489 
Mathematics. — “On the values which the function §(s) assumes 
for s positive and odd.” By J. G. vaN DER Corput. (Commu- 
nicated by Prof. J. C. KrLUYver). 
(Communicated in the meeting of February 26, 1916). 
This article is intended to deduce some formulae that may be used 
to calculate $(s) for odd values of the argumentum s > 1; for this 
purpose we will express these ¢ functions in the quantities /, (m, 7), 
[, (m,n) and [(n, a), which have the following significance: 
1 
Eme nr= af (1 — y)" cotg my dy = 
0 
RER 1 : S B, a 
nt m(m-+ 1)... (m+n) “1 (2%) /(2%--m) (2x-+-m-+1)...(2x%+m-+n) 
for m and ” integer and positive, 
1 
siete, 
1, (mn) = 5 fo" — yy co dy = 
Ee 
1 
mlm +t Im) 4 (2x)!(2%-+m)(2% +m + 1). (2e 4+ m+n) 
for m and n ih aa m > 0 and n 20, 
& 
EEE 
Dol A 
hen 
I (n, a) Ly)" : gra cotg ™ dy =n! © Es an)" 
oef aes za alg nay ) dy Me (Oey Ona 
for 1 >> a > 0, and» integer > O and also for a = 1 and » integer >0. 
In order to connect the § function with the quantities /, (m,n), 
1, (m,n) and I (n, ea) we will use the method indicated by Professor 
Dr. J. C. Kivyver in the article: “Sur les valeurs que prend la 
fonction §(s) de Riemann pour s entier positif et impair’. (Bulletin 
des Sciences Mathematiques, 1896). 
If f(y) represents a polynomium in y, which becomes zero for 
y =O and for y=—1, the uniform function 
is holomorphous in the domain of the rectangle bounded by the 
lines c=0, y=0, «=8, y= 2x. By applying the theorem of 
Caucny and then putting 8 =o, we therefore find the relation 
