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Level A · Machine-checkable Hard Number theory P-ramanujan-tau

Ramanujan τ-function

Lehmer's conjecture: τ(n) ≠ 0 for all n > 0.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite 
@misc{cairn-ramanujan-tau,
  title        = {Ramanujan τ-function},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/ramanujan-tau}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

Lehmer's conjecture: for all .

There are two conjectures related to the Ramanujan τ-function:

  • Ramanujan-Petersson conjecture: For every prime p, the absolute value of the Ramanujan τ-function at p is bounded by 2 * p^(11/2).
  • Lehmer's conjecture: The Ramanujan τ-function is never zero for any positive integer n.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.RamanujanTau.

theorem lehmer_ramanujan_tau : ∀ n > 0, τ n ≠ 0

What counts as progress

  • A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
  • Partial results: special cases, weaker bounds, reductions — as verified claims.
  • Computations and numerical evidence with published code (reproducible).
  • Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Source and licence

Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.