Related papers: El Logaritmo Integral: N\'umeros primos y algo m\'…
Sturm's theorem (1829/35) provides an elegant algorithm to count and locate the real roots of any real polynomial. In his residue calculus (1831/37) Cauchy extended Sturm's method to count and locate the complex roots of any complex…
The concept of number and its generalization has played a central role in the development of mathematics over many centuries and many civilizations. Noteworthy milestones in this long and arduous process were the developments of the real…
Integration, just as much as differentiation, is a fundamental calculus tool that is widely used in many scientific domains. Formalizing the mathematical concept of integration and the associated results in a formal proof assistant helps in…
Bieberbach's conjecture was very important in the development of Geometric Function Theory, not only because of the result itself, but also due to the large amount of methods that have been developed in search of its proof, it is in this…
In this paper, we discuss the value-distribution of the Riemann zeta-function. The authors give some results for the discrepancy estimate and large deviations in the limit theorem by Bohr and Jessen.
The Gauss-Bonnet Formula is a significant achievement in 19th century differential geometry for the case of surfaces and the 20th century cumulative work of H. Hopf, W. Fenchel, C. B. Allendoerfer, A. Weil and S.S. Chern for…
Because of its relation to the distribution of prime numbers, the Riemann zeta function {\zeta} (s) is one of the most important functions in mathematics. The zeta function is defined by the following formula for any complex number s with…
Motivated by previous work leveraging factorizations of second- and fourth-order differential operators, a general integral inequality involving higher order derivatives is proven by elementary means. It is then shown how this framework…
We generalize several integrals studied by Glaisher. These ideas are then applied to obtain an analog of an integral due to Ismail and Valent.
This article describes some aspects of Cauchy integrals and related geometry of sets and measures in Euclidean spaces, etc.
Let $\chi$ be a primitive Dirichlet character whose conductor $q$ is a prime number. For the certain averages of values of $\log |L(s, \chi)|$ in $q$-aspect at a fixed $s=\sigma>1/2$, under Generalized Riemann Hypothesis (GRH), we explain…
We discuss relationships between the McShane, Pettis, Talagrand and Bochner integrals. A large number of different methods of integration of Banach-space-valued functions have been introduced, based on the various possible constructions of…
Let $s:[1,\infty) \to \C $ be a locally integrable function in Lebesgue's sense. The logarithmic (also called harmonic) mean of the function $s$ is defined by [\tau(t) := \frac 1{\log t} \int_1^t \frac {s(x)}{x} dx, \qquad t>1,] where the…
This survey will appear in Vol. VII of the Hendbook of Teichm{\"u}ller theory (European Mathematical Society Publishing House, 2020). It is a commentary on Teichm{\"u}ller's paper "Einfache Beispiele zur Wertverteilungslehre", published in…
We describe methods to evaluate elementary logarithmic integrals. The integrand is the product of a rational function and a linear polynomial in ln x.
The key operation in Bayesian inference, is to compute high-dimensional integrals. An old approximate technique is the Laplace method or approximation, which dates back to Pierre- Simon Laplace (1774). This simple idea approximates the…
In 1826 Cauchy presented an Integral over the real line. Al and I thought a derivation would be mighty fine. So we packed our contour integral bags that day, and we now present an analytic continuation this time.
Numerical study of the distribution of the Riemann zeros differences following the work [1] shows the significance of the function for which the prime sum expression is proposed. Computational results related to this definition explored…
We develop efficient numerical integration methods for computing an integral whose integrand is a product of a smooth function and the Gaussian function with a small standard deviation. Traditional numerical integration methods applied to…
This is the first installment in a series of papers devoted to examining certain aspects of the asymptotic value distribution and distribution of zeros manifested by members of a broad class of linear combinations of L-functions in the…