Subconvexity for ζ(1/2+it): reducing Bourgain's exponent μ(1/2) ≤ 13/84 toward the Lindelöf value 0
A century of work moved the exponent from 1/4 to 13/84 ≈ 0.155, each step needing a new idea, so 74: above the zero-density problem (61) because pointwise bounds on the critical line are harder than averaged ones, and below the parity-limit prime-gap problem (80). Application value 11 is indirect (divisor and circle problems, moment bounds). Leverage 49 covers subconvexity, moments and lattice-point problems that take μ(1/2) as input.
The problem
Show the zeta function grows more slowly along the critical line than the current record allows.
- 1Measure how big zeta gets on the line
- 2Easy argument gives exponent 1/4
- 3Clever sum bounds lower it to 13/84
- 4Lindelöf predicts the truth is zero
Definition. Let . This is the smallest exponent such that . The says , that is, for every . The follows from the functional equation and the Phragmén–Lindelöf principle, a kind of maximum principle for analytic functions. The problem asks for any improvement on Bourgain's , via exponential-sum or decoupling methods, or any new bound on the -th moments .
Built from
earlier problems → this oneThe problems this one is built from: progress or tools it needs (builds on), special cases that come first (generalizes), and weaker results on the way (implies). On the right, what builds on it.
- Start with
- Step 2
- Subconvexity for ζ(1/2+it): reducing Bourgain's exponent μ(1/2) ≤ 13/84 toward the Lindelöf value 0This problem
- Then
- Riemann Hypothesis: all non-trivial zeros of ζ(s) lie on Re s = 1/2Number Theory · D96
Path to understanding
foundations → problemThe research entries this problem rests on, in the order to read them. Each links to its explainer.
- Start with
- Harmonic analysisTheory
- Step 2
- Step 3
- The Riemann zeta function and prime distributionTheory · host
- Then
- Subconvexity for ζ(1/2+it): reducing Bourgain's exponent μ(1/2) ≤ 13/84 toward the Lindelöf value 0The problem
State of the art
Open. Hardy and Littlewood (around 1921) proved μ(1/2) ≤ 1/6 by Weyl differencing; Huxley (2005) reached 32/205 ≈ 0.1561 with the Bombieri–Iwaniec method; Bourgain (arXiv:1408.5794, J. Amer. Math. Soc. 2017) proved 13/84 ≈ 0.1548 by applying ℓ² decoupling to the first spacing problem of the Bombieri–Iwaniec method. Bourgain and Watt (arXiv:1709.04340) obtained related mean-square bounds with applications to the circle and divisor problems. The Lindelöf hypothesis is implied by RH and implies the density hypothesis. Recent papers (2024–2026) still cite 13/84 as the record, and no improvement was found as of October 2026.
Realistic applications
Subconvexity feeds bounds for L-functions, lattice-point discrepancies, and moments, and underlies bounds on the divisor problem; there is no engineering application.
What counts as solved
A proved bound μ(1/2) < 13/84, or the Lindelöf hypothesis μ(1/2) = 0, accepted after refereeing.
How it gets verified
Expert reviewOnly expert review of the argument is possible today.
The target is a subconvexity exponent mu(1/2) < 13/84 or the Lindelof hypothesis mu(1/2) = 0. Mathlib has the zeta function but not exponential-sum estimates of this type; Formal Conjectures has zeta-moment statements (Paper/RiemannZetaMoments.lean) but not this bound. Resolution is refereed acceptance of the proof.
Claims under verification
+ add a claimNo claimed results on the Verification board for this problem yet.
Where the work is
New analytic ideas needed; computation only guides exponent optimization.
Bottlenecks
ranked by votes- #1TheoryExponential-sum methods plateau
Van der Corput, exponent pairs, Bombieri–Iwaniec and decoupling each gain a little; all lose to the lack of square-root cancellation for exponential sums Σ n^{it} of length near t^(1/2).
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