Access the full text.
Sign up today, get DeepDyve free for 14 days.
R. Vaughan, T. Wooley (1995)
On a certain nonary cubic form and related equationsDuke Mathematical Journal, 80
G. Greaves (1994)
On the representation of a number as a sum of two fourth powersMathematical Notes, 55
(1995)
Sums of two kth powers
(1971)
On some rules of Laguerre’s, and systems of equal sums of like powers
C. Hooley (1981)
On Another Sieve Method and the Numbers that are a Sum of Two hth PowersProceedings of The London Mathematical Society
(1981)
The Hardy–Littlewood method (Cambridge
C. Hooley (1980)
On the numbers that are representable as the sum of two cubes.Journal für die reine und angewandte Mathematik (Crelles Journal), 1980
(1996)
II ’, J
Abstract Investigations concerning the generating function associated with the k th powers, fP(α)=∑1≤n≤Pe(αnk), originate with Hardy and Littlewood in their famous series of papers in the 1920s, ‘On some problems of “Partitio Numerorum”’ (see [ 7 , Chapters 2 and 4]). Classical analyses of this and similar functions show that when P is large the function approaches P in size only for α in a subset of (0, 1) having small measure. Moreover, although it has never been proven, there is some expectation that for ‘most’ α, the generating function is about P in magnitude. The main evidence in favour of this expectation comes from mean value estimates of the form ∫01|fP(α)|sdα˜Γ(12s+1)Ps/2. An asymptotic formula of the shape (1.2), with strong error term, is immediate from Parseval's identity when s = 2, and follows easily when s = 4 and k > 2 from the work of Hooley [ 2 , 3 , 4 ], Greaves [ 1 ], Skinner and Wooley [ 5 ] and Wooley [ 9 ]. On the other hand, (1.2) is false when s > 2 k (see [ 7 , Exercise 2.4]), and when s = 4 and k = 2. However, it is believed that when t < k , the total number of solutions of the diophantine equation x1k+···+xtk=y1k+···+ytk, with 1 ≤ xj , yj ≤ P (1 ≤ j ≤ t ), is dominated by the number of solutions in which the xi are merely a permutation of the yj , and the truth of such a belief would imply that (1.2) holds for even integers s with 0 ≤ s < 2 k . The purpose of this paper is to investigate the extent to which knowledge of the kind (1.2) for an initial segment of even integer exponents s can be used to establish information concerning the general distribution of fP (α), and the behaviour of the moments in (1.2) for general real s . 1991 Mathematics Subject Classification 11L15. © London Mathematical Society
Bulletin of the London Mathematical Society – Oxford University Press
Published: Mar 1, 1998
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.