scholarly journals Some problems of analytic number theory III

1981 ◽  
Vol Volume 4 ◽  
Author(s):  
R Balasubramanian ◽  
K Ramachandra
Keyword(s):  
Number Theory ◽  
Abelian Group ◽  
Analytic Number Theory ◽  
Divisor Problem ◽  
Main Theme ◽  
Group Problem ◽  
Circle Problem ◽  
Analytic Number ◽  
International Audience

International audience The main theme of this paper is to systematize the Hardy-Landau $\Omega$ results and the Hardy $\Omega_{\pm}$ results on the divisor problem and the circle problem. The method of ours is general enough to include the abelian group problem and the results of Richert and the later modifications by Warlimont, and in fact theorem 6 of ours is an improvement of their results. All our results are effective as in our earlier paper II with the same title. Some of our results are new.

scholarly journals Shifted convolution and the Titchmarsh divisor problem over 𝔽 q [ t ]

2015 ◽  
Vol 373 (2040) ◽  
pp. 20140308 ◽  
Author(s):  
J. C. Andrade ◽  
L. Bary-Soroker ◽  
Z. Rudnick
Keyword(s):  
Number Theory ◽  
Finite Field ◽  
Function Field ◽  
Analytic Number Theory ◽  
Divisor Problem ◽  
Divisor Functions ◽  
Analytic Number

In this paper, we solve a function field analogue of classical problems in analytic number theory, concerning the autocorrelations of divisor functions, in the limit of a large finite field.

scholarly journals Mathematical Reminiscences: How to Keep the Pot Boiling

2013 ◽  
Vol Volume 34-35 ◽  
Author(s):  
K Ramachandra
Keyword(s):  
Number Theory ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Diophantine Equations ◽  
Analytic Number Theory ◽  
Transcendental Numbers ◽  
Analytic Number ◽  
The Riemann Zeta Function ◽  
International Audience ◽  
Riemann Zeta

International audience Analytic number theory deals with the application of analysis, both real and complex, to the study of numbers. It includes primes, transcendental numbers, diophantine equations and other questions. The study of the Riemann zeta-function $\zeta(s)$ is intimately connected with that of primes. \par In this note, edited specially for this volume by K. Srinivas, some problems from a handwritten manuscript of Ramachandra are listed.

scholarly journals Some problems of Analytic number theory IV

2002 ◽  
Vol Volume 25 ◽  
Author(s):  
R Balasubramanian ◽  
K Ramachandra
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Asymptotic Formula ◽  
Kernel Function ◽  
Generating Functions ◽  
Analytic Number Theory ◽  
Second Order ◽  
Analytic Number ◽  
International Audience ◽  
Error Terms

International audience In the present paper, we use Ramachandra's kernel function of the second order, namely ${\rm Exp} ((\sin z)^2)$, which has some advantages over the earlier kernel ${\rm Exp} (z^{4a+2})$ where $a$ is a positive integer. As an outcome of the new kernel we are able to handle $\Omega$-theorems for error terms in the asymptotic formula for the summatory function of the coefficients of generating functions of the ${\rm Exp}(\zeta(s)), {\rm Exp\,Exp}(\zeta(s))$ and also of the type ${\rm Exp\,Exp}((\zeta(s))^{\frac{1}{2}})$.

scholarly journals On the Half Line: K Ramachandra.

2013 ◽  
Vol Volume 34-35 ◽  
Author(s):  
Nilotpal Kanti Sinha
Keyword(s):  
Twentieth Century ◽  
Number Theory ◽  
Analytic Number Theory ◽  
Half Line ◽  
Biographical Note ◽  
Analytic Number ◽  
International Audience

International audience This is a short biographical note on the life and works of K. Ramachandra, one of the leading mathematicians in the field of analytic number theory in the second half of the twentieth century.

Titchmarsh's theorem of Hankel transform

2019 ◽  
pp. 11-24
Author(s):  
Mohamed-Ahmed Boudref
Keyword(s):  
Number Theory ◽  
Data Analysis ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Lipschitz Condition ◽  
Analytic Number Theory ◽  
Hankel Transform ◽  
Partial Differential ◽  
Analytic Number ◽  
Bessel Transform

Hankel transform (or Fourier-Bessel transform) is a fundamental tool in many areas of mathematics and engineering, including analysis, partial differential equations, probability, analytic number theory, data analysis, etc. In this article, we prove an analog of Titchmarsh's theorem for the Hankel transform of functions satisfying the Hankel-Lipschitz condition.

scholarly journals A Survey of Some Recent Developments on Higher Transcendental Functions of Analytic Number Theory and Applied Mathematics

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2294
Author(s):  
Hari Mohan Srivastava
Keyword(s):  
Number Theory ◽  
Special Functions ◽  
Applied Mathematics ◽  
Analytic Number Theory ◽  
Review Article ◽  
Mathematical Functions ◽  
Analytic Number ◽  
Recent Developments ◽  
Transcendental Functions ◽  
Mathematical Sciences

Often referred to as special functions or mathematical functions, the origin of many members of the remarkably vast family of higher transcendental functions can be traced back to such widespread areas as (for example) mathematical physics, analytic number theory and applied mathematical sciences. Here, in this survey-cum-expository review article, we aim at presenting a brief introductory overview and survey of some of the recent developments in the theory of several extensively studied higher transcendental functions and their potential applications. For further reading and researching by those who are interested in pursuing this subject, we have chosen to provide references to various useful monographs and textbooks on the theory and applications of higher transcendental functions. Some operators of fractional calculus, which are associated with higher transcendental functions, together with their applications, have also been considered. Many of the higher transcendental functions, especially those of the hypergeometric type, which we have investigated in this survey-cum-expository review article, are known to display a kind of symmetry in the sense that they remain invariant when the order of the numerator parameters or when the order of the denominator parameters is arbitrarily changed.

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