相关论文: A Survey on the Special Function "Shin"
Different types of sinc integrals are investigated when the standard sine function is replaced by the generalised $\sin_{p,q}$ in two parameters. A striking generalisation of the improper Dirichlet integral is achieved. A second surprising…
We develop a discrete spectral framework for Dirichlet $L$-functions that reveals a combinatorial structure underlying their special values and connects this to their zeros. Our approach approximates the classical Dirichlet series by finite…
The present article is devoted to the description of further investigations of the author of this article. These investigations (in terms of various representations of real numbers) include the generalized Salem functions and…
We introduce the Schur class of functions, discrete analytic on the integer lattice in the complex plane. As a special case, we derive the explicit form of discrete analytic Blaschke factors and solve the related basic interpolation…
In this manuscript, the author derived a definite integral involving the logarithmic function, function of powers and polynomials in terms of the Lerch function. A summary of the results is produced in the form of a table of definite…
In this paper, we investigate the determinants involving some trigonometric functions. We establish a connection between these determinants and the special values of Dirichlet L-functions, thereby extending Guo's results to arbitrary…
We review the physics at the end of the nineteenth century and summarize the process of the establishment of Special Relativity by Albert Einstein in brief. Following in the giant's footsteps, we outline the scientific method which helps to…
The numbers of even and odd permutations with a given ascent number are investigated using an operator that was previously introduced by the author. Their difference is called a signed Eulerian number. By means of the operator the…
In the reference Phys. Rev. Lett. 132, 233801 (2024), the authors claim to have introduced a ''real-space spin Chern number'' as well as a ''Spin Berry connection'' and a ''Spin Berry curvature''. The main finding of their letter is the…
We analyze the Wigner function constructed on the basis of the discrete rotation and displacement operators labeled with elements of the underlying finite field. We separately discuss the case of odd and even characteristics and analyze the…
The authors survey recent results in special functions of classical analysis and geometric function theory, in particular the circular and hyperbolic functions, the gamma function, the elliptic integrals, the Gaussian hypergeometric…
A large number of publications are available for the Optical Character Recognition (OCR). Significant researches, as well as articles are present for the Latin, Chinese and Japanese scripts. Arabic script is also one of mature script from…
Adolf Hurwitz is rather famous for his celebrated contributions to Riemann surfaces, modular forms, diophantine equations and approximation as well as to certain aspects of algebra. His early work on an important generalization of…
Boolean functions are mathematical objects used in diverse domains and have been actively researched for several decades already. One domain where Boolean functions play an important role is cryptography. There, the plethora of settings one…
N. G. de Bruijn (1958) studied the asymptotic expansion of iterates of sin$(x)$ with $0 < x \leq \pi/2$. Bencherif & Robin (1994) generalized this result to increasing analytic functions $f(x)$ with an attractive fixed point at 0 and $x >…
In this note, we study the arithmetic function $f : \mathbb{Z}_+^* \to \mathbb{Q}_+^*$ defined by $f(2^k \ell) = \ell^{1 - k}$ ($\forall k, \ell \in \mathbb{N}$, $\ell$ odd). We show several important properties about that function and then…
We introduce and study a family of functions we call the "mock characters". These functions satisfy a number of interesting properties, and of all completely multiplicative arithmetic functions seem to come as close as possible to being…
A method of estimating sums of multiplicative functions braided with Dirichlet characters is demonstrated, leading to a taxonomy of the characters for which such sums are large.
The Riemann zeta function, and more generally the L-functions of Dirichlet characters, are among the central objects of study in number theory. We report on a project to formalize the theory of these objects in Lean's "Mathlib" library,…
In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are…