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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

The number of distinct subsums of $ \sum \sb{1}\sp{N}\,1/i$


Authors: M. N. Bleicher and P. Erdős
Journal: Math. Comp. 29 (1975), 29-42
MSC: Primary 10A25
MathSciNet review: 0366795
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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

$\displaystyle S(N) \geqslant e\left( {\frac{{N\log 2}}{{\log N}}\prod\limits_3^{k + 1} {{{\log }_j}N} } \right)$

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

$\displaystyle \frac{N}{{\log N}}\prod\limits_{i = 1}^{k + 1} {{{\log }_i}N \leq... ...}}N}}} \right)} \frac{N}{{\log N}}\prod\limits_{i = 3}^{k + 1} {{{\log }_i}N} .$

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{displaymath}\begin{array}{*{20}{c}} {{e_0}(x) = x,} \hfill & {{e_{i + 1}}... ...\log }_{i + 1}}x = \log ({{\log }_i}x),} \hfill \\ \end{array} \end{displaymath}

where $ \log x$ denotes the logarithm to the base e.

References [Enhancements On Off] (What's this?)

  • [1] M. N. Bleicher, A new algorithm for the expansion of Egyptian fractions, J. Number Theory 4 (1972), 342–382. MR 0323696 (48 #2052)
  • [2] M. N. BLEICHER & P. ERDÖS, "Denominators of Egyptian fractions. II" (To appear.)
  • [3] M. N. BLEICHER & P. ERDÖS, "The number of distinct subsums of $ 1 + 1/2 + 1/3 + 1/4 + \cdots + 1/N$ Notices Amer. Math. Soc., v. 20, 1973. Abstract #706-10-3.
  • [4] J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 0137689 (25 #1139)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1975-0366795-4
PII: S 0025-5718(1975)0366795-4
Keywords: Harmonic series, squarefree integers, prime numbers, distinct subsums
Article copyright: © Copyright 1975 American Mathematical Society