Related papers: Application of the Argument Principle to Functions…
In this paper, we show some expressions of certain $q$-multiple zeta-star values at roots of unity. These explicit formulas are expressed by using the determinants or Bell polynomials. Explicit formulas for other types of values can be…
This paper presents a theory of non-linear integer/real arithmetic and algorithms for reasoning about this theory. The theory can be conceived as an extension of linear integer/real arithmetic with a weakly-axiomatized multiplication…
An important step in the efficient computation of multi-dimensional theta functions is the construction of appropriate symplectic transformations for a given Riemann matrix assuring a rapid convergence of the theta series. An algorithm is…
We give an exposition of the Newton-Leibniz calculus. We begin by defining the integral as a limit of Riemann sums, verify the integrals of the standard catalog of functions by direct manipulation, prove the substitution lemmas as theorems…
In this note we explore the relationship between the operation of convolution of functions and the Eulerian integrals. This approach allow us to obtain some expressions for the convolution of a certain class of functions in terms of the…
A new mathematical notation is proposed for the iteration of functions. It facilitates the application of the iteration of functions in mathematical and logical expressions, definitions of sets, and formulations of algorithms. Illustrations…
The addition relation for the Riemann theta functions and for its limits, which lead to the appearance of exponential functions in soliton type equations is discussed. The presented form of addition property resolves itself to the…
We describe an algorithm for arbitrary-precision computation of the elementary functions (exp, log, sin, atan, etc.) which, after a cheap precomputation, gives roughly a factor-two speedup over previous state-of-the-art algorithms at…
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…
We present a rational version of the classical Landen transformation for elliptic integrals. This is employed to obtain explicit closed-form expressions for a large class of integrals of even rational functions and to develop an algorithm…
Additive utility function models are widely used in multiple criteria decision analysis. In such models, a numerical value is associated to each alternative involved in the decision problem. It is computed by aggregating the scores of the…
A simple and elementary derivation of values at integer points for the Riemann's zeta and related functions is reported.
We give a new perspective on the Lorentzian OPE inversion formula of arXiv:1703.00278, building on arXiv:2302.06469. We introduce an ``auxiliary'' fourpoint function that can be related to the traditionally defined ones via a Radon…
We propose a second-order method for unconditional minimization of functions $f(z)$ of complex arguments. We call it the Mixed Newton Method due to the use of the mixed Wirtinger derivative $\frac{\partial^2f}{\partial\bar z\partial z}$ for…
This paper presents a reformulation of the Leibniz product rule as a finite sum that expresses the fractional derivative of the product of two differentiable functions. This paper then proves the cases for when the product consists of an…
Reverse Mathematics is a program in the foundations of mathematics which provides an elegant classification of theorems of ordinary mathematics based on computability. Our aim is to provide an alternative classification of theorems based on…
We study the incomplete Mellin transformation of the fractional part and the related log-sine function when composed by an affine complex map. We evaluate the corresponding integral in two different ways which yields equalities with series…
We describe an algorithm for obtaining explicit expressions for lower terms for the conjectured full asymptotics of the moments of the Riemann zeta function, and give two distinct methods for obtaining numerical values of these…
In this paper, we introduce a method of converting implicit equations to the usual forms of functions locally without differentiability. For a system of implicit equations which are equipped with continuous functions, if there are unique…
In this article we obtain, using an expression of the digamma function $\psi(x)$ due to Mikolas, integral representations of the zeta function of odd arguments $\zeta(2p+1)$ for any positive value of $p$. The integrand consists of the…