Mr Hardy, On a theorem of Mr 0. Polya 



61 



This he proves by observing that the modulus of J^^ does not 



exceed 



n\ M{r) ^ V{n + \)V{n) 



(r-l)(r-2)...(r-w) r(2n) ^ '' 



and by an application of Stirling's Theorem. In order to com- 

 plete the proof in this manner it is necessary to assume the 

 condition (1). 



If however we suppose only that 



lim2-'-ilf(r) = (2), 



or J'f(r) = o(2'-) (2'), 



the proof may be completed as follows. We have 



where ^= 2ne'^. Now 



\x — s\ = V(4?r — 4?2.s cos 6 + 8^)'^ 2n — s cos 

 for 1 <s <.n, so that 



de 



.){a)-2)...{a;-n)\\ ' 



U{x- s) 



1 



if cos ^ > 0, and 



> n (2n - s cos 6) = (cos ^f 11 (2« sec O-s) 



n (x - s) 

 1 



^ n (2n - s cos ^) = I cos ^ i" n (27* | sec ^ | + s) 

 1 1 



if cos ^ < 0. Hence 



where Kn = nl2^''j 



Jn = {Kn) + (X„), 



''' T( 2na-n) 

 r {2n(r) 



a'^dO, 



i„ = „,2»|'V„_£(2^iV,.«rf«, 



and a = sec 0. 



A straightforward application of Stirling's Theorem shows that 



uniformly in 6, where 



^ = <t> (0) = {2a - 1) log (2o- - 1) - 2a- log 2o- + log o" + 2 log 2, 

 ^ = ^ (^) = 2(T log 2o- - (2o- + 1 ) log (2a + 1) + log o" + 2 log 2. 



