Chap. VIII] 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 homo- 
geneous functions 
p-2 
Xak,hah {k = l,. . .,p-m-2) 
h=0 
which remain divisible by p after applying all cyclic permutations of the 
ah, so that 
sV»a,„^0 (mod p) filo/iV.'.^.T™-!)- 
A simple proof of this corollary is given. 
L. Gegenbauer^^ noted that the niunber of roots of /(a^)=0 (mod k) is 
since D{k) = 1 or according as /(a:) is divisible by k or not. Let ki,...,ka 
be a series of increasing positive integers and g{x) any function. In the 
first equation take k = ki, multiply by g{ki) and sum iorl = l,. . .,8. Revers- 
ing the order of the summation indices I, x in the new right-hand member, 
we get 
i \f(x), h\g{k) = S G, G=XD{ix)g{f,), 
1=1 1=0 
where in G the summation index n takes those of the values ki,...,ki which 
exceed x. Thus G represents the sum G{f{x)] ki,..., k^; x) of the values 
oi fifx) when ij, ranges over those of the numbers ki,. . ., ks which exceed x 
and are divisors oif{x). In particular, if ^(x) = 1, G becomes the number x}/ 
of the k's which exceed x and divide /(a:). 
Let f{x)=vi^nx. Then f{x) = (mod k) 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, 71] m) denote {k, n) or in the respective cases. Then 
S {ki, n; m) g{ki) = s G(m^nx; ki,. . ., k^; x). 
'-1 x=0 
Let G{a, h) denote the sum of the values of g(ii) when n ranges over all 
the divisors >6 of a; xl/{a, h) the number of divisors >b of a. Taking 
ki = l ioT 1=1,. . ., d, we deduce 
S 5-1 
S (Z, n; m)^(0= S {G{m^nx, x)-G{m^nx, b)\. 
1=1 z=0 
For gil) = 1, this reduces to Lerch's^°° relation (16) in Ch. X. Again, 
a b 
2 {G{m+nx, x — 1) —Gim+nx,h+x)\ = S {G{m—nfx,tx) — Giin—nfx,ti-\-a)\, 
X-l u=0 
"SitzuDgsberichte Ak. Wiss. Wien (Math.), 98, Ila, 1889, 28-36. 
