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N. M. Khodzhaev

On a Sprindzhuk problem

We consider estimates of the function

equal to the square-free part of the positive integer argument t . V. G. Sprindzhuk posed the following problem. Is there a constant c > 0 such that the inequality

S((n + 1) ... (n + k)) < k^k

is fulfilled for an infinite number of pairs of positive integers n and k such that k < ln^cn? We prove that there exist positive constants c₇,..., c₁₀ such that for n ≥ c₇

In the paper, we obtain several other estimates of the function S(t) and discuss some conjectures concerning S(t) and derive corollaries of those conjectures.

Discrete Mathematics and Applications, Walter de Gruyter

Print ISSN: 0924-9266
Volume: 13, 06/2003
Pages: 189 - 208

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