scholarly journals The definite integral $\int\limits_{ - \infty }^\infty {\tfrac{{e^{ax^2 + bx} }}{{e^{ax} + d}}da} $ and the analytic theory of numbersand the analytic theory of numbers

1933 ◽  
Vol 61 (0) ◽  
pp. 323-360 ◽  
Author(s):  
L. J. Mordell
Keyword(s):  
Analytic Theory ◽  
Definite Integral ◽  
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 Corrigenda

1931 ◽  
Vol 27 (1) ◽  
pp. 25-25
Keyword(s):  
Fourier Series ◽  
Analytic Continuation ◽  
Analytic Theory ◽  
Theory Of Numbers

Some applications of Fourier series in the analytic theory of numbers.Page 589, equation (3–10), after “k>0” insert “and 0<R(s)< 1,” and for “2nπi/k” read “2nπix/k.”Page 589, equation (3·11), forAdd also ‘The evaluation of the integrals given in (3·11) is obvious when 0 < R (s) < 1, and then holds also for 0 < R (s) < 2 by the theory of analytic continuation.”

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