320 HlSTOKY OF THE THEOEY OF NUMBERS. [CHAP. X
and hence deduced the law of a recursion formula for An. The law of a recursion formula for Bn = 4 { T3(ri) — TI (n) } is obtained from
2 BnsBn 2 s(2n+1)' cos(2n+l)~= 2 (2n+l)s(2n+1)2sin(2n+l)7,
n«0 n=0 4 «»0 4
with J50= 1, which was found by use of Jacobins 0(J, s). Next,
is shown to satisfy the functional equation
If a convergent series 2cnsn is a solution <£(s) of the latter, the coefficients are uniquely determined by the c4k,B(k = 1, 2, . . . ) , which are arbitrary. Hence the function Bn is determined for all values of n by its values for 7i = 4fc~ 3 (fc = l,2,...).
S. Wigert149 proved that, for sufficiently large values of n, r(n)<2', where <=(l+e) log n-^log log n, for every e >0; while there exist certain values of n above any limit for which r(n) > 2*, 5 = (1 — €) log n -r-log log n.
J. V. Pexider150 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.60 Taking a = [-\/n] , we have the second equation (11). Proof is given of the first equation (11) and
where d ranges over the divisors of [n].
0. Meissner151 noted that, if m =piei. . .pnCn, where pi is the least of the distinct primes pi, . . . , pn> then
i — 1 m log m
where 6r is finite and independent of m. If fc> 1, ak(m)/mk is bounded.
W. Sierpinski152 proved that the mean of the number of integers whose squares divide n, of their sum, and of the greatest of them, are TC 1 , ,3^ 3 . , 9C . 36 " log s
""
respectively, where C is Euler's constant.
J. W. L. Glaisher153 derived formulas differing from his110 earlier ones only in the replacement of d by ( — l)d~ld, i. e., by changing the sign of each
149Arkiv for mat., aat., fys., 3, 1906-7, No. 18, 9 pp.
15QRendiconti Circolo Mat. Palermo, 24, 1907, 58-63.
»lArchiv Math. Phys., (3), 12, 1907, 199.
152Sprawozdania Tow. Nank. (Proc. Sc. Soc. Warsaw), 1, 1908, 215-226 (Polish).
163Proc. London Math. Soc., (2), 6, 1908, 424r467.