta function admits a meromorphic continuation to the entire complex plane, with only one pole at

s=1. s=1.

Such continuation reveals deep arithmetic information. The zeros and poles of these analytic continuations govern prime distribution and asymptotic behavior of arithmetic functions.

Role in Number Theory

Dirichlet series translate arithmetic into analysis.

Prime factorization becomes Euler products. Divisor sums become products of series. Average orders become properties of poles and zeros.

This analytic viewpoint is one of the central ideas of modern number theory. It allows problems about integers to be studied through complex functions, contour integration, and analytic continuation.

The bridge between arithmetic and analysis built by Dirichlet series is the foundation of analytic number theory.