# 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**

### Riemann–Siegel formula

**approximate functional equationRiemann-Siegel formulaZ(1)**

### 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.