The zeta function is a complex function whose zeros are connected to the distribution of prime numbers.The Riemann zeta function is defined for complex s with real part greater than 1 as ζ(s) = ∑ 1/n^s and analytically continues to the entire complex plane except s=1. Its zeros are divided into trivial ones (at negative even integers) and non-trivial ones. The Riemann Hypothesis states that all non-trivial zeros have real part 1/2. The zeta function factors into the Euler product over primes, linking it to number theory. The distribution of its zeros governs deviations in the prime number theorem. A deep connection with matrices: in the 1970s, Hugh Montgomery discovered that the pair correlation of zeros matches that of eigenvalues of random Hermitian matrices, giving rise to the conjecture of a link with quantum chaos.
Euler in 1737 derived the product over primes, establishing the fundamental connection. Riemann in 1859 introduced analytic continuation and formulated the hypothesis. In the 20th century, Hardy proved the infinitude of zeros on the critical line, Selberg proved that a positive proportion of zeros lie on it. In 1972, Montgomery identified statistical regularities of zeros, and Odlyzko confirmed them numerically on a huge scale.
The functional equation ζ(s) = 2^s π^{s-1} sin(πs/2) Γ(1-s) ζ(1-s) is symmetric about the critical line. The explicit Riemann–von Mangoldt formula expresses the sum over prime powers in terms of the sum over non-trivial zeros of ζ(s). This means each prime 'contributes' to the oscillations of ζ(s), and each zero modulates the deviations in the distribution of primes from the main term. The density of zeros at height T grows as (T/2π) log(T/2πe). The Prime Number Theorem is equivalent to the absence of zeros on the line Re(s)=1. Through this lens, we see how primes, like keys, fit the locks of cryptography.
Testing the reliability of cryptographic keys, analyzing random processes, quantum chaos — wherever we need to look into the depths of numerical disorder.
The famous millennium problem — proving the Riemann Hypothesis — is directly related to the security of modern cryptosystems, since a proof could accelerate unveiling the distribution of prime numbers.
| β_ρ | variable | beta function for ρ joule per cubic metre |
| M | variable | renormalization scale — energy joule |
| ρ | variable | running parameter joule per cubic metre |
| ∂/∂M | operator | partial derivative with respect to scale M — partial derivative |
