# 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.

### List of unsolved problems in mathematics

Unsolved problems in mathematicsopenopen problem

### Van der Corput's method

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