328 
h=s == 
3 09 _ FT (1 se Jer 
djg d h=1 Ph 
and therefore, if «—41, for at least one value of & other than zero 
_L(a@0*) must tend to infinity. 
3. It will be readily seen that the method used in obtaining the 
asymptotic expansion of L(z) can be applied also to the series 
n= zn 
NEE bu, 
if only the coefficient 6, is a simple analytical function of the index 
n. If we choose, for instance, 
1 
bn = _- Sa) 
n 
we have 
n= | n n= o 
M (2) = — J — = = log (l—z") = log M(1— 2") 
n= N Wen n=1 it 
and we can put 
h=q—1 
M (« Or) — M (er) = logg+ = Vi (e,p), 
it 
where 
Vile ap) = = {log (L—ara +h Oly) — log (1—2"dGhp) }. 
n=0 
Operating as before we shall find 
| A=q—1 
M (« Or) — M (a?) =logg—— = 
7 
h h 
h log 2 sin? | + zina) 
= | en q 
jl 1 I 1 k=2m—1 1 k ; I ili k ‘ 1 \2m 
en uses = Al! ed A oaf ‚gam 
ni eee (3) ators) + ua (10g ) 
x 
where 
h=q—-1 
h 
Ay aie (~) (D&—-) cote) zap, 
h=1 q U pie 
and K, has a finite value independent of «. 
Now for Eurer’s product ILA”) we have, when 0<2 <1), 
1 
4n3 
— 
] 1 : 1 se" lege 
AES tr rr SORT ADE — + Mle *), 
6 log — 
z 
