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.