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On the Distribution of Generating Functions

On the Distribution of Generating Functions 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 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Oxford University Press

On the Distribution of Generating Functions

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References (8)

Publisher
Oxford University Press
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/S002460939700386X
Publisher site
See Article on Publisher Site

Abstract

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

Journal

Bulletin of the London Mathematical SocietyOxford University Press

Published: Mar 1, 1998

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