298 SECTIONAL TRANSACTIONS.— A. 



Department of Mathematics (A*). 

 (Morning Session.) 



Mr. E. H. LiNFOOT. — A Problem in the Analytic Theory of Numbers. 



Waring's problem, perhaps the most famous in the modern theory of numbers, 

 deals with the representation of an integer as a sum of kth powers, k being fixed. 

 Evidently it is one of a whole series of problems ; given any class M of positive integers, 

 we may investigate the representation of an integer as a sum of ' 31 numbers.' Mr. 

 Evelyn and I have considered* the problem which arises when M is the class of numbers 

 not divisible by kth poivers, k as before being fixed for all. We ask : In how many ways 

 can a large number n be represented as the sum of s J/-numbers ? The answer is that 

 the number of representations is asymptotic to 



_L j^ 11 (^, , (-ir'\ n f,,j^)'^ 



X,s{k) (.s-l)! pt+„V (J'''-1)V^''|« V (P'-l)"'"' 



asw-^co. We prove this when s >^ 3 by an application of the powerful Hardy- 

 Littlewood method, which consists in investigating the function 



f(x) = S*-" {\x\ < 1) 



near its barrier of singularities |a;| = l. Here M runs through the ' Jf-numbers ' ; 

 the power series, which clearly has \x\ = 1 as its circle of convergence, cannot be continued 

 beyond this circle. When 5=2 the analytic method fails, but in this case we are able 

 to establish the result by an elementary though difficult argument. J 



* Hath. Zeitschrift 30 (1929), 433-448. J JoMrnaZ/MrJ/a^A. 164 (unpublished). 



Dr. L. S. BoSANQUET. — The Summability of Fourier Series. 



Let f{t) be integrable — Lebesgue and periodic with period 2tc, and let 



^(t) = i{f(x+t)+f{x-t)-2s}. 



Hardy and Littlewood and their pupils have proved : * 



I. If <p(<) -^ 0, (C, a), then the Fourier series of f{t) is summable (C, a + S) to s, 

 for t = X, where a ^ and 8 > 0. 



II. If the Fourier series oif(t) is summable (C, a) to s, for t = x, then m (t) — >- 0, 

 (C, a + 1 + S), where a > - 1 and S > 0. 



There are similar results for Allied series and Power series. The theorems remain 

 true if the Denjoy integral is employed, provided in I that a + S > 1. They are 

 false in certain cases with 8=0. 



A problem arises of defining a scale of two-parameter summability, reducing to 

 Cesaro summability when the second parameter is zero, and satisfving conditions of 

 consistency. 



A typical result is 



III. If (f(t)-^O, (CO), then the Fourier series of f{t) is summable (0, 1 +S) to 

 «, for t = X, where S > 0. 



This is false with S = 0. 



* The following definitions are assumed known : cp(i!) ->- I, (C, a), as ->- 0, where 

 7. > 0, means 



r a-i 



a I (1 - m) q) {tu)du ■ 



J 



■ ? as / -> ; 



cp(<) -^ I, (C, 0) means q5(<) ->- 1 in the elementary sense ; 

 2«/i is summable (C, a) to s, where a > 0, means 



( 1 - - ) a„ — >- a as M) ->- CO. 



2 



n<w 



