Elementary Methods in the Analytic Theory of Numbers

1967 ◽  
Vol 35 (2) ◽  
pp. 201
Author(s):  
P. A. B. Pleasants ◽  
A. O. Gel'fond ◽  
Yu. V. Linnik
Keyword(s):  
Analytic Theory ◽  
Elementary Methods ◽  
Theory Of Numbers

Some Applications of Fourier series in the Analytic Theory of Numbers

1928 ◽  
Vol 24 (4) ◽  
pp. 585-596 ◽  
Author(s):  
L. J. Mordell
Keyword(s):  
Fourier Series ◽  
Class Number ◽  
Lattice Point ◽  
Zeta Functions ◽  
Gauss Sums ◽  
Analytic Theory ◽  
Quadratic Fields ◽  
Lattice Point Problems ◽  
Theory Of Numbers

It is a familiar fact that an important part is played in the Analytic Theory of Numbers by Fourier series. There are, for example, applications to Gauss' sums, to the zeta functions, to lattice point problems, and to formulae for the class number of quadratic fields.

Ω theorems for the complex divisor function

1994 ◽  
Vol 115 (1) ◽  
pp. 145-157
Author(s):  
R. R. Hall
Keyword(s):  
Analytic Theory ◽  
Divisor Function ◽  
Image Position ◽  
Theory Of Numbers

This paper is a sequel to [6] and concerns the complex divisor functionwhich has had a number of applications in the analytic theory of numbers. Thus Hooley's Δ-function [8] defined bysatisfies the inequalityand Erdös' τ+-function, defined bysatisfies

Partition theory and thermodynamics of multi-dimensional oscillator assemblies

1951 ◽  
Vol 47 (3) ◽  
pp. 591-601 ◽  
Author(s):  
V. S. Nanda
Keyword(s):  
Generating Function ◽  
Statistical Thermodynamics ◽  
Analytic Theory ◽  
Close Similarity ◽  
Thermodynamic Approach ◽  
Large Integer ◽  
Partition Problem ◽  
Partition Theory ◽  
Theory Of Numbers

The close similarity between the basic problems in statistical thermodynamics and the partition theory of numbers is now well recognized. In either case one is concerned with partitioning a large integer, under certain restrictions, which in effect means that the ‘Zustandsumme’ of a thermodynamic assembly is identical with the generating function of partitions appropriate to that assembly. The thermodynamic approach to the partition problem is of considerable interest as it has led to generalizations which so far have not yielded to the methods of the analytic theory of numbers. An interesting example is provided in a recent paper of Agarwala and Auluck (1) where the Hardy Ramanujan formula for partitions into integral powers of integers is shown to be valid for non-integral powers as well.

scholarly journals Elementary methods in the theory of numbers

1939 ◽  
Vol 31 ◽  
pp. xvi-xxiii
Author(s):  
S. A. Scott
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Exponential Function ◽  
Monotone Sequence ◽  
Binomial Theorem ◽  
Irrational Numbers ◽  
Early Stages ◽  
Elementary Methods ◽  
Usual Notation ◽  
Theory Of Numbers

§ 1. The importance of proving inequalities of an essentially algebraic nature by “elementary” methods has been emphasised by Hardy (Prolegomena to a Chapter on Inequalities), and by Hardy, Littlewood and Polya (Inequalities). The object of this Note is to show how some of the results in the early stages of Number Theory can be obtained by making a minimum appeal to irrational numbers and the notion of a limit. We use the elementary notion of a logarithm to a base “a” > 1, and make no appeal to the exponential function. The Binomial Theorem is only used for a positive integer index. Our minimum appeal rests in the assumption that a bounded monotone sequence tends to a limit. We adopt throughout the usual notation. Finally, it need scarcely be added that the methods employed are not claimed to be new.

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