OPEN
This is open, and cannot be resolved with a finite computation.
Let $(\epsilon_k)_{k\geq 0}$ be independently uniformly chosen at random from $\{-1,1\}$. If $R_n$ counts the number of real roots of $f_n(z)=\sum_{0\leq k\leq n}\epsilon_k z^k$ then is it true that, almost surely,\[\lim_{n\to \infty}\frac{R_n}{\log n}=\frac{2}{\pi}?\]
Erdős and Offord
[EO56] showed that the number of real roots of a random degree $n$ polynomial with $\pm 1$ coefficients is $(\frac{2}{\pi}+o(1))\log n$.
It is ambiguous in
[Er61] whether Erdős intended the coefficients to be uniformly chosen from $\{-1,1\}$ or $\{0,1\}$. In the latter case, the constant $\frac{2}{\pi}$ should be $\frac{1}{\pi}$ (see the discussion in the comments).
In the case of $\{-1,1\}$ Do
[Do24] proved that, if $R_n[-1,1]$ counts the number of roots in $[-1,1]$, then, almost surely,\[\lim_{n\to \infty}\frac{R_n[-1,1]}{\log n}=\frac{1}{\pi}.\]See also
[522].
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This page was last edited 19 October 2025.
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T. F. Bloom, Erdős Problem #521, https://www.erdosproblems.com/521, accessed 2026-01-17
