Problems In Analytic Number Theory Research Articles

On the distribution of αp^2 + β modulo one for primes p such that p + 2 has no more two prime divisors

A classical problem in analytic number theory is to study the distribution of frac- tional part αp^k + β, k ≥ 1 modulo 1, where α is irrational and p runs over the set of primes. We consider the subsequence generated by the primes p such that p + 2 is an almost-prime (the existence of infinitely many such p is another topical result in prime number theory) and prove that its distribution has a similar property.

The method for the heat-diffusion equation in analytic number theory

This paper addresses the entire functions via theory of the tempered xi function. The zeros of the entire functions by the Fourier sine and cosine integrals are stud?ied in detail. The method for the heat-diffusion equation is proposed to structure the connection between Fourier analysis and heat-diffusion equation show the open problems in analytic number theory.

Sums of certain fractional parts

In this note, an upper bound for the sum of fractional parts of certain smooth functions is given. Such sums arise naturally in numerous problems of analytic number theory. The main feature is here an improvement of the main term due to the use of Weyl’s bound for exponential sums and a device used by Popov.

Power moments and value distribution of functions

In this paper we study various which one can associate to a given function $f$, or rather to the power moments of $f$. These are motivated by long standing open problems in analytic number theory. We show how these abscissae connect to the distribution of values of $f$ in a very elegant way using convex conjugates. This connection allows us to show which abscissae are realizable for both general and more specific arithmetical functions. Further it may give a new approach to, for example, Dirichlet's divisor problem.

The Gaussian core model in high dimensions

We prove lower bounds for energy in the Gaussian core model, in which point particles interact via a Gaussian potential. Under the potential function $t \mapsto e^{-\alpha t^2}$ with $0 < \alpha < 4\pi/e$, we show that no point configuration in $\mathbf{R}^n$ of density $\rho$ can have energy less than $(\rho+o(1))(\pi/\alpha)^{n/2}$ as $n \to \infty$ with $\alpha$ and $\rho$ fixed. This lower bound asymptotically matches the upper bound of $\rho (\pi/\alpha)^{n/2}$ obtained as the expectation in the Siegel mean value theorem, and it is attained by random lattices. The proof is based on the linear programming bound, and it uses an interpolation construction analogous to those used for the Beurling-Selberg extremal problem in analytic number theory. In the other direction, we prove that the upper bound of $\rho (\pi/\alpha)^{n/2}$ is no longer asymptotically sharp when $\alpha > \pi e$. As a consequence of our results, we obtain bounds in $\mathbf{R}^n$ for the minimal energy under inverse power laws $t \mapsto 1/t^{n+s}$ with $s>0$, and these bounds are sharp to within a constant factor as $n \to \infty$ with $s$ fixed.

Heini Halberstam, 1926-2014

Heini Halberstam, 1926-2014

A generalised Gauss circle problem and integrated density of states

Counting lattice points inside a ball of large radius in Euclidean space is a classical problem in analytic number theory, dating back to Gauss. We propose a variation on this problem: studying the asymptotics of the measure of an integer lattice of affine planes inside a ball. The first term is the volume of the ball; we study the size of the remainder term. While the classical problem is equivalent to counting eigenvalues of the Laplace operator on the torus, our variation corresponds to the integrated density of states of the Laplace operator on the product of a torus with Euclidean space. The asymptotics we obtain are then used to compute the density of states of the magnetic Schrödinger operator.

Behavior of quasi-particles on hybrid spaces. Relations to the geometry of geodesics and to the problems of analytic number theory

We review our recent results concerning the propagation of “quasi-particles” in hybrid spaces — topological spaces obtained from graphs via replacing their vertices by Riemannian manifolds. Although the problem is purely classical, it is initiated by the quantum one, namely, by the Cauchy problem for the time-dependent Schrodinger equation with localized initial data.We describe connections between the behavior of quasi-particles with the properties of the corresponding geodesic flows. We also describe connections of our problem with various problems in analytic number theory.

