Erdős problem 394 · problems.science

For the least $t_k(n)$ with $n \mid t_k(n)(t_k(n)+1)\cdots(t_k(n)+k-1)$ , do the conjectured logarithmic-saving and adjacent-length estimates hold on average? Both answered affirmatively, with $c = 1/2048$ admissible in the $t_2$ bound.

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10 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

394.lean

Retained formal statement5 of 9

In [ErGr80] they mention a conjecture of Erdős that the sum is $o(x^2)$ . This was proved by Erdős and Hall [ErHa78], who proved that in fact $\sum_{n\leq x}t_2(n)\ll \frac{\log\log\log x}{\log\log x}x^2.$

FormalConjectures/ErdosProblems/394.leanErdos394.erdos_394.variants.hall_bound2 linesExact file

1(fun x => ∑ n ∈ Finset.Icc 1 ⌊x⌋₊, ↑(Erdos394.t 2 n)) =O[Filter.atTop] fun x =>2  ↑x ^ 2 * (Real.log (Real.log (Real.log ↑x)) / Real.log (Real.log ↑x))

SolvedStatement only, no proof

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