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324                   HISTORY OF THE THBOBY OF NXJMBBBS.               [CHA*. x
. . .2>ap, while ap=l except when JV=4 or 36. The value of X for which a2>O3> . . . >a\ is investigated at length. The ratio of two consecutive highly composite numbers N tends to unity. There is a table of N's up to r(N) = 10080. An N is called a superior highly composite number if there exists a positive number e such that
for all values of NI and N2 such that N2>N>Ni.   Properties of r(N) are found for (superior) highly composite .numbers.
Ramanujan177 gave for the zeta function (12) the formula
and found asymptotic formulae for
2 r'( j),           2 r(jt>+c),           n<r.(j),           £ ^0>i(j),           D. (n),
-1                    y-l                          ;-l                    j-1
for a=0 or 1, where
D.(n) «
summed.for the divisors rf of v.   If 5 is a common divisor of li, v,
E. Landau177" gave another asymptotic formula for the number of decompositions of the numbers j£ x into k factors, k S 2. Ramanujan178 wrote «r,(0) =if (— «) and proved that
or.-   -   r(r+s+2)

for positive odd integers r, s.   Also that there is no error term in the right member if r=l, 5 = 1, 3, 5,7, 11; r = 3, s=3, 5, 9; r = 5, s = 7.
J. G. van der Corput179 wrote s for the g. c. d. of the exponents en, a2, . . . in m=*TLp*i and expressed in terms of zeta function f (i), i=2, . . . ,
if A;>1; the sum being 1 — C if fc= —1, where C is Euler's constant.
"'Messenger Math., 45, 1915-6, 81-84.. "wSitzungsber. Ak. Wiss. Mtinchen, 1915, 317-28. 178Trans. Cambr. Phil. Soc., 22, 1916, 159-173. "•Wiskundige Opgaven, 12, 1916, 116-7.