Definition of the Zeta Function

Infinite Series and Prime Numbers

One of the central objects of analytic number theory is the Riemann zeta function. It connects infinite series, prime numbers, complex analysis, and arithmetic structure into a single analytic framework.

The zeta function begins with the series

\sum_{n=1}^{\infty}\frac1{n^s},

where $s$ is a complex variable.

This function first appeared in the work of entity[“people”,“Leonhard Euler”,“Swiss mathematician”] during the study of infinite series. Later, entity[“people”,“Bernhard Riemann”,“German mathematician”] extended it into the complex plane and revealed its deep connection with prime distribution.

Definition

For a complex number

s=\sigma+it,

with real part

\sigma>1,

the Riemann zeta function is defined by

\zeta(s) = \sum_{n=1}^{\infty}\frac1{n^s}.

Using exponential notation,

n^{-s}=e^{-s\log n}.

Thus

\zeta(s) = 1+\frac1{2^s}+\frac1{3^s}+\frac1{4^s}+\cdots.

The series converges absolutely when

\operatorname{Re}(s)>1.

Absolute Convergence

To study convergence, consider

\left|\frac1{n^s}\right| = \frac1{n^\sigma}.

Therefore

\sum_{n=1}^{\infty}\left|\frac1{n^s}\right| = \sum_{n=1}^{\infty}\frac1{n^\sigma}.

This is a $p$ -series, which converges precisely when

\sigma>1.

Hence the zeta series converges absolutely in the half-plane

\operatorname{Re}(s)>1.

Absolute convergence permits rearrangement and termwise analytic operations.

Special Values

For real values $s>1$ , the zeta function becomes a positive real number.

Examples include

\zeta(2) = \sum_{n=1}^{\infty}\frac1{n^2},

and

\zeta(3) = \sum_{n=1}^{\infty}\frac1{n^3}.

Euler famously proved

\zeta(2)=\frac{\pi^2}{6}.

More generally, he derived formulas for all even positive integers:

\zeta(2k) = \frac{(-1)^{k+1}B_{2k}(2\pi)^{2k}}{2(2k)!},

where $B_{2k}$ are Bernoulli numbers.

Odd values such as

\zeta(3)

are much more mysterious.

Divergence at $s=1$

s=1,

the zeta series becomes the harmonic series:

\zeta(1) = \sum_{n=1}^{\infty}\frac1n.

Since the harmonic series diverges,

\zeta(1)

is not finite.

In fact, the zeta function has a simple pole at $s=1$ . This singularity is closely connected with the density of prime numbers.

Complex Variable Viewpoint

The zeta function is fundamentally a function of a complex variable:

s=\sigma+it.

Thus

n^{-s} = n^{-\sigma}e^{-it\log n}.

The factor

e^{-it\log n}

introduces oscillation, making the zeta function highly analytic and deeply connected with harmonic analysis.

The variable $t$ controls oscillatory behavior, while $\sigma$ governs convergence.

Euler Product

The defining series factors into an infinite product over primes:

\zeta(s) = \prod_p \left(1-\frac1{p^s}\right)^{-1},

valid for

\operatorname{Re}(s)>1.

This formula follows from unique prime factorization.

It reveals that the zeta function encodes the arithmetic structure of the prime numbers. Much of analytic number theory comes from extracting information about primes from analytic properties of $\zeta(s)$ .

Analytic Continuation

Although the defining series converges only for

\operatorname{Re}(s)>1,

Riemann showed that the zeta function extends to a meromorphic function on the entire complex plane.

The continuation has only one singularity: a simple pole at

s=1.

This extension allows one to study zeros and analytic behavior far beyond the original region of convergence.

Zeros of the Zeta Function

The equation

\zeta(s)=0

defines the zeros of the zeta function.

The negative even integers

-2,-4,-6,\ldots

are called the trivial zeros.

The remaining zeros lie inside the critical strip

0<\operatorname{Re}(s)<1.

These nontrivial zeros govern the fine structure of prime distribution.

The Riemann Hypothesis asserts that all nontrivial zeros satisfy

\operatorname{Re}(s)=\frac12.

This is one of the most important open problems in mathematics.

Importance in Number Theory

The zeta function stands at the center of analytic number theory because it transforms arithmetic questions into analytic ones.

Through its Euler product, it encodes primes. Through its zeros, it controls fluctuations in prime distribution. Through analytic continuation and functional equations, it connects arithmetic with complex analysis and spectral theory.

Modern generalizations of the zeta function include:

Dirichlet $L$ -functions,
Dedekind zeta functions,
modular $L$ -functions,
automorphic $L$ -functions.

These objects form the foundation of much of modern number theory.