Related papers: Some Results on Analysis and number theory

Counterparts of several classical results of number theory are proven for the ring of polynomials with coefficients in a number field. A theorem of Milnor that determines the Witt ring of a function field is applied to prove an analogue of…

In a multidimensional infinite layer bounded by two hyperplanes, the Poisson equation with the polynomial right-hand side is considered. It is shown that the Dirichlet boundary value problem and the mixed Dirichlet-Neumann boundary value…

Multiplicative arithmetic functions satisfying the parallelogram functional equation on prime numbers are investigated. It is derived that the unique solution is a quadratic function by the Goldbach's conjecture.

In this expository article we provide an elegant proof of the one-sided Ingham-Karamata Tauberian theorem. As an application, we present a short deduction of the prime number theorem.

We survey the classical results on the prime number theorem

We give complete, finite quasiequational axiomatisations for algebras of unary partial functions under the operations of composition, domain, antidomain, range and intersection. This completes the extensive programme of classifying algebras…

Let $x$ denote a sufficiently large integer. We show that the recent result of Grimmelt and Merikoski actually yields the largest prime factor of $n^2 +1$ is greater than $x^{1.317}$ infinitely often. As an application, we give a new upper…

We investigate here the representability of integers as sums of triangular numbers, where the $n$-th triangular number is given by $T_n = n(n + 1)/2$. In particular, we show that $f(x_1,x_2,..., x_k) = b_1 T_{x_1} +...+ b_k T_{x_k}$, for…

We propose a new class of fundamental solutions for the numerical analysis of boundary value problems for the Maxwell equations. We prove completeness of systems of such fundamental solutions in appropriate Sobolev spaces on a smooth…

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, four classes of monomial complete permutation polynomials and one class of…

This is the second part of a work dedicated to the study of Bernstein-Sato polynomials for several analytic functions depending on parameters. In this part, we give constructive results generalizing previous ones obtained by the author in…

We study numerically the $\alpha$- and $\omega$-limits of the Newton maps of two of the most elementary families of polynomial transformations on the plane: those with a linear component and those with both components of degree two. Our…

We consider some combinatorial problems on matrix polynomials over finite fields. Using results from control theory we give a proof of a result of Helmke, Jordan and Lieb on the number of linear unimodular matrix polynomials over a finite…

A recent heuristic argument based on basic concepts in spectral analysis showed that the twin prime conjecture and a few other related primes counting problems are valid. A rigorous version of the spectral method, and a proof of the more…

We give formulas for the number of polynomials over a finite field with given root multiplicities, in particular in cases when the formula is surprisingly simple (a power of q). Besides this concrete interpretation, we also prove an…

Let Q be a non-singular quadratic form with integer coefficients. When Q is indefinite we provide new upper bounds for the least non-trivial integral solution to the equation Q=0. When Q is positive definite we provide improved upper bounds…

Assuming two deep but standard conjectures from the Langlands Programme, we prove that the asymptotic Fermat's Last Theorem holds for imaginary quadratic fields Q(\sqrt{-d}) with -d=2, 3 mod 4. For a general number field K, again assuming…

We introduce sequences of functions orthogonal on a finite interval: proper orthogonal rational functions, orthogonal exponential functions, orthogonal logarithmic functions, and transmuted orthogonal polynomials

We give a purely combinatorial proof for a two-fold generalization of van der Waerden-Brauer's theorem and Hindman's theorem. We also give tower bounds for a finite version of it.

The aim of this paper is to show that there exists a deterministic algorithm that can be applied to compute the factors of a polynomial of degree 2, defined over a finite field, given certain conditions.