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This is a case in which even the best bound that can be proved using the Riemann hypothesis is far weaker than what seems true: Cramér's conjecture implies that every gap is O((log p) 2), which, while larger than the average gap, is far smaller than the bound implied by the Riemann hypothesis. Numerical evidence supports Cramér's conjecture.
The Riemann hypothesis is one of the most important conjectures in mathematics.It is a statement about the zeros of the Riemann zeta function.Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar to the Riemann zeta-function.
The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that: The real part of every nontrivial zero of the Riemann zeta function is 1/2. The Riemann hypothesis is that all nontrivial zeros of the analytical continuation of the Riemann zeta function have a real part of 1 / 2 .
The Conjecture is that this is true for all natural numbers (positive integers from 1 through infinity). ... Specifically, the Riemann Hypothesis is about when 𝜁(s)=0; the official statement is ...
For example, the Riemann hypothesis is a conjecture from number theory that — amongst other things — makes predictions about the distribution of prime numbers. Few number theorists doubt that the Riemann hypothesis is true.
In 1859 Bernhard Riemann used complex analysis and a special meromorphic function now known as the Riemann zeta function to derive an analytic expression for the number of primes less than or equal to a real number x. Remarkably, the main term in Riemann's formula was exactly the above integral, lending substantial weight to Gauss's conjecture.
In the event that the Riemann hypothesis is false, the argument is much simpler, essentially because the terms for zeros violating the Riemann hypothesis (with real part greater than 1 / 2 ) are eventually larger than (/).
In mathematics, Weil's criterion is a criterion of André Weil for the Generalized Riemann hypothesis to be true. It takes the form of an equivalent statement, to the effect that a certain generalized function is positive definite. Weil's idea was formulated first in a 1952 paper.