Chap. X] SuM AND NuMBEE OF DiVISORS. 323 
P. Bachmann^^^ proved the final formula of Busche.^^' 
K. Knopp^^^ studied the convergence of 26„a:V(l — 2:"), including the 
series of Lambert^, and proved that the function defined in the unit circle 
by Euler's^ product (1) can not be continued beyond that circle. 
E. T. BelP'° proved that, if P is the product of all the distinct prime 
factors of m, and X is their number, and d ranges over all divisors of m, 
6^Sr(d)T(^^ =r{m)T{Pm)T{P''m). 
J. F. Steffensen^'^^ proved that,^° if Ix denotes log x, 
S. Wigert^^^ proved, for the sum n's{n) of the divisors of n, 
(1 — €)e^ log log n< s{n) < {l-\-e)e^ log log n, 
^s{n) = '^x-^l^{x), rP{x) = x^ 1 + 2 Ip(^), 
for €> and p{x)=x — [x]. For x sufficiently large, 
(i-e) log x<xP{x)<(l+e) log x. 
Besides results on Ss(a^)(x— n)*, lls{n) log x/n, he proved that 
X ns{n)=^+xlh\ogx-rP{x)}+0{x). 
E. Landau^^^ gave corrections and simplifications in the proofs by 
Wigert."2 
E. T. Bell^^^ introduced a function including as special cases the functions 
treated by Liouville,^^"-^ restated his theorems and gave others. 
J. G. van der Corput^^^ proved, for ix(d) as in Chapter XIX, 
Sd'')u(d)So-„(A;)=x. 
S. Ramanujan^'^® proved that t{N) is always less than 2* and 2', where^" 
^ = lWV+« { (IsSpf '=^*-('°^ ^)+^f'°« iVe— .-!, 
for Li(x) as in Ch. XVIII, and for a a constant. Also, t(N) exceeds 2*'and 
2' for an infinitude of values of N. A highly composite number N is one 
for which TiN)>T{n) when N>n', if Ar = 2''^3"». . .p"p, then aa^as^ag^ 
"SArchiv Math. Phys., (3), 21, 1913, 91. 
"9Jour. flir Math., 142, 1913, 283-315; minor errata, 143, 1913, 50. 
""Amer. Math. Monthly, 21, 1914, 130-1. 
i"Acta Math., 37, 1914, 107. Extract from his Danish Diss., "Analytiske Studier med Anven- 
delser paa Taltheorien," Kopenhagen, 1912. 
"HUd., 113-140. 
"^Gottingsche gelehrte Anzeigen, 177, 1915, 377-414. 
"<Univ. of Washington PubUcations Math. Phys., 1, 1915, 6-8, 38-44. 
'"Wiskundige Opgaven, 12, 1915, 182-4. 
"oProc. London Math. Soc, (2), 14, 1915, 347-409. 
