On the reciprocal sum of lcm of k-tuples

Abstract

We prove that the reciprocal sum $S_k(x)$ of the least common multiple of $k\ge 3$ positive integers in $\mathbb{N}\cap [1,x]$ satisfies $$S_k(x)=P_{2^k-1}(\log x)+O(x^{-{\theta }_k+\epsilon })$$ where $P$ is a polynomial of degree $2^k-1$ and ${\theta }_k=\frac{2^k}{(k+1{)}^{\frac{k+1}{2}}}\cdot \frac{3}{2^k+6k-5}$. This was conjectured in Hilberdink, Luca, and Tóth~\cite[Remark 2.4]{HLT}. We also prove asymptotic formulas for similar sums conjectured there.

16 pages, 2 figures, an error in the previous version in section 3.3 is fixed, the results are made weaker

References in corpus (1)

• Pairwise Relative Primality of Positive Integers