er — , ‘ iy prponiz 1 
0 0 
1 1 1 
5 Lz af 5 i EN ir 
= AO) vee — + of ( -— Ct cot EA dy—xia | (f(y)—f(O)) dy 4- 
ee esd J \Y 
0 0 0 
; 1 
0 he 1 
— ise 1) ) a v—| Ty) 6 — xa cotg ze Jay 
a 2 
0 0 
1 
1 
(0 
=f (0) log 2ma +} zi f (0) — zie f. fy “f! ola os ) 
0 
1 
- fi Ty) (- — xa cotg ze) dy. 
0 
As for n integer and positive 
ee) 
i eha 0 COS U 1et 2 SIM Zare 
| vj de gmt gere 
(n ) ezp2niz — | pee yn ah zn 
0 
eo cos 2x7Tet wo sin Zune 
many relations and: > > are 
- 1 yn 1 xl 
to be deduced from this formula. But we will restrict ourselves 
to the ¢-function and therefore write f(y) = (J—y)", in which » 
represents an integer positive number; if the real part of acomplex 
number y is indicated by R(y) it ensues from what preceded that 
4 \2n 
o Ri 1 +——]} —1 
f ( a 2) Ge ard on ( | \ 
3 geel ine F (2ror rn |) ‘i 
0 0 
2n. 1 
S= log 21a —— = —— Emek as is se, (3) 
1 % 
In order to find relations for the &-function by means of this 
formula, the following auxiliary proposition may be used: 
If a represents an integer number > 2 and o describes half the 
reduced rest system, modulo a, between O and a, and that in such 
a way that the series of the numbers, of which the values are 
successively assumed by gv, does not contain two numbers, the sum 
of which is equal to a, then : | 
