Riemann hypothesis

Hiervon wäre allerdings ein strenger Beweis zu wünschen; ich habe indess die Aufsuchung desselben nach einigen flüchtigen vergeblichen Versuchen vorläufig bei Seite gelassen, da er für den nächsten Zweck meiner Untersuchung entbehrlich schien.... it is very probable that all roots are real.

Of course one would wish for a rigorous proof here; I have for the time being, after some fleeting vain attempts, provisionally put aside the search for this, as it appears dispensable for the immediate objective of my investigation.At the death of Riemann, a note was found among his papers, saying "These properties of ζ(s) (the function in question) are deduced from an expression of it which, however, I did not succeed in simplifying enough to publish it."

Riemann's formula is then where the sum is over the nontrivial zeros of the zeta function and where Π0 is a slightly modified version of Π that replaces its value at its points of discontinuity by the average of its upper and lower limits: The summation in Riemann's formula is not absolutely convergent, but may be evaluated by taking the zeros ρ in order of the absolute value of their imaginary part.

He checked that a few of the zeros lay on the critical line with real part 1/2 and suggested that they all do; this is the Riemann hypothesis.

The statement that the equation is valid for every s with real part greater than 1/2, with the sum on the right hand side converging, is equivalent to the Riemann hypothesis.

From this we can also conclude that if the Mertens function is defined by then the claim that for every positive ε is equivalent to the Riemann hypothesis (J. E. Littlewood, 1912; see for instance: paragraph 14.25 in Titchmarsh (1986)).

Littlewood's result has been improved several times since then, by Edmund Landau,[6] Edward Charles Titchmarsh,[7] Helmut Maier and Hugh Montgomery,[8] and Kannan Soundararajan.

A typical example is Robin's theorem,[10] which states that if σ(n) is the sigma function, given by then for all n > 5040 if and only if the Riemann hypothesis is true, where γ is the Euler–Mascheroni constant.

The Riemann hypothesis also implies quite sharp bounds for the growth rate of the zeta function in other regions of the critical strip.

Brad Rodgers and Terence Tao discovered the equivalence is actually Λ = 0 by proving zero to be the lower bound of the constant.

!Care should be taken to understand what is meant by saying the generalized Riemann hypothesis is false: one should specify exactly which class of Dirichlet series has a counterexample.

This is the conjecture (first stated in article 303 of Gauss's Disquisitiones Arithmeticae) that there are only finitely many imaginary quadratic fields with a given class number.

The grand Riemann hypothesis extends it to all automorphic zeta functions, such as Mellin transforms of Hecke eigenforms.

For instance, the fact that the Gauss sum, of the quadratic character of a finite field of size q (with q odd), has absolute value

Assuming a functional equation and meromorphic continuation, the generalized Riemann hypothesis for the L-factor states that its zeros inside the critical strip

The main conjecture of Iwasawa theory, proved by Barry Mazur and Andrew Wiles for cyclotomic fields, and Wiles for totally real fields, identifies the zeros of a p-adic L-function with the eigenvalues of an operator, so can be thought of as an analogue of the Hilbert–Pólya conjecture for p-adic L-functions.

Odlyzko (1987) showed that the distribution of the zeros of the Riemann zeta function shares some statistical properties with the eigenvalues of random matrices drawn from the Gaussian unitary ensemble.

The analogy with the Riemann hypothesis over finite fields suggests that the Hilbert space containing eigenvectors corresponding to the zeros might be some sort of first cohomology group of the spectrum Spec (Z) of the integers.

[25] Zagier (1981) constructed a natural space of invariant functions on the upper half plane that has eigenvalues under the Laplacian operator that correspond to zeros of the Riemann zeta function—and remarked that in the unlikely event that one could show the existence of a suitable positive definite inner product on this space, the Riemann hypothesis would follow.

Cartier (1982) discussed a related example, where due to a bizarre bug a computer program listed zeros of the Riemann zeta function as eigenvalues of the same Laplacian operator.

Schumayer & Hutchinson (2011) surveyed some of the attempts to construct a suitable physical model related to the Riemann zeta function.

The Lee–Yang theorem states that the zeros of certain partition functions in statistical mechanics all lie on a "critical line" with their real part equal to 0, and this has led to some speculation about a relationship with the Riemann hypothesis.

But Haselgrove (1958) proved that T(x) is negative for infinitely many x (and also disproved the closely related Pólya conjecture), and Borwein, Ferguson & Mossinghoff (2008) showed that the smallest such x is 72185376951205.

Louis de Branges (1992) showed that the Riemann hypothesis would follow from a positivity condition on a certain Hilbert space of entire functions.

Despite this obstacle, de Branges has continued to work on an attempted proof of the Riemann hypothesis along the same lines, but this has not been widely accepted by other mathematicians.

When one goes from geometric dimension one, e.g. an algebraic number field, to geometric dimension two, e.g. a regular model of an elliptic curve over a number field, the two-dimensional part of the generalized Riemann hypothesis for the arithmetic zeta function of the model deals with the poles of the zeta function.

Related conjecture of Fesenko (2010) on the positivity of the fourth derivative of a boundary function associated to the zeta integral essentially implies the pole part of the generalized Riemann hypothesis.

This inequality follows by taking the real part of the log of the Euler product to see that where the sum is over all prime powers pn, so that which is at least 1 because all the terms in the sum are positive, due to the inequality The most extensive computer search by Platt and Trudgian[17] for counterexamples of the Riemann hypothesis has verified it for |t| ≤ 3.0001753328×1012.

This allows one to verify the Riemann hypothesis computationally up to any desired value of T (provided all the zeros of the zeta function in this region are simple and on the critical line).

The consensus of the survey articles (Bombieri 2000, Conrey 2003, and Sarnak 2005) is that the evidence for it is strong but not overwhelming, so that while it is probably true there is reasonable doubt.