324 
History of the Theory of Numbers. 
[Chap. X 
. . . ^flp, while ap= 1 except when A'' = 4 or 36. The value of X for which 
a2>ai> . . . >ax is investigated at length. The ratio of two consecutive 
highly composite numbers A'' tends to unity. There is a table of A"s up 
to t{N) = 10080. An N is called a superior highly composite number k 
there exists a positive number e such that 
N.^ 
N' = Ni^ 
for all values of A^ and No such that A'2> A'> A^i. Properties of t{N) are 
found for (superior) highly composite numbers. 
Ramanujan^" gave for the zeta function (12) the formula 
and found asjTnptotic formulae for 
j=i 
S r'(j), 
=1 
Sr(jr+c), 
S(7„(j>i(i), 
A(n), 
for a = or 1 , where 
A(n) = S^r(i.) =SM(d)r (0Z),(^), 
summed for the di\isors d of v. If 5 is a common di\isor of u, v, 
xM=iMW.g)rg)=2.W.@x(0. 
E. Landau^'^^'' gave another asjinptotic formula for the number of de- 
compositions of the numbers ^ x into k factors, A' ^ 2. 
Ramanujan^'* wrote c^O) =^^("5) and proved that 
2,,(,>.(n-,) ^^Pm^ ■ ^^l^lV 
J=0 
r(r+s + 2) f(r+s+2) ^^+*+^ 
r(i-r)+r(i-^) 
(n) 
/Z(r,+,_i(n)+0(n2'^+«+^^/^), 
for positive odd integers r, s. Also that there is no error term in the right 
member if r=l, s = 1, 3, 5, 7, 11; r = 3, s=3, 5, 9; r = 5, s = 7. 
J. G. van der Corput^"^ wrote s for the g. c. d. of the exponents ai, a-z,... 
in m='n.pi''i and expressed in terms of zeta function f(i), i=2, . . ., k-\-l, 
2 {a,{s)-l]/m 
m=2 
if A' > 1 ; the sum being 1 — CifA=— 1, where C is Euler's constant. 
"'Messenger Math., 45, 1915-6, 81-84. 
"'"Sitzungsber. Ak. Wiss. Miinchen, 1915, 317-28. 
i^Trans. Cambr. Phil. Soc, 22, 1916, 159-173. 
"•Wiskundige Opgaven, 12, 1916, 116-7. 
