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PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Vuluffle 66, Numb,, 1. S~Pt,u,b~, 1977
SHORTER NOTES
The purpose of this department is to publish very short papers of an unusually elegant and polished Character. for which there is no other outlet.
DISTINCT SUMS OVER SUBSETS
F. RANSON, J. M. STEELE AND F. STENGER
ABsTRAcT. A finite set of integers with distinct subset sums has a precisely bounded Dirichlet series.
Let 1 < a, < a2 < . . . < a. be a set of integers for which all of the sums I" jejai, cj = 0 or 1, are distinct. It was conjectured by P. Erdds and proved by C. Ryavec that
< 2.
j., ai
We will show that for all real s > 0,
n
<
a, 12
The hypothesis clearly implies for 0 < x < 1 that
X 00 k > (1 + Xtr,)(1 + Xa2) . . . (1 + X, )
2 X
k O
*4 and
n
log(l X) > log(l + X"),
as was observed in 1
The crucial idea here is that the form llog xl,' dxlx is changed only by a constant factor under the substitution y = x', so integrating we have
flog(, _ x), log xle dx
0 X
n
>fllog(l +Y)11ogyl's 1
0 y i 1 a,
Received by the editors May 13, 1977. AMS (MOS) subject classifficaflow (1970). Primary I0J99, 10 01; Secondary 10A40, 10HOS.
0 Arnerican Mathmatical Society 1977
179
180 F. ~SON, J. M. STEELE AND F. STENGER
To calculate the first integral we substitute x = e u, expand log(l e'),
and integrate term by terin to obtain F(P + 1)~(,8 + 2), where ~ is the
Riemann zeta function. In the same way the second integral is found to be
F( P + 1) + 2)(1 (.1),8 `). These calculations are vali~ for all 6 > 1,
2
so the theorem follows.
BIBLIOGRAPHY
1. S. J. Benkowski and P. Erd6s, On weird and pseudoperfect nuntbers, Math. Comp. 20 (1974), 617 623.
DEPARTmENT OF MATHEMMICS, UNWERSM OF BRinsH CoLuhatA, VANcouvER, B. C, CANADA V6T 1W5
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