The number of distinct subsums of $\sum _{1}^{N} 1/i$

Abstract:

In this paper we improve the lower bounds for the number, $S(N)$, of distinct values obtained as subsums of the first N terms of the harmonic series. We obtain a bound of the form \[ S(N) \geqslant e\left ( {\frac {{N\log 2}}{{\log N}}\prod \limits _3^{k + 1} {{{\log }_j}N} } \right )\] whenever ${\log _{k + 1}}N \geqslant k + 1$, for $k \geqslant 3$. Slight modifications are needed for $k = 1,2$. We begin by discussing the number ${Q_k}(N)$ of integers $n \leqslant N,n = {p_1}{p_2} \cdots {p_k}$, where ${p_i} > {e^{\alpha {p_{i - 1}}}},i = 2, \cdots ,k$. We prove that \[ \frac {N}{{\log N}}\prod \limits _{i = 1}^{k + 1} {{{\log }_i}N \leqslant {Q_k}(N) \leqslant \left ( {1 + \frac {k}{{{{\log }_{k + 1}}N}}} \right )} \frac {N}{{\log N}}\prod \limits _{i = 3}^{k + 1} {{{\log }_i}N} .\] This bound is valid for ${\log _{k + 1}}N \geqslant k + 1$ and for $1 \leqslant \alpha \leqslant 2(1 - {e_2}(4)/{e_3}(4))$. The symbols ${\log _i}x$ and ${e_i}(x)$ are defined by \[ \begin {array}{*{20}{c}} {{e_0}(x) = x,} \hfill & {{e_{i + 1}}(x) = {e^{{e_i}(x)}},} \hfill \\ {{{\log }_0}x = x,} \hfill & {{{\log }_{i + 1}}x = \log ({{\log }_i}x),} \hfill \\ \end {array} \] where $\log x$ denotes the logarithm to the base e.