MATHEMATICS: HARDY AND LITTLEWOOD 
585 
when 0- ^ 0, the second of these formulae meaning^ that F (a) is not 
of the form o (o-~^), and the two together that 
0<h=]k^CTPF(<T)< 00. (3.2) 
These relations all hold uniformly in /. It follows that, if p > 0 and 
r = \ z\-^ ly the function f (z) is exactly of the order (1 — r)"^, and this 
uniformly in 6. Incidentally, of course, it follows that every point of 
the unit circle is a singular point: but this is known already.^ 
The series furnishes an example in which the orders in the unit circle 
of the functions f (z) = X aj^ and g (s) = S | a„ | / differ by exactly J, 
the maximum possible.^ 
When p = 0, / (z) is bounded, but does not tend to a limit when z 
approaches any point of the unit circle along a radius vector. We know 
of no other example of a function possessing this property. When p < 0, 
/ (2) is continuous for | s | < 1 . 
4. Let 
» 
,p-\ ^otik log k + Idirik . (4 1) 
and suppose first that p > 0. Then it is easy to deduce from the results 
of §3 that Sn is of the form 12 {n^) when n-^ ^ . The corresponding 
^O' result lies a little deeper: all that can be proved in this manner is^ 
that s„ = 0 (n^ log n). But a direct investigation, modelled on that of 
the early part of our second paper in the Acta Mathematica, shows that 
the factor log n may be omitted. It should be observed that an essential 
step in our argument depends on an important lemma due to Landau, ^ 
according to which 
^7 ^z'^ log (7;;.)^^ 
<23X^+^ (4.2) 
for Z > 1. 7 >0, >0. We thus find that Sn is, for every positive value 
of a, exactly of the order n^, and this uniformly in d. The series 
^ain log n + lOTrin 
is never convergent, or summahle by any of Cesdro^s means. 
When p = 0, s„ is bounded, but the series is never convergent or 
summable. When p < 0 it is convergent; and uniformly in 6. 
5. For further applications it is necessary to consider the real and 
imaginary parts of our function and series separately, and this is most 
easily effected by introducing some restriction as to the value of a- 
Suppose that a is an integer, not of the form 4 ^ + L Thus we may take 
a = 2, a = 27r/log2. Then the results of §§3-4 hold for the real and 
