On the Average Order of the Divisor Function, Lattice Point Functions, and Other Arithmetical Functions

Hafner, James Lee

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Description

Title

On the Average Order of the Divisor Function, Lattice Point Functions, and Other Arithmetical Functions

Author(s)

Hafner, James Lee

Issue Date

1980

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

Mathematics

Language

eng

Abstract

• We consider a large class of arithmetical functions generated by Dirichlet series satisfying a very general functional equation with gamma factors. We obtain one and two-sided (OMEGA)-theorems for the error terms associated with certain weighted averages of these arithmetical functions. For example, if {a(n)} is such an arithmetical function generated by the Dirichlet series* for (//R) e s sufficiently large, then the weighted summatory function A(,(rho))(x) is defined for (rho) (GREATERTHEQ) 0 by*
• An asymptotic equivalent Q(,(rho))(x) is easily obtained by standard techniques of contour integration. We are concerned with the order of magnitude of the error terms P(,(rho))(x) defined by P(,(rho))(x) = A(,(rho))(x) - Q(,(rho))(x).
• For all the classical problems, our theorems yield results which are at least as good as those previously obtained by other authors. In particular, for the Dirichlet divisor problem and the problem of lattice points in a circle, our results are new. We describe briefly these results.
• The error terms, when (rho) = 0, for these two problems are denoted by (DELTA)(x) and P(x), respectively. The problem of estimating the exact order of magnitude of these error terms dates back to the 1850's. It is easily shown that they are both O(x('1/2)). Many authors have since reduced this exponent of 1/2. The best estimate of this type known to date, for (DELTA)(x), is (DELTA)(x) = O(x('a)(log x)('b)) where a = 346/1067 and b = 211/100. Hardy and Landau first studied the (OMEGA)-problem. They showed that both (DELTA)(x) and P(x) are (OMEGA)(,(+OR-)) (x('1/4)). Later, in 1916 Hardy gave better one-sided results, namely, (DELTA)(x) = (OMEGA)(,+) ((x log x)('1/4) loglog x) and P(x) = (OMEGA)(,-) ((x log x)('1/4)).
• Up to the present time, Hardy's results have not been improved. We, however, have improved both of these results by a power of loglog x. Thus we show that there exist positive absolute constants A and B such that (DELTA)(x) = (OMEGA)(,+) ((x log x)('1/4)(loglogx('(3+2 log 2)/4). (.)exp{-ASQRT.(logloglogx}), (1) and P(x) = (OMEGA)(,-) ((x log x)('1/4)(loglog x)('(log 2)/4). (.)exp{-BSQRT.(logloglogx}). (2) We have two proofs of these results. The first proof follows along the same lines as those of Hardy, but there are significant differences. Dirichlet's approximation theorem is applied only in those instances where the arithmetical functions involved are "large". One must first determine precisely how to interpret quantitatively this concept of "large". Then, the sums that arise from this are considerably more difficult to estimate than Hardy's sums.
• Results (1) and (2) are also obtained as special cases of our general (OMEGA)-theorems. Furthermore, we obtain new results in these problems when (rho) > 0, and new results of this type in the more general Piltz divisor problem when (rho) (GREATERTHEQ) 0. It appears that with all the methods used to date, our results in these particular problems are the best one can obtain.
• *Please refer to dissertation for diagrams.

Graduation Semester

1980

Type of Resource

text

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