Claims

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1980_03_01_wagner: Wagner proves that for every sequence on the unit circle the maximum modulus of the partial products exceeds a fixed power of log n infinitely often, so is unbounded; answers the first question, refereed and credited by the site.

1991_11_01_beck: Beck proves that for every sequence on the unit circle the largest of the first N maximum moduli exceeds N^c for all large N, with an absolute c>0; answers the second question, refereed in the Annals, credited by the site.

2026_07_14_korsky: Korsky, working with GPT 5.6-Pro, proves that for every sequence on the unit circle the sum of the first N maximum moduli is at least a constant times N^(5/4)/sqrt(log N), answering the third question; accepted by the site.

2026_08_29_korsky: Korsky, working with GPT-5.6 Pro, claims that for every sequence on the unit circle the first N maximum moduli sum to at least (e^(-1/2)+o(1)) N^(3/2), believed sharp; a shared write-up on the site's thread, credited by the site.

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