CHAP, vni] NUMBER OF ROOTS OF CONGRUENCES. 227 There exist exactly p— m— 2 roots of (2), distinct from one another and from zero, if and only if there exist exactly p—m—2 distinct linear homogeneous functions p~2 2 akthah (& = !,. . ., p-ro-2) A=O which remain divisible by p after applying all cyclic permutations of the ah, so that A simple proof of this corollary is given. L. Gegenbauer29 noted that the number of roots of f(x)z=Q (mod k) is since D(k) = 1 or 0 according as/(x) is divisible by k or not. Let fcj, . . . , k8 be a series of increasing positive integers and g(x) any function. In the first equation take k = kh multiply by g(kt) and sum for I = 1, . . . , 5. Reversing the order of the summation indices I, x in the new right-hand member, we get S \f(x), kl\g(kl] = 2 0, G •where in (r the summation index JJL takes those of the values /c1; . . . , ks which exceed x. Thus G .represents the sum G(f(x)'} klt . . ., ks; x) of the values of /(p) when /z ranges over those of the numbers ki, . . . , ks which exceed x and are divisors olf(x). In particular, if g(x) = 1, (r becomes the number ^ of the k's which exceed x and divide /(x). Let f(x)=m=*=nx. Then f(x) = 0 (mod fc) has (k, n) roots or no root according as m is or is not divisible by the g. c. d. (k, n) of k and n; let (k, n] m) denote (k, n) or 0 in the respective cases. Then 6 k-l ; klt . . ., k5; x). Let G(a, 6) denote the sum of the values of g(tL) when M ranges over all the divisors >b of a; ^(a, 6) the number of divisors >6 of a. Taking ki = I for I — 1, . . . , 5, we deduce 2 (Z, n; ?n)0(Z) = S {ff(m=t=nx, x) -G(m=fcna;, 5) [ . /-I z-O For g(l) = 1, this reduces to Lerch's100 relation (16) in Ch. X. Again, +a;) = S G(m~n^^-G(m-n "Sitzungsberichte Ak. Wiss. Wien (Math.), 98, Ila, 1889, 28-36.