Riemann hypothesis

Riemann hypothesis. Proposed by the German mathematician Bernhard Riemann in his seminal 1859 paper, this conjecture concerns the behavior of a specific complex function. It posits that all non-trivial zeros of the Riemann zeta function have a real part equal to one-half, placing them on a vertical line in the complex plane. The hypothesis is central to number theory and has profound implications for understanding the distribution of prime numbers.

Statement of the hypothesis

The Riemann zeta function is defined for complex numbers with a real part greater than one by the infinite series ζ(s) = Σ 1/n^s. Through analytic continuation, it can be extended to the entire complex plane except for a simple pole at s = 1. The function's non-trivial zeros are those complex numbers s = σ + it for which ζ(s) = 0, excluding the negative even integers known as trivial zeros. The conjecture states that every non-trivial zero lies on the **critical line**, where the real part σ is exactly 1/2. This is in contrast to the **critical strip**, the region where 0 < σ < 1, which is known to contain all non-trivial zeros. The statement is equivalent to precise bounds on the growth of various arithmetic functions, such as the Möbius function.

Connection to prime numbers

The profound link between the hypothesis and primes arises from the deep connection between the zeta function and the distribution of prime numbers, initially discovered by Leonhard Euler. Riemann himself demonstrated that the distribution of primes is intimately tied to the locations of the zeta function's zeros. A key formula, the explicit formula, relates a sum over these zeros to a sum over prime powers. If the hypothesis is true, it implies the strongest possible form of the prime number theorem, giving an extremely tight bound on the error term π(x) - Li(x). This would confirm that primes are distributed as regularly as possible, with the difference from the logarithmic integral growing very slowly. The hypothesis also governs the oscillations in functions like Chebyshev's ψ function.

Zeros of the zeta function

Extensive computational work, beginning with calculations by J. E. Littlewood and later advanced by Alan Turing using early computers, has verified the hypothesis for vast numbers of zeros. Projects like the ZetaGrid distributed computing initiative and work by researchers such as Andrew Odlyzko have checked that the first several trillion non-trivial zeros lie precisely on the critical line. These zeros are not randomly placed; their imaginary parts exhibit a statistical distribution predicted by the GUE conjecture from random matrix theory. The study of these zeros involves sophisticated techniques from complex analysis and has connections to phenomena in quantum chaos. The hypothetical linear independence of the imaginary parts over the rationals is another area of intense study.

Attempts at proof and verification

Numerous attempts to prove or disprove the hypothesis have been made by leading mathematicians, though none have yet succeeded. Early significant work was done by G. H. Hardy, who proved in 1914 that infinitely many zeros lie on the critical line. Later, Atle Selberg made crucial advances in understanding the distribution of these zeros. The search for a proof is a central goal in modern mathematics and is one of the Clay Institute's seven Millennium Prize Problems. While a general proof remains elusive, some progress has been made on related analogues, such as the proof of the Weil conjectures for varieties over finite fields by Pierre Deligne. False proofs are periodically announced but do not withstand scrutiny from the mathematical community.

Consequences and generalizations

The truth of the hypothesis would settle numerous open problems across mathematics. In number theory, it would imply the Lindelöf hypothesis and strengthen results on the smallest gaps between primes. It also has consequences for the calculation of arithmetic functions like the summatory function of the Möbius function. The hypothesis has inspired major generalizations, such as the Generalized Riemann Hypothesis for Dirichlet L-functions and other L-functions in the Langlands program. These extended conjectures are of fundamental importance in areas like algebraic number theory and the study of elliptic curves. Its resolution, one way or another, would represent a monumental event in the history of mathematical science. Category:Number theory Category:Mathematical conjectures Category:Millennium Prize Problems