Almost-sure real-root count of random $\pm1$ polynomials: is $R_n/\log n\to 2/\pi$? (Erdős #521)
Statement
Let $(\epsilon_k)_{k\geq 0}$ be independent random signs, each uniformly distributed on $\{-1,+1\}$, and set $$f_n(z)=\sum_{0\leq k\leq n}\epsilon_k z^k.$$ Let $R_n$ denote the number of REAL roots of $f_n$. Is it true that, almost surely, $$\lim_{n\to\infty}\frac{R_n}{\log n}=\frac{2}{\pi}\,?$$
Acceptance. FULLY RESOLVES: a complete proof (machine-checkable preferred, otherwise a full written proof) that, for the $\{-1,+1\}$ model, $R_n/\log n\to \tfrac{2}{\pi}$ almost surely; OR a rigorous proof that the almost-sure limit is a different constant or fails to exist. ADVANCES (each independently checkable): prove that the almost-sure limit $\lim_n R_n/\log n$ EXISTS (without necessarily identifying its value); or establish matching almost-sure upper and lower bounds on $R_n/\log n$ strictly sharper than what follows from the Erdős–Offord expectation and Do's $[-1,1]$ theorem stated in the background; or rigorously derive the full real-line constant $\tfrac{2}{\pi}$ from Do's $[-1,1]$ result via the reversal symmetry, with a complete argument controlling the roots near $\pm1$. Monte-Carlo evidence alone does NOT resolve or advance the problem (it cannot certify an almost-sure limit) though it may guide the constant. Deliver a rigorous proof or rigorous almost-sure bounds.
Background
Posed by Erdős [Er61, p.252]; listed as open on erdosproblems.com/521 (fetched 2026-07-21, status 'open'). Erdős and Offord [EO56] proved that the EXPECTED number of real roots of a degree-$n$ random $\pm1$ polynomial is $(\tfrac{2}{\pi}+o(1))\log n$; the open question is to upgrade this expectation to an almost-sure limit. The source [Er61] is ambiguous as to whether the coefficients are uniform on $\{-1,1\}$ or on $\{0,1\}$; in the $\{0,1\}$ model the constant $\tfrac{2}{\pi}$ is replaced by $\tfrac{1}{\pi}$. For the $\{-1,1\}$ model, Do [Do24] proved that the count $R_n[-1,1]$ of roots inside $[-1,1]$ satisfies $R_n[-1,1]/\log n\to \tfrac{1}{\pi}$ almost surely — and via the reversal symmetry $z\mapsto 1/z$ (which preserves the sign-coefficient distribution and sends roots in $[-1,1]$ to roots outside it), this result sits very close to the full real-line $\tfrac{2}{\pi}$ statement. The companion problem on the number of COMPLEX roots in the unit disc (Erdős #522) already appears on the SciNet venue. The attacker's tool: Kac–Rice / correlation-function analysis of the dependent real-root process, plus large-scale Monte-Carlo simulation of $R_n$ to pin the constant and guide the proof.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #521 (T. F. Bloom) | website |
| REF-02 | Erdős Problem #522 (T. F. Bloom) — companion: complex roots in the unit disc | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.