Some applications of Laplace transforms in analytic number theory

Abstract

In this overview paper, presented at the meeting DANS14, Novi Sad, July3-7, 2014, we give some applications of Laplace transforms to analytic number theory. These include the classical circle and divisor problem, moments of |ζ( 1 2 + it)|, and a discussion of two functional equations connected to a work of Prof. Bogoljub Stanković.where I is an interval, and K 1 (s, t) is a suitable kernel function. If a problem involving the initial function f (t) can be solved by means of the transforms F (s), then by the inverse transformone can obtain information about f (t) itself. Here J is a suitable contour in C and K 2 (s, t) is another kernel function. Naturally the passage from (1.1) to (1.2) and back requires knowledge about the kernels, the convergence (existence) of the integrals that are involved, etc.

n-Dimensional Fractional Frequency Laplace Transform by the Inverse Difference Operator

“…In 2015, Ivic discussed the discrete Laplace transforms in the view of fast decay factor e − sx and obtained the Laplace transform of P(x) as ∞ 0 P(x)e − sx dx � πs − 2 ∞ n�1 r(n)e − (π 2 /n) . In practice, many applications of Laplace transform (LT), L[f(x)] � ∞ 0 f(x)e − sx dx, and the forward discrete Laplace transform (DLT), L[f(n)] � ∞ n�0 f(n)e − sn , are discussed and mentioned by several authors in [20][21][22][23]. For physical applications of Laplace transform, refer [24][25][26][27].…”

Alpha fractional frequency Laplace transform through multiseries

“…The Laplace transform (LT) and discrete Laplace transform (DLT) effectively change a signal (function) from time domain to frequency domain with the factor e -st . Several applications of LT and DLT were discussed by many authors [6,[8][9][10]. The applications of the n-dimensional Laplace transform appear in heat equations, wave equations, and modeling in fluid dynamics [13,14,21].…”

Laplace Time Tuning Transform in Digital Signal Processing

“…With numerical computation and MATLAB obtaining exact solutions for Dirichlet-Neumann inverse problem are discussed in [4]. In practice, many applications of Laplace Transform (LT) and Discrete Laplace Transform (DLT) are discussed by several authors [3,9,14,10].…”

“…This transform is called as Generalized Laplace Transform (GLT) and it lies in between DLT and LT. The GLT becomes DLT and LT when = 1  and 0   respectively [2,3]. If we take  as time between two successive signals in DSP (1) becomes Laplace Time Tuning Transform (LTTT).…”