Riemann Zeta Function

The Riemann zeta function, denoted $\zeta(s)$, is a function of a complex variable that connects number theory with complex analysis. Initially defined by the sum of reciprocal powers of the positive integers, it extends to the complex plane except for a simple pole at $s=1$. Its product representation involves every prime number, and the locations of its zeros are closely related to the distribution of primes. (dlmf.nist.gov)

Definition and convergence

For $s=\sigma+it$ with real part $\sigma>1$, the defining infinite series is

This is a Dirichlet series with all coefficients equal to $1$. It converges absolutely in this half-plane: the absolute value of its $n$-th term is $n^{-\sigma}$, and the corresponding real series converges when $\sigma>1$. At $s=1$, the series becomes the divergent harmonic series. (dlmf.nist.gov)

Elsewhere, $\zeta(s)$ is defined by analytic continuation, not by ordinary summation of the defining series. Its continuation is a meromorphic function with exactly one pole, at $s=1$, whose residue is $1$. In particular,

Thus, the domain where the original series converges must be distinguished from the domain where the continued function is defined. (dlmf.nist.gov)

Euler product and prime factorization

For $\operatorname{Re}s>1$, the zeta function has the Euler product

The connection with primes follows from the fundamental theorem of arithmetic. Expanding each factor as a geometric series gives

Multiplying these expansions produces one term $n^{-s}$ for each positive integer, because every such integer has a unique prime factorization. Absolute convergence justifies this expansion and rearrangement. (dlmf.nist.gov)

The Euler product is more than an alternative formula: it expresses the multiplicative structure of the integers through an analytic function. It also implies that $\zeta(s)$ has no zeros in $\operatorname{Re}s>1$. The product in this form cannot simply be used outside that half-plane. (dlmf.nist.gov)

Analytic continuation and integral representations

A useful representation employs the Dirichlet eta function, defined for $\operatorname{Re}s>0$ by

It satisfies

Consequently, $\eta(s)/(1-2^{1-s})$ extends the original zeta series into a larger region. At zeros of the denominator other than $s=1$, the quotient is interpreted by its removable limiting value. (dlmf.nist.gov)

An integral representation involving the gamma function is

To obtain it, expand $(e^x-1)^{-1}$ as $\sum_{n\ge1}e^{-nx}$, then integrate each term. The resulting integral is $\Gamma(s)n^{-s}$. This representation connects reciprocal-power sums with exponential integrals; other integral formulas extend into wider regions of the complex plane. (dlmf.nist.gov)

Functional equation

The zeta function satisfies the functional equation

It relates values at $s$ to values at $1-s$. Where individual factors have poles or zeros, the identity is understood meromorphically, with cancellations evaluated by limits. (dlmf.nist.gov)

A symmetric formulation uses the Riemann xi function:

The apparent singularities cancel, so $\xi(s)$ is analytic throughout the complex plane, and

This normalization makes the symmetry around the vertical line $\operatorname{Re}s=\tfrac12$ explicit. Its zeros are precisely the nontrivial zeros of $\zeta(s)$. (dlmf.nist.gov)

Special values

At positive even integers, the zeta function can be evaluated using Bernoulli numbers:

Examples include

The first identity evaluates the reciprocal-square series known as the Basel problem. (dlmf.nist.gov)

At nonpositive integers,

In particular,

A related derivative value is

These identities concern the analytically continued function. The value $\zeta(-1)=-1/12$ does not mean that $1+2+3+\cdots$ converges to a negative number: its ordinary partial sums increase without bound. Analytic continuation and ordinary convergence are different operations. (dlmf.nist.gov)

Zeros and the Riemann hypothesis

The zeros at $-2,-4,-6,\ldots$ are called trivial zeros. The functional equation explains their occurrence through its sine factor. All remaining zeros lie in the critical strip

There are infinitely many nontrivial zeros, and their set is symmetric about both the real axis and the critical line $\operatorname{Re}s=\tfrac12$. (dlmf.nist.gov)

The Riemann hypothesis asserts that every nontrivial zero lies on the critical line. Infinitely many zeros are known to lie there, but this does not establish that all do. The hypothesis remains unsolved according to the Clay Mathematics Institute. Finite numerical verification, however extensive, cannot by itself prove a statement about every zero. (dlmf.nist.gov)

Connection with the distribution of primes

The prime number theorem states that

where $\pi(x)$ counts the primes not exceeding $x$, and $\sim$ means that the ratio tends to $1$. The classical analytic proofs depend on the absence of zeta zeros on $\operatorname{Re}s=1$. This zero-free boundary was established independently by Jacques Hadamard and Charles-Jean de la Vallée Poussin in 1896. (dlmf.nist.gov)

The locations of zeros also govern the error in approximations to prime counts. In particular, the Riemann hypothesis is equivalent to the estimate

where one may take

Here big-O notation bounds the magnitude of the error up to a constant factor. The prime number theorem describes the leading average behavior; the hypothesis would impose a substantially sharper bound on deviations from that behavior. (dlmf.nist.gov)

Historical development

The function’s name commemorates Bernhard Riemann, whose 1859 paper On the Number of Primes Less Than a Given Magnitude investigated it as a function of a complex variable. Riemann connected its zeros with prime counting and formulated the hypothesis about their real parts. His paper established the framework in which the analytic behavior of the zeta function became a means of studying the distribution of primes. (claymath.org)

Computation and related functions

Numerical evaluation requires formulas appropriate to the argument. Direct summation is available in $\operatorname{Re}s>1$, while Euler–Maclaurin expansions provide finite sums with correction terms and controlled remainders. On the critical line at large height $t$, an approximate functional equation uses two sums of about $\sqrt{t/(2\pi)}$ terms rather than the divergent defining series. The Riemann–Siegel formula refines this approach for computations involving zeros. (dlmf.nist.gov)

The Hurwitz zeta function generalizes the defining sum by shifting its terms:

For positive real $a$, this series converges when $\operatorname{Re}s>1$, and $\zeta(s,1)=\zeta(s)$. Like the Riemann zeta function, it has a meromorphic continuation in $s$. (dlmf.nist.gov)

The zeta function also serves as a Dirichlet generating function for arithmetic identities. For example,

where the Möbius function $\mu(n)$ is zero when $n$ has a squared prime factor and otherwise equals $(-1)^k$ when $n$ has $k$ distinct prime factors. Such identities connect analytic operations on $\zeta(s)$ with divisibility and multiplicative arithmetic. (dlmf.nist.gov)

References

1. DLMF: §25.2 Definition and Expansionsdlmf.nist.gov
2. DLMF: §27.4 Euler Products and Dirichlet Seriesdlmf.nist.gov
3. DLMF: §25.5 Integral Representationsdlmf.nist.gov
4. DLMF: §25.4 Reflection Formulasdlmf.nist.gov
5. DLMF: §25.6 Integer Argumentsdlmf.nist.gov
6. DLMF: §25.10 Zerosdlmf.nist.gov
7. Riemann Hypothesis — Clay Mathematics Instituteclaymath.org
8. DLMF: §27.12 Asymptotic Formulas: Primesdlmf.nist.gov
9. Riemann's 1859 Manuscript — Clay Mathematics Instituteclaymath.org
10. DLMF: §25.9 Asymptotic Approximationsdlmf.nist.gov
11. DLMF: §25.11 Hurwitz Zeta Functiondlmf.nist.gov