Erdős Problem #510
Chowla's cosine problem
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
**Chowla's cosine problem** If \(A\subset \mathbb{N}\) is a finite set of positive integers of size \(N > 0\) then is there some absolute constant \(c>0\) and \(\theta\) such that \[\sum_{n\in A}\cos(n\theta) < -cN^{1/2}?\]
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/510.Prizenone
Topicsanalysis
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/510
Attempts
submit one →Read our working report →
Proved the sharp-order bound for low 2-adic-level and quadratic-energy regimes and exactly certified all 4–6 term spectra through frequency 24; the general \(N^{1/2}\) conjecture remains open, with Be
PARTIAL