327 
we have finally 
k=2mn—2 k+1 Be 
L (20?) —qIAet*) = — Hal) SE (5) (loes) ‘| 
T 
k=0 
| \ ml 
+ kK, gem (tog =) : 
wv 
and again A, is finite and independent of z. 
From (7) we conclude that, if # —1, the function Z (v/v) tends 
to infinity in the same manner as 
(7) 
1 
C — log log— — 2 log g 
wv 
ae) - —— +49, 
q log — 
. v 
and that 
Lim { L («6r) — gL («*)} = —1iq—1)—— = hcot—. (B) 
11 2q =d q 
Thus we have the rather remarkable result that it is only the 
real part of (v6?) that increases indefinitely in a manner quite 
independent of p. 
2. If we only wish to shew that, when « —1, the function 
L(v0*) cannot remain finite for all non zero values of &, an element- 
k=q-—1 
ary discussion of the sum > LZ (ef!) suffices to obtain this result. 
k=0 
We have at once 
k=q—1 n= o 
> L(#0k) =q = t(ng) ero, 
k==0 n= 
and, denoting by D the greatest common measure of n and q, we 
may substitute 
n q 
t = > (dara — 
ao (3) (3) 
a Lee Eno! ) Leers. 
thus DE 
k=0 
Hence, making « tend to 1, we get 
k=q—1 
Lim SL (Gey 
zi Lle) po ra d 
where  (d) denotes the number of integers less than d and prime 
to d. 
Supposing q to be equal to the product p, ! p‚?...pss, we have 
