Lindelöf hypothesis

In mathematics, the Lindelöf hypothesis is a conjecture by Finnish mathematician Ernst Leonard Lindelöf (see ) about the rate of growth of the Riemann zeta function on the critical line.wikipedia
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Riemann hypothesis

Critical line theoremcritical line1st
This hypothesis is implied by the Riemann hypothesis.
The Riemann hypothesis has various weaker consequences as well; one is the Lindelöf hypothesis on the rate of growth of the zeta function on the critical line, which says that, for any ε > 0,

Ernst Leonard Lindelöf

Ernst LindelöfLindelöfLindelöf, Ernst Leonard
In mathematics, the Lindelöf hypothesis is a conjecture by Finnish mathematician Ernst Leonard Lindelöf (see ) about the rate of growth of the Riemann zeta function on the critical line.

Prime gap

prime gapsgapgaps
However, this result is much worse than that of the large prime gap conjecture.
The Lindelöf hypothesis would imply that Ingham's formula holds for c any positive number: but even this would not be enough to imply that there is a prime number between n 2 and (n + 1) 2 for n sufficiently large (see Legendre's conjecture).

Mathematics

mathematicalmathmathematician
In mathematics, the Lindelöf hypothesis is a conjecture by Finnish mathematician Ernst Leonard Lindelöf (see ) about the rate of growth of the Riemann zeta function on the critical line.

Riemann zeta function

zeta functionRiemann zeta-functionRiemann's zeta function
In mathematics, the Lindelöf hypothesis is a conjecture by Finnish mathematician Ernst Leonard Lindelöf (see ) about the rate of growth of the Riemann zeta function on the critical line.

Big O notation

Obig-O notationlittle-o notation
as t tends to infinity (see O notation).

Infimum and supremum

supremuminfimumleast upper bound
If σ is real, then μ is defined to be the infimum of all real numbers a such that ζ(σ + iT) = O(T a ).

Functional equation

functional equationsfunctionalAbel's functional equation
It is trivial to check that μ = 0 for σ > 1, and the functional equation of the zeta function implies that μ = μ(1 − σ) − σ + 1/2.

Phragmén–Lindelöf principle

Phragmén–Lindelöf theoremPhragmen- Lindelöf type theoremsPhragmen-Lindelof principle
The Phragmén–Lindelöf theorem implies that μ is a convex function.

Convex function

convexconvexitystrictly convex
The Phragmén–Lindelöf theorem implies that μ is a convex function.

G. H. Hardy

G.H. HardyHardyGodfrey Harold Hardy
The upper bound of 1/4 was lowered by Hardy and Littlewood to 1/6 by applying Weyl's method of estimating exponential sums to the approximate functional equation.

John Edensor Littlewood

J. E. LittlewoodLittlewoodJ.E. Littlewood
The upper bound of 1/4 was lowered by Hardy and Littlewood to 1/6 by applying Weyl's method of estimating exponential sums to the approximate functional equation.

Hermann Weyl

WeylH. WeylHermann Klaus Hugo Weyl
The upper bound of 1/4 was lowered by Hardy and Littlewood to 1/6 by applying Weyl's method of estimating exponential sums to the approximate functional equation.

Exponential sum

Weyl sumexponential
The upper bound of 1/4 was lowered by Hardy and Littlewood to 1/6 by applying Weyl's method of estimating exponential sums to the approximate functional equation.

Riemann–Siegel formula

approximate functional equationRiemann-Siegel formulaZ(1)
The upper bound of 1/4 was lowered by Hardy and Littlewood to 1/6 by applying Weyl's method of estimating exponential sums to the approximate functional equation.

Random matrix

random matricesrandom matrix theory random matrix theory
: for the leading coefficient when k is 6, and used random matrix theory to suggest some conjectures for the values of the coefficients for higher k.

Young tableau

Young diagramYoung tableauxstandard Young tableaux
The leading coefficients are conjectured to be the product of an elementary factor, a certain product over primes, and the number of n by n Young tableaux given by the sequence

Albert Ingham

Albert Edward InghamInghamA.E. Ingham
Denoting by p n the n-th prime number, a result by Albert Ingham shows that the Lindelöf hypothesis implies that, for any ε > 0,

Eventually (mathematics)

sufficiently largeeventuallylarge enough
if n is sufficiently large.

Van der Corput's method

exponent pairExponent pair conjectureexponent pairs
This conjecture implies the Lindelöf hypothesis.