Math 2.0

Number theory Analytic number theory Conjecture

Lindelöf hypothesis

Builds on Riemann zeta functionBig O notationRiemann hypothesis

Statement

Let ζ\zeta be the Riemann zeta function. For every ε>0\varepsilon > 0 there is a constant CεC_\varepsilon such that ∣ζ(12+it)∣≤Cε∣t∣ε\abs{\zeta(\tfrac12 + it)} \le C_\varepsilon \abs{t}^{\varepsilon} for all real tt with ∣t∣≥1\abs{t} \ge 1.

Remarks

The conjecture is open. The Riemann hypothesis implies it. The functional equation and the Phragmén–Lindelöf principle give the convexity bound ζ(12+it)=O(∣t∣1/4+ε)\zeta(\tfrac12 + it) = O(\abs{t}^{1/4 + \varepsilon}). Every improvement of the exponent 1/41/4 is called a subconvexity bound; Hardy and Littlewood obtained 1/6+ε1/6 + \varepsilon with Weyl's method, and Bourgain obtained 13/84+ε13/84 + \varepsilon. An equivalent form: for every integer k≥1k \ge 1 and every ε>0\varepsilon > 0, 1T∫0T∣ζ(12+it)∣2k dt=O(Tε).\frac{1}{T} \int_{0}^{T} \abs{\zeta(\tfrac12 + it)}^{2k} \, dt = O(T^{\varepsilon}).

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Big O notation, concept, no evidence levelBig O notationconceptGamma function, concept, no evidence levelGamma functionconceptMeromorphic function, concept, no evidence levelMeromorphic functionconceptRiemann zeta function, concept, no evidence levelRiemann zeta functionconceptFunctional equation of the Riemann zeta function, statement, no evidence levelFunctional equation o…statementRiemann hypothesis, conjecture, no evidence levelRiemann hypothesisconjectureLindelöf hypothesis, conjecture, no evidence levelLindelöf hypothesisconjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Big O notation concept
  2. Gamma function concept
  3. Meromorphic function concept
  4. Riemann zeta function concept
  5. Functional equation of the Riemann zeta function statement
  6. Riemann hypothesis conjecture
  7. Lindelöf hypothesis conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Lindelöf_hypothesis
About this page
TagPV82BVXE1A
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Kindconjecture
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MSC11M06, 11M26
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AboutRiemann zeta function
UsesRiemann zeta function, Big O notation, Riemann hypothesis
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Changed2 November 2026

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