Let z_i be an infinite sequence of complex numbers such that | z_i|=1 for all i≥ 1, and for n≥ 1 let p_n(z)=∏_i≤ n (z-z_i). Let M_n=max_| z|=1| p_n(z)|. Is it true that limsup M_n=∞? Is it true that there exists c>0 such that for infinitely many n we have M_n > n^c? Is it true that there exists c>0 such that, for all large n, ∑_k≤ nM_k > n^1+c?

Erdős Problem #119 — Let $z_i$ be an infinite sequence of complex numbers such that $\lvert z_i\rvert=1$ for all $i\geq 1$ , and for $n\geq 1$ let $p_n(z)=\prod_{i\leq n} (z-z_i).$ Let…

About 52 years old · traced to 1974

Let $z_i$ be an infinite sequence of complex numbers such that $\lvert z_i\rvert=1$ for all $i\geq 1$ , and for $n\geq 1$ let $p_n(z)=\prod_{i\leq n} (z-z_i).$ Let $M_n=\max_{\lvert z\rvert=1}\lvert p_n(z)\rvert$ . Is it true that $\limsup M_n=\infty$ ? Is it true that there exists $c>0$ such that for infinitely many $n$ we have $M_n > n^c$ ? Is it true that there exists $c>0$ such that, for all large $n$ , $\sum_{k\leq n}M_k > n^{1+c}?$

References

Primary source

ULAM AI, UnsolvedMath dataset, Erdős Problems set, revision b9437975f3c873f635a13c48f8b022f5ba80898a; CC BY 4.0.

Additional references

UnsolvedMath, Erdős Problems set, ULAM AI, licensed CC BY 4.0.

analysis polynomials

InErdős Problems

Posted 2 months ago · edited 3 weeks ago

Progress summary

Refreshed 1 month ago

Open

The conjecture is still open: known results show occasional large values, but no one has proved the required overall growth.

Attributed to Erdős and recorded as Problem 4.1 in 1974, the conjecture asks whether maxima associated with every unit-circle zero sequence have the required cumulative growth.

Known results

Erdős constructed examples with $M_n\le n+1$ .
Linden (1977) obtained $M_n\ll n^{1-c}$ for some $c>0$ .
Wagner (1980) proved $M_n>(\log n)^c$ infinitely often for some $c>0$ .
Beck (1991) proved $\max_{n\le N}M_n>N^c$ for some $c>0$ . \nThese bounds do not establish the cumulative assertion; the Lean formalization remains marked by sorry.

Current status (as of September 2026): Individual-maxima lower bounds are known, but the cumulative-maxima conjecture remains open, with no recorded proof or counterexample.

GPT-5.6 SolOpenAIsolvedevidence

Sources

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