On some mean values for the divisor function and the Riemann zeta-function

Let Δ(x) and E(x) denote respectively the error terms in the summatory formula for the divisor function and in the mean square formula for ζ(s) on the critical line. We consider some general mean values for Δ(x) and E(x) and discover interesting differences between these two functions. In particular, this yields evidence that E(x) is more negative than Δ(x).     

Discrete Universality in the Selberg Class

The Selberg class S consists of functions L(s) that are defined by Dirichlet series and satisfy four axioms (Ramanujan conjecture, analytic continuation, functional equation, and Euler product). It has been known that functions in S that satisfy the mean value condition on primes are universal in the sense of Voronin, i.e., every function in a sufficiently wide class of analytic functions can be approximated by the shifts L(s + iτ ), τ ∈ R. In this paper we show that every function in the same class of analytic functions can be approximated by the discrete shifts L(s + ikh), k = 0, 1,..., where h > 0 is an arbitrary fixed number.     

On the k-polygonal numbers and the mean value of Dedekind sums

For any positive integer k ≥ 3, it is easy to prove that the k-polygonal numbers are a_n(k) = (2n+n(n?1)(k?2))/2. The main purpose of this paper is, using the properties of Gauss sums and Dedekind sums, the mean square value theorem of Dirichlet L-functions and the analytic methods, to study the computational problem of one kind mean value of Dedekind sums S(a_n(k)ā_m(k), p) for k-polygonal numbers with 1 ≤ m, n ≤ p ? 1, and give an interesting computational formula for it.     

On Extraction of Smooth Solutions of a Class of Almost Hypoelliptic Equations with Constant Power

A linear differential operator P(x, D) = P(x₁,... x _n , D₁,..., D _n ) = ∑_αγ_α(x)D^α with coefficients γ_α(x) defined in E ⁿ is called formally almost hypoelliptic in E ⁿ if all the derivatives D^ν_ξP(x, ξ) can be estimated by P(x, ξ), and the operator P(x, D) has uniformly constant power in Eⁿ. In the present paper, we prove that if P(x, D) is a formally almost hypoelliptic operator, then all solutions of equation P(x, D)u = 0, which together with some of their derivatives are square integrable with a specified exponential weight, are infinitely differentiable functions.     

The strong <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-<Emphasis Type="Italic">greedy</Emphasis> property of the Walsh system

For any 0 < ? < 1 one can find a measurable set E ? [0, 1] with the measure |E| > 1 ? ? such that for each function f(x) ε L ¹ (0, 1) a function g(x) ε L ¹ (0, 1) exists such that it coincides with f (x) on E, its Fourier—Walsh series converges to it in the metric of L ¹ (0, 1), and all nonzero terms of the sequence of Fourier coefficients of the new function obtained by the Walsh system have the modulo decreasing order; consequently, the greedy algorithm for this function converges to it in the L ¹ (0, 1)-norm.     

A tracing of the fractional temperature field

This paper is devoted to a study of L~q-tracing of the fractional temperature field u(t, x)—the weak solution of the fractional heat equation(?_t +(-?_x)~α)u(t, x) = g(t, x) in L~p(R_+~(1+n)) subject to the initial temperature u(0, x) = f(x) in L~p(R~n).     

An integro-local theorem applicable on the whole half-axis to the sums of random variables with regularly varying distributions

We obtain an integro-local limit theorem for the sum S(n) = ξ(1)+?+ξ(n) of independent identically distributed random variables with distribution whose right tail varies regularly; i.e., it has the form P(ξ≥t) = t ^?β L(t) with β > 2 and some slowly varying function L(t). The theorem describes the asymptotic behavior on the whole positive half-axis of the probabilities P(S(n) ∈ [x, x + Δ)) as x → ∞ for a fixed Δ > 0; i.e., in the domain where the normal approximation applies, in the domain where S(n) is approximated by the distribution of its maximum term, as well as at the “junction” of these two domains.     

Congruent elliptic curves with non-trivial Shafarevich-Tate groups: Distribution part

Given a large positive number x and a positive integer k, we denote by Qk(x) the set of congruent elliptic curves E(n): y2= z3- n2 z with positive square-free integers n x congruent to one modulo eight,having k prime factors and each prime factor congruent to one modulo four. We obtain the asymptotic formula for the number of congruent elliptic curves E(n)∈ Qk(x) with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to(Z/2Z)2. We also get a lower bound for the number of E(n)∈ Qk(x)with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to(Z/2Z)4. The key ingredient of the proof of these results is an independence property of residue symbols. This property roughly says that the number of positive square-free integers n x with k prime factors and residue symbols(quadratic and quartic) among its prime factors being given compatible values does not depend on the actual values.     

