History Of The Theory Of Numbers - I

444 HlSTOBY OF THE THEOKY OF NUMBERS. [CHAP. XIX
Those divisible by ai and 02, but by no one of a3) . . . , ak, form the sub-set /(03, . . ., OA). Finally, those divisible by ai. . ., ak form the sub-set designated /(O). Thus
F(alt 02, .--,«*) =/(<*!, 02, • • . , «*) +S/(02, 03, . . . , a*) +|/(os, 04, . . . , ak)
where 2 indicates that the summation extends over all combinations of 01, . . . , or taken* &— r at a time.
When we have any such set or function /(o1; . . . , a*), uniquely determined by ab . . . , ak, independently of their order, and we define F by the foregoing formula, then we have the inverse formula
, 02, . . . , a,) =/?7(a1, 02, . . . , an) -2F(a2, a3, . . . , an) +SF(a3, a4, . . . ,an)
2
n-l
where 2J now indicates that the summation extends over all the combina-
r
tions of Oj,. . ., an taken n— r at a time. The proof is just like that by B. Merry for Dedekind's formula.
To give an example, let n = 2, al=2J a2 = 3, £=3, 4, 6, 8. Then F(al) =3, 6; /(aO =3, /(O) =6; F(a2) =4, 6, 8; /(a2) =4, 8. Thus
F(<h,cd-F(aJ-F(ad+F(Q)=S-(Z, 6) -(4, 6, 8)+6=0=/(al, a2).
A. Berger17a called /i conjugate to /2 if S/i(d)/2(d) = 1 for ft = 1, 0 for &> 1, when d ranges over the divisors of k. Let g(mn)—g(m)g(n)f ^(l) = l. Write h(k)=^f(d)fl(8)g(d), where dS = A;. Then f(k)=2f2(d)g(d)h(d). Dedekind^s inversion formula is a special case. For, if /x(n) = l, then f2(n)=n(ri).
K. Zsigmondy18 stated that if, for every positive integer r,
?/W=.F«,
where c ranges over all combinations of powers ^r of the relatively prime
positive integers nb . . ., np, while rc denotes the greatest integer ^r/c, then
/(r) =F(r) -2F(rn) + S, F^) - . . . ,
» WfTl
where the summation indices n, n', . . . range over the combinations of HI, . . . , nfi taken 1,2,... at a time.
R. D. von Sterneck19 noted that, if d ranges over the divisors of n =^(n) implies that
Taking m=l,. . ., n and solving, we get ^(n) expressed as a determinant of order n, whence
if n = p!aj. . .p/" and dp is derived from n by reducing p exponents by unity.
"Here and in the statement of the theorem occur confusing misprints for fc and n. 17aNova Acta Regiae Soc. Sc. Upsaliensis, (3), 14, 1891, No. 2, 46, 104. "Jour, fiir Math., Ill, 1893, 346. Applied in Ch. V, Zsigmondy.77
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