584 MATHEMATICS: HARDY AND LITTLEWOOD 
of Taylor's series and trigonometrical series. We have since discovered 
that even simpler and more elegant illustrations may be derived from 
the series 
'^^^ ^airin log n + lQirin (12) 
This series behaves, for different values of the parameters a and 0, 
far more regularly than does the series (1.1). To put the matter roughly, 
the behaviour of the series does not, in its most essential features, depend 
upon the arithmetic nature of a. 
2. Our fundamental formula is 
Here fl^ > 1, p is real, and R {y) > 0. The formula becomes illusory 
when p is zero or a negative integer, but the alterations required are of 
a trivial character. The formula is easily proved by means of Cauchy's 
Theorem: similar formulae were proved by one of us in a paper pub- 
lished in 1907.2 
We now write y = a it, where t > 0, suppose that a -^0, and ap- 
proximate to the series of Gamma-functions by means of Stirling's 
Theorem. We thus obtain 
f£^e-'^-^f(z)=F(a)+^(a), (2:2) 
log a 
where 
CO 
f{z) = 2 nP-" z"; (2.21) 
1 
2- ^ = ^ , z = re^', r = e = alog( ^\ (2.22) 
log a' (loga)^+^ 
so that r-^1 when — ^o- 0; 
F(cr) = (2.23) 
0 
and <f> (a) is of one or other of the forms 
A+o(l),o(\og^yOia-''+*), 
according as p< |, p = or p >i. 
3. It is known^ that, if p>0, 
. F{<r)=0(a-''),F{<r) = a{<r-n> (3-1) 
