Erdős problem 394

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.

Sources

The source's own material, retained exactly.

10 retained statements36c6d0fef71a

FormalConjectures/ErdosProblems/

394.lean

Retained formal statement6 of 9

Erdős and Hall conjecture that the sum is $o(x^2/(\log x)^c)$ for any $c<\log 2$ .

FormalConjectures/ErdosProblems/394.leanErdos394.erdos_394.variants.hall_conjecture1 lineExact file

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

OpenStatement only, no proof