Arithmetic Correlations Over Large Finite Fields

The auto-correlations of arithmetic functions, such as the von Mangoldt function, the M\"obius function and the divisor function, are the subject of classical problems in analytic number theory. The function field analogues of these problems have recently been resolved in the limit of large finite field size $q$. However, in this limit the correlations disappear: the arithmetic functions become uncorrelated. We compute averages of terms of lower order in $q$ which detect correlations. Our results show that there is considerable cancellation in the averaging and have implications for the rate at which correlations disappear when $q \rightarrow\infty$; in particular one cannot expect remainder terms that are of the order of the square-root of the main term in this context.

Shifted convolution and the Titchmarsh divisor problem over 𝔽q[t

In this paper, we solve a function field analogue of classical problems in analytic number theory, concerning the autocorrelations of divisor functions, in the limit of a large finite field.

Weyl asymptotics for tensor products of operators and Dirichlet divisors

We study the counting function of the eigenvalues for tensor products of operators, and their perturbations, in the context of Shubin classes and closed manifolds. We emphasize connections with problems of analytic number theory, concerning in particular generalized Dirichlet divisor functions.

Weyl asymptotics of bisingular operators and Dirichlet divisor problem

We consider a class of pseudodifferential operators, with crossed vector valued symbols, defined on the product of two closed manifolds. We study the asymptotic expansion of the counting function of positive selfadjoint operators in this class. Using a general Theorem of Aramaki, we can determine the first term of the asymptotic expansion of the counting function and, in a special case, we are able to find the second term. We give also some examples, emphasizing connections with problems of analytic number theory, in particular with Dirichlet divisor function.

On the distribution of αp modulo one for primes p of a special form

Abstract A classical problem in analytic number theory is to study the distribution of αp modulo 1, where α is irrational and p runs over the set of primes. We consider the subsequence generated by the primes p such that p+2 is an almost-prime (the existence of infinitely many such p is another topical result in prime number theory) and prove that its distribution has a similar property.

Some problems of Analytic number theory IV

In the present paper, we use Ramachandra's kernel function of the second order, namely ${\rm Exp} ((\sin z)^2)$, which has some advantages over the earlier kernel ${\rm Exp} (z^{4a+2})$ where $a$ is a positive integer. As an outcome of the new kernel we are able to handle $\Omega$-theorems for error terms in the asymptotic formula for the summatory function of the coefficients of generating functions of the ${\rm Exp}(\zeta(s)), {\rm Exp\,Exp}(\zeta(s))$ and also of the type ${\rm Exp\,Exp}((\zeta(s))^{\frac{1}{2}})$.

On the distribution in short intervals of products of a prime and integers from a given set

A classical problem in analytic number theory is the distribution in short intervals of integers with a prescribed multiplicative structure, such as primes, almost-primes, k-free numbers and others. Recently, partly due to applications to cryptology, much attention has been received by the problem of the distribution in short intervals of integers without large prime factors, see Lenstra–Pila–Pomerance [3] and section 5 of the excellent survey by Hildebrand–Tenenbaum [1].In this paper we deal with the distribution in short intervals of numbers representable as a product of a prime and integers from a given set [Sscr ], defined in terms of cardinality properties. Our results can be regarded as an extension of the above quoted results, and we will provide a comparison with such results by a specialization of the set [Sscr ].

On the accuracy of Zernike moments for image analysis

We give a detailed analysis of the accuracy of Zernike moments in terms of their discretization errors and the reconstruction power. It is found that there is an inherent limitation in the precision of computing the Zernike moments due to the geometric nature of a circular domain. This is explained by relating the accuracy issue to a celebrated problem in analytic number theory of evaluating the lattice points within a circle.

On the Sphere Problem

One of the oldest problems in analytic number theory consists of counting points with integer coordinates in the d-dimensional ball. It is very easy to find a main term for the counting function, but the size of the error term is difficult to estimate (...).

Addendum to Ramachandra's paper "Some problems of analytic number theory, I"

Addendum to Ramachandra's paper "Some problems of analytic number theory, I"

SLn technique and the density conjecture

The possibility of the application of the SLn spectral technique to certain important problems of analytic number theory is discussed.

The Hilbert-Kamke problem in analytic number theory

The Hilbert-Kamke problem in analytic number theory