Related papers: The Bateman-Horn constant for $x^3+x+1$
We estimate the error term in the asymptotic formula of the Sidon constant for (ordinary) Dirichlet polynomials by providing explicit lower and upper bounds. The lower bound is already implicitly known, but we supply the necessary…
Some additive reverses of the continuous triangle inequality for Bochner integral of vector-valued functions in Hilbert spaces are given. Applications for complex-valued functions are provided as well.
In this paper, we investigate the integral of $x^n\log^m(\sin(x))$ for natural numbers $m$ and $n$. In doing so, we recover some well-known results and remark on some relations to the log-sine integral…
Let $x \geq 1$ be a large number, let $f(x) \in \mathbb{Z}[x]$ be a prime polynomial of degree $\text{deg}(f)=m$, and let $u\ne \pm 1, v^2$ be a fixed integer. Assuming the Bateman-Horn conjecture, an asymptotic counting function for the…
An analytical expression for the third coefficient of the Jones Polynomial $P_J[\gamma,\, {\em e}^q]$ in the variable $q$ is reported. Applications of the result in Quantum Gravity are considered.
A general solution to the Complex Bateman equation in a space of arbitrary dimensions is constructed.
We prove that under certain explicit conditions, the Mahler measure of a three-variable polynomial can be expressed in terms of elliptic curve $L$-values and Bloch-Wigner dilogarithmmic values, conditionally on Beilinson's conjecture. In…
In this note we prove an explicit binomial formula for Jack polynomials and discuss some applications of it.
For the phonon Boltzmann equation with only pair collisions we characterize the set of all collisional invariants under some mild conditions on the dispersion relation.
We study the behaviour of the constant that is provided in the articles [12] and [13], which is connected with the determination of the Bellman function of three integral variables of the dyadic maximal operator. More precisely we study the…
Necessary and sufficient conditions on $A,B\in \mathbb{F}_{q^3}^*$ for $f(X)=X^{q^2-q+1}+AX^{q^2}+BX$ being a permutation polynomial of $\mathbb{F}_{q^3}$ are investigated via a connection with algebraic varieties over finite fields.
In more than two variables, detection of the bifurcation set of polynomial mapping $\bR^n \to \bR^p$, $n\ge p$, is a still unsolved problem. In this note we provide a solution for $n= p+1 \ge 3$.
The value of the proton singlet axial constant is estimated. It has been shown that the axial anomaly plays a crucial role in this calculation. Obtained result is sufficiently suppressed in comparison with the naively expected one. The…
The Hamburger moment problem for the $q$-Lommel polynomials which are related to the Hahn-Exton $q$-Bessel function is known to be indeterminate for a certain range of parameters. In this paper, the Nevanlinna parametrization for the…
We extend the authors' previous work on Wiener-Wintner double recurrence theorem to the case of polynomials.
We determine the p-exponent in many of the coefficients in the power series (log(1+x)/x)^t, where t is any integer. In our proof, we introduce a variant of multinomial coefficients. We also characterize the power series x/log(1+x) by…
A particulare case of the three-body problem, in the PPN formalism, is presented. The Hamiltonian function is obtained and the problem is reduced to a perturbed two-body one.
We solve a Lehmer-type question about the Mahler measure of integer-valued polynomials.
Much work has been done attempting to understand the dynamic behaviour of the so-called "3x+1" function. It is known that finite sequences of iterations with a given length and a given number of odd terms have some combinatorial properties…
We survey various classical results on invariants of polynomials, or equivalently, of binary forms, focussing on explicit calculations for invariants of polynomials of degrees 2, 3, 4.