Chap. X] SuM AND NUMBER OF DiVISORS. 307 
@(m)=or(l)+(7(3)+(r(5) + . . .+(7(m)=S^[^^], 
where t ranges over the odd integers ^ m. For the K and L of Lipschitz^ 
and G{m) =(r(l)+o-(2) + . . . -\-<T(m), it is shown that 
LW^(?(m)^[Vm] + [^|], !r(m)^[v^] (mod 2). 
J. Hacks^^ gave a geometrical proof of (11) and of Dirichlet's^^ expression 
for T{n), just preceding (7). He proved that the smn of all the divisors, 
which are exact ath powers, of 1, 2, . . . , m is 
m 
S{1^+2«+...+[a/^H. 
3 = 1 
He gave (pp. 13-15) several expressions for his^^ i^M, &(m), K{m). 
L. Gegenbauer^^" gave simple proofs of the congruences of Hacks. ^^ 
M. Lerch^^ considered the number \p{a, h) of divisors >6 of a and proved 
that 
[n/2] n 
(15) X \}/{n—p, p) =71, Si/'(n4-p, p)=2n. 
p=0 p=0 
A. Strnad^^ considered the same formulas (15). 
M. Lerch^°° considered the number x(«j b) of the divisors ^6 of a and 
proved that 
[(m-l)/a] 
S {\J/{m—aa,k+a)—xi'm—aa,a)] 
<7 = 
k 
+ S {\J/{m+\a, X-l)-x(m+Xa, a)) =0. 
x=i 
This reduces to his (15) for a = l, k = l orm + 1. Let (k, n; m) denote the 
g. c. d. {k, n) of k, n or zero, according as {k, n) is or is not a divisor of m. 
Then 
a— 1 a 
(16) S {i/'(m4-an, a)— ;/'(m+an, a)] = S (A;, n; m). 
In case m and n are relatively prime, the right member equals the number 
0(a, n) of integers^ a which are prime to n. Finally, it is stated that 
(17) S 1/^(772 — an, a) = S x(^ — ctn, n), c= . 
a = a = L n J 
Gegenbauer,^^ Ch. VIII, proved (16) and the formula preceding it. 
"Acta Math., 10, 1887, 9-11. 
"oSitzungsber. Ak. Wiss. Wien (Math.), 95, 1887, II, 297-8. 
"Prag Sitzungsberichte (Math.), 1887, 683-8. 
"Casopis mat. fys., 18, 1888, 204. 
""Compt. Rend. Paris, 106, 1888, 186. Bull, des sc. math, et astr., (2), 12, 1, 1888, 100-108, 121-6. 
