14 Mr Landau, Note on Mr Hardy's extension of a theorem, etc. 



Note on Mr Hardy's extension of a theorem of Mr Poly a. 

 By Edmund Landau. (Communicated by Prof. G. H. Hardy.) 



[Received 10 December 1919. Read 26 January 1920.] 



In a recent note in these Proceedings* Mr Hardy has estab- 

 lished an improved form of a theorem of Mr Pdlya, viz. : 



Suppose that g {x) is an integral function, and M (r) the maxi- 

 mum of I ^ (^) I for \x\^r. Suppose further that g {x) is an integer 

 for x = 0, 1, 2, 3, ..., and that 



M{r)=o{2'). 

 Then g (x) is a polynomial. 



As Mr Hardy remarks at the beginning of his note, it is 

 sufficient (after the analysis given already by Mr Pdlya), to prove 

 the one formula 



^,2-r _ — ^ — =0(1). 



— TT 



n (2n - s cos 6) 



Mr Hardy's proof of this formula may be replaced by the following 

 shorter proof 

 Since 



nl2^n^ = nl2^n(^^-^,2^nn^l=OWn), 



n(2n-s) ^ ^' 



s=l 



it is enough to prove 



'[ir{e,n)de==o{^), 



where 



ylr(e,n)=u( ^ ^^ ~ ' . 

 ^ ^ ' s=i\2n — scos6 



Now 



l-cos6/_2-24 + ---^2 24^2 24 ~ 12 ' 

 for — TT < ^ ^ TT, and 



1 + 2/ 



* Vol. XIX. (1919), pp. 60-63. 



