Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 298.10012
Autor: Bleicher, Michael N.; Erdös, Paul
Title: The number of distinct subsums of sum1N 1/i. (In English)
Source: Math. Comput. 29, 29-42 (1975).
Review: In this paper we improve the lower bounds given in ``Denominators of Egyptian fractions. II'', Ill. J. Math 20, 598-613 (1976; Zbl 336.10007) ans Notices Am. Math. Soc. 20, \# 706-10-3 (1973)] for the number, S(N), of distinct values obtained as subsums of the first N terms of the harmonic series. The estimates in J. Number Theory 8, 157-168 (1976; Zbl 328.10010) and the before mentioned articles were derived because the upper bound was needed for lower estimates of the denominators of Egyptian fractions. In this paper we concentrate on the lower bounds. We obtain a bound of the form S(N) \geq e ({N log 2 \over log N} prodk+13 logjN ) whenever logk+1N \geq k+1, for k \geq 3. Slight modifications are needed for k = 1,2; see Corollaries 1, 2, 3 and 4 for more details. In order to do this we begin by discussing the number Qk(N) of integers n \leq N, n = p1,p2 ... pk where pi > e\alpha pi-1, i = 2, ... ,k. We first prove that
{N \over log N} prod k+1i = 3 logiN \leq Qk(N) \leq (1+{k \over logk+1N} ) {N \over log N} prodk = 1i = 3 logiN. This bound is valid for logk+1N \geq k+1 and for 1 \leq \alpha \leq 2(1-e2(4)/e3(4)). The bounds on N and \alpha are for convenience in evaluating the range of validity and the constants in the inequality, not for essential reasons. The symbols logix and ei(x) are defined by eo(x) = x, ei+1(x) = eei(x), logox = x, logi+1x = log(logix), where log x denotes the logarithm to the base e.
Classif.: * 11A99 Elementary number theory
11A41 Elemementary prime number theory
11D61 Exponential diophantine equations
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