] SUM AND NUMBER OF DIVISOES. 323
P. Bachmann168 proved the final formula of Busche.147 K. Knopp169 studied the convergence of S&n£n/(l— of), including the series of Lambert7, and proved that the function defined in the unit circle by Euler's1 product (1) can not be continued beyond that circle.
E. T. Bell170 proved that, if P is the product of all the distinct prime factors of w, and \ is their number, and d ranges over all divisors of m,
= r(m) r(Pm) r(P2m). J. F. Steffensen171 proved that,90 if Ix denotes log x,
S. Wigert172 proved, for the sum n-s(ri) of the divisors of n, (1 - e)ec log log n< s(n) < (1 +c)ec log log n,
for e>0 and p(x) = z— [*]. For x sufficiently large,
(i-«) log x<^(a;)<(f+€) log a?. Besides results on 2s(x)(x— n)*, S*(n) log a;/n, he proved that
E. Landau173 gave corrections and simplifications in the proofs by Wigert.172
E. T. Bell174 introduced a function including as special cases the functions treated by Liouville,26"29 restated his theorems and gave others.
J. G. van der Corput175 proved, for p(d) as in Chapter XIX,
x/d
'
S. Ramanujan176 proved that r(N) is always less than 2k and 2', where90
for Li(x) as in Ch. XVIII, and for a a constant. Also, r(N) exceeds 2fcand 2* for an infinitude of values of N. A highly composite number N is one for which r(N)>r(n) when N>n; if A/r = 2a»30'. . .p^, then
««Archiv Math. Phys., (3), 21, 1913, 91.
169Jour. ftir Math., 142, 1913, 283-315; minor errata, 143, 1913, 50.
"0Amer. Math. Monthly, 21, 1914, 130-1.
1MActa Math., 37, 1914, 107. Extract from his Danish Diss., "Analytiske Studier med Anven-
delser paa Taltheorien," Kopenhagen, 1912. i»/6id., 113-140.
17*G6ttingsche gelehrte Anzeigen, 177, 1915, 377-414. 17*Univ. of Washington Publications Math. Phys., 1, 1915, 6-8, 38-44. 175Wiskundige Opgaven, 12, 1915, 182-L "«Proc. London Math. Soc., (2), 14, 1915, 347-409.