320 History of the Theory of Numbers. [Chap, x 
and hence deduced the law of a recursion formula for A„. The law of a 
recursion formula for B„ = 4{T2{7i) — Ti{n)\ is obtained from 
S ^y S s''"+'^' cos(2n+l)^= i; (2n+l)s<2n+i)'sijj(2n+l)7, 
n=0 n=0 4 n=0 4 
with Bo=l, which was found by use of Jacobi's d{\, s). Next, 
is shown to satisfy the functional equation 
$(ts)=lj$(s)-$(-s)}-$(s2)+2$(s*). 
If a convergent series liens'" is a solution $(s) of the latter, the coefficients are 
uniquely determined by the C4i_3(/j = 1, 2, . . . ), which are arbitrary. Hence 
the function 5„ is determined for all values of n by its values forn = 4/c— 3 
(A: = l, 2,...). 
S. Wigert^^^ proved that, for sufficiently large values of n, r(n)<2', 
where f = (l + e) log n^-log log n, for every e >0; while there exist certain 
values of n above any limit for which riri) >2', s = (1 — e) log n -i-log log n. 
J. V. Pexider^^° proved that, if a, n are positive, a an integer, 
by the method used, for the case in which n is an integral multiple of a, 
by E. Cesaro.^° Taking a = [Vn], we have the second equation (11). Proof 
is given of the first equation (11) and 
S.[g=2.W, 2[2][!^]=S<?-.W, 
where d ranges over the divisors of [n]. 
0. Meissner^" noted that, if m =pi". . .p/", where pi is the least of the 
distinct primes pi, . . . , 7?„, then 
,=iPj — 1 m ,=2Pt — 1 w log m 
where G is finite and independent of m. If /v> 1, (Tk{m)/rn!' is bounded. 
W. Sierpinski^^^ proved that the mean of the number of integers whose 
squares divide n, of their sum, and of the greatest of them, are 
x^ 1, .3^ 3 , , 9C , 36 - logs 
— , -logn+^C, -:2lognH — rA — -^Z ^^, 
respectively, where C is Euler's constant. 
J. W. L. Glaisher^^^ derived formulas differing from his^^° earlier ones 
only in the replacement of d by { — lY~^d, i. e., by changing the sign of each 
"»Arkiv for mat., ast., fys., 3, 1906-7, No. 18, 9 pp. 
»"Rendiconti Circolo Mat. Palermo, 24, 1907, 58-63. 
"'Archiv Math. Phys., (3), 12, 1907, 199. 
"^prawozdania Tow. Nank. (Proc. Sc. Soc. Warsaw), 1, 1908, 215-226 (Polish). 
i"Proc. London Math. Soc, (2), 6, 1908, 424-^67. 
