454 SECTIONAL TRANSACTIONS.—aAt, A*. 
Wednesday, September 13. 
Visit to Messrs. Taylor, Taylor and Hobson’s Optical Works, Leicester. 
DEPARTMENT OF MATHEMATICS (A*). 
Thursday, September 7. 
Dr. E. H. Lrnroot.—On the dissection of large numbers (11.0). 
Much of the recent work in the theory of numbers has been concerned 
with the representation of a positive integer as a sum of integers of specified 
type. The Waring and Goldbach problem: are of course the outstanding 
examples, but there are several others of a less formidable nature which are 
of considerable interest. One of these is the representation of a number 
as the sum of kth power-free numbers (numbers not containing any kth 
power greater than 1 as a factor) ; this problem yields up its main results 
to purely arithmetical arguments, though there are some cases in which 
the Winogradoff method, based on the Farey dissection of an interval, is 
needed to obtain the sharpest error terms. 
The theorems discussed are all of purely arithmetical nature. The 
following are two examples. (1) We ask whether every sufficiently large 
number can be diss cted into, say, a square and a quadratfrei number. 
Estermann showed that it can, and gave an asymptotic formula for the 
number of dissections. We then ask whether the dissection can still be 
made if the two parts are restricted to be ‘ almost in a given ratio ’—that 
is to say, whether for all 7 greater than some number mo the equation 
n = m? + q (q = quadratfrei) 
always has a solution satisfying 
m? = dyn + O(n 8); g =Agn + O(n" 4), 
where Aj, A» >O; Ay +A, =1; 0o<B<1. It will be shown that such is 
the case provided B <i, and an asymptotic formula for the number of 
representations will be given. 
(2) A similar theorem holds for dissections into two quadratfrei 
numbers almost in a given ratio; in this case the asymptotic formula is 
valid and significant for all values of B in the range (0, 1). 
Dr. L. S. Bosanquet.—The absolute summability of Fourier series (11.30). 
A series Lay is said to be absolutely summable (A) if Lanx” converges to 
f(x) for o<x<1 and f(x*) is of bounded variation in (o, 1). The sum is 
then _ lim of (*)- 
i= 
The Pasties series of an even function 9(¢), integrable L, is absolutely 
summable (A) to zero at the point t = 0 if, for example, 
(1) @a(t)/t is integrable L in (0, yn) for some « > 0, or 
(2) 9.(t) is of bounded variation in (0, 1) ie some « >0, and 9,(t) > 0 
as t-> o, where 
t 
Qa(t) = “| (t — u)*-1 o(u)du, «>0, 
° 
Pa(t) = et). 
The second condition includes the first. Special cases were given” by 
J. M. Whittaker and B. N. Prasad. j 
