Riemann Hypothesis
Statement
The Riemann zeta function \(\zeta(s)=\sum_{{n=1}}^\infty n^{{-s}}\) has all non-trivial zeros on the critical line Re(s)=1/2. One of the Millennium Prize Problems.
Examples
Example 1. What are the trivial zeros of ζ(s)?
Solution. Negative even integers: −2,−4,−6,…
In Depth
The Riemann zeta function \(\zeta(s)=\sum_{n=1}^\infty n^{-s}\) converges for \(\text{Re}(s)>1\) and extends analytically to the entire complex plane (except a simple pole at \(s=1\)). Its zeros control the distribution of primes: the explicit formula \(\pi(x)=\text{Li}(x)-\sum_\rho\text{Li}(x^\rho)+\ldots\) sums over the non-trivial zeros \(\rho\).
The Riemann Hypothesis (RH), stated in 1859, asserts all non-trivial zeros satisfy \(\text{Re}(\rho)=1/2\). It has been verified for the first \(10^{13}\) zeros. If true, RH implies the sharpest possible error bound in the PNT: \(|\pi(x)-\text{Li}(x)|=O(\sqrt{x}\ln x)\).
RH is one of the seven Millennium Prize Problems (\$1 million prize). It has hundreds of equivalent formulations and conditional results — many theorems in analytic number theory are proved 'assuming RH'. Its resolution would have profound implications for cryptography, since the distribution of primes affects the security of many cryptographic systems.
Further Reading & Context
The Riemann zeta function \(\zeta(s)=\sum_{n=1}^\infty n^{-s}\) converges for \(\text{Re}(s)>1\) and extends analytically to all \(s\neq1\). The Euler product \(\zeta(s)=\prod_p(1-p^{-s})^{-1}\) encodes the distribution of primes. The functional equation \(\xi(s)=\xi(1-s)\) (where \(\xi\) is a completed version) relates values at \(s\) and \(1-s\).
The non-trivial zeros of \(\zeta(s)\) lie in the critical strip \(0<\text{Re}(s)<1\). The Riemann hypothesis (RH) asserts they all lie on the critical line \(\text{Re}(s)=1/2\). Over \(10^{13}\) zeros have been computed and all lie on the critical line, but no proof exists.
RH implies the sharpest known error term in the prime number theorem: \(\pi(x)=\text{Li}(x)+O(\sqrt{x}\ln x)\). It also implies that primes are equidistributed in arithmetic progressions with optimal error bounds, and that the Mertens function satisfies \(M(x)=O(x^{1/2+\epsilon})\).
Equivalent formulations: RH is equivalent to Robin's inequality \(\sigma(n)
The Generalized Riemann Hypothesis (GRH) extends RH to all Dirichlet \(L\)-functions. Many results in number theory are proved conditionally under GRH, including the best bounds on primitive roots, the distribution of primes in arithmetic progressions, and the complexity of certain algorithms.