On boundary value problems for systems of nonlinear generalized ordinary differential equations

A general theorem (principle of a priori boundedness) on solvability of the boundary value problem dx = dA(t) · f(t, x), h(x) = 0 is established, where f: [a, b]×R ⁿ → R ⁿ is a vector-function belonging to the Carathéodory class corresponding to the matrix-function A: [a, b] → R ^n×n with bounded total variation components, and h: BV_s([a, b],R ⁿ ) → R ⁿ is a continuous operator. Basing on the mentioned principle of a priori boundedness, effective criteria are obtained for the solvability of the system under the condition x(t₁(x)) = B(x) · x(t ₂(x))+c ₀, where t _i: BV_s([a, b],R ⁿ ) → [a, b] (i = 1, 2) and B: BV_s([a, b], R ⁿ ) → R ⁿ are continuous operators, and c ₀ ∈ R ⁿ .     

Congruent elliptic curves with non-trivial Shafarevich-Tate groups

We study 2-primary parts ⅢX(E~((n))/Q)[2~∞] of Shafarevich-Tate groups of congruent elliptic curves E~((n)): y~2= x~3-n~2x, n ∈Q~×/Q~(×2). Previous results focused on finding sufficient conditions for ⅢX(E~((n))/Q)[2~∞]trivial or isomorphic to(Z/2Z)~2. Our first result gives necessary and sufficient conditions such that the 2-primary part of the Shafarevich-Tate group of E~((n))is isomorphic to(Z/2Z)～2 and the Mordell-Weil rank of E~((n)) is zero,provided that all prime divisors of n are congruent to 1 modulo 4. Our second result provides sufficient conditions for ⅢX(E~((n))/Q)[2~∞]■(Z/2Z)~(2k), where k≥2.     

Equiconvergence of the trigonometric Fourier series and the expansion in eigenfunctions of the Sturm-Liouville operator with a distribution potential

We consider the Sturm-Liouville operator L = ?d ²/dx ² + q(x) with the Dirichlet boundary conditions in the space L ₂[0, π] under the assumption that the potential q(x) belongs to W ₂ ^?1 [0, π]. We study the problem of uniform equiconvergence on the interval [0, π] of the expansion of a function f(x) in the system of eigenfunctions and associated functions of the operator L and its Fourier sine series expansion. We obtain sufficient conditions on the potential under which this equiconvergence holds for any function f(x) of class L ₁. We also consider the case of potentials belonging to the scale of Sobolev spaces W ₂ ^?θ [0, π] with ½ < θ ≤ 1. We show that if the antiderivative u(x) of the potential belongs to some space W ₂ ^θ [0, π] with 0 < θ < 1/2, then, for any function in the space L ₂[0, π], the rate of equiconvergence can be estimated uniformly in a ball lying in the corresponding space and containing u(x). We also give an explicit estimate for the rate of equiconvergence.     

Series by Haar system with monotone coefficients

Let χ = {χ _n } _n=0 ^∞ be the Haar system normalized in L ²(0, 1) and M = {M _s } _s=1 ^∞ be an arbitrary, increasing sequence of nonnegative integers. For any subsystem of χ of the form {φ _k } = χS = {χ _n } _n∈S , where S = S(M) = {n _k } _k=1 ^∞ = {n ∈ V[p]: p ∈ M}, V[0] = {1, 2} and V[p] = {2 ^p + 1, 2 ^p + 2, …, 2 ^p+1} for p = 1, 2, … a series of the form Σ _i=1 ^∞ a _i φ _i with a _i ↘ 0 is constructed, that is universal with respect to partial series in all classes L ^r (0, 1), r ∈ (0, 1), in the sense of a.e. convergence and in the metric ofL ^r (0, 1). The constructed series is universal in the class of all measurable, finite functions on [0, 1] in the sense of a.e. convergence. It is proved that there exists a series by Haar system with decreasing coefficients, which has the following property: for any ? > 0 there exists a measurable function µ(x), x ∈ [0, 1], such that 0 ≤ µ(x) ≤ 1 and |{x ∈ [0, 1], µ(x) ≠ = 1}| < ?, and the series is universal in the weighted space L _µ[0, 1] with respect to subseries, in the sense of convergence in the norm of L _µ[0, 1].     

On the equiconvergence rate of trigonometric series expansions and eigenfunction expansions for the Sturm-Liouville operator with a distributional potential

We study the Sturm-Liouville operator L = ?d ²/dx ² + q(x) in the space L ₂[0, π] with the Dirichlet boundary conditions. We assume that the potential has the form q(x) = u′(x), u ∈ W ₂ ^θ [0, π], 0 < θ < 1/2. We consider the problem on the uniform (on the entire interval [0, π]) equiconvergence of the expansion of a function f(x) in a series in the system of root functions of the operator L with its Fourier expansion in the system of sines. We show that if the antiderivative u(x) of the potential belongs to any of the spaces W ₂ ^θ [0, π], 0 < θ < 1/2, then the equiconvergence rate can be estimated uniformly over the ball u(x) ∈ B _R = {v(x) ∈ W ₂ ^θ [0, π] | ∥v∥_{W 2} ^θ ≤ R} for any function f(x) ∈ L ₂[0, π].     

Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces

The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces L_p(R^d) (in the case p > 1), but (in the case when 1/p(·) is log-Hölder continuous and p- = inf{p(x): x ∈ R^d > 1) on the variable Lebesgue spaces L_p(·)(R^d), too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type (1, 1). In the present note we generalize Besicovitch’s covering theorem for the so-called γ-rectangles. We introduce a general maximal operator M_s^γδ, and with the help of generalized Φ-functions, the strong- and weak-type inequalities will be proved for this maximal operator. Namely, if the exponent function 1/p(·) is log-Hölder continuous and p- ≥ s, where 1 ≤ s ≤ ∞ is arbitrary (or p- ≥ s), then the maximal operator M_s^γδ is bounded on the space L_p(·)(R^d) (or the maximal operator is of weak-type (p(·), p(·))).     

On the Joint Distribution of First-passage Time and First-passage Area of Drifted Brownian Motion

For drifted Brownian motion X(t) = x-µ t + B _t (µ > 0) starting from x > 0, we study the joint distribution of the first-passage time below zero ,t(x), and the first-passage area ,A(x), swept out by X till the time t(x). In particular, we establish differential equations with boundary conditions for the joint moments E[t(x) ^m A(x) ⁿ ], and we present an algorithm to find recursively them, for any m and n. Finally, the expected value of the time average of X till the time t(x) is obtained.     

Equiconvergence of expansions in multiple Fourier series and in fourier integrals with “lacunary sequences of partial sums”

We investigate the equiconvergence on T^N = [?π, π)^N of expansions in multiple trigonometric Fourier series and in the Fourier integrals of functions f ∈ L_p(T^N) and g ∈ L_p(R^N), p > 1, N ≥ 3, g(x) = f(x) on T^N, in the case where the “partial sums” of these expansions, i.e., S_n(x; f) and J_α(x; g), respectively, have “numbers” n ∈ Z^N and α ∈ R^N (n_j = [α_j], j = 1,..., N, [t] is the integral part of t ∈ R¹) containing N ? 1 components which are elements of “lacunary sequences.”     

The density of zeros of the Riemann zeta-function in the critical strip

A new proof of Ingam’s theorem on the density of zeros of the Riemann zeta-function in the critical strip is given basing on an idea of H. Bohr and F. Carlson. Multiplication of segments of the Dirichlet series for the functions ζ(s) and 1/ζ(s) is used, which permits to simplify the proof.     

Short cubic exponential sums over primes

For y ≥ x ^4/5 L ^8B+151 (where L = log(xq) and B is an absolute constant), a nontrivial estimate is obtained for short cubic exponential sums over primes of the form S ₃(α; x, y) = ∑ _x?y<n≤x Λ(n)e(αn ³), where α = a/q + θ/q ², (a, q) = 1, L ^32(B+20) < q ≤ y ⁵ x ^?2 L ^?32(B+20), |θ| ≤ 1, Λ is the von Mangoldt function, and e(t) = e ^2πit.     

A New Class of Generalized Polynomials Associated with Hermite and Euler Polynomials

In this paper, we introduce a new class of generalized polynomials associated with the modified Milne-Thomson’s polynomials \({\Phi_{n}^{(\alpha)}(x,\nu)}\) of degree n and order α introduced by Dere and Simsek. The concepts of Euler numbers E _n , Euler polynomials E _n (x), generalized Euler numbers E _n (a, b), generalized Euler polynomials E _n (x; a, b, c) of Luo et al., Hermite–Bernoulli polynomials \({{_HE}_n(x,y)}\) of Dattoli et al. and \({{_HE}_n^{(\alpha)} (x,y)}\) of Pathan are generalized to the one \({ {_HE}_n^{(\alpha)}(x,y,a,b,c)}\) which is called the generalized polynomials depending on three positive real parameters. Numerous properties of these polynomials and some relationships between E _n , E _n (x), E _n (a, b), E _n (x; a, b, c) and \({{}_HE_n^{(\alpha)}(x,y;a,b,c)}\) are established. Some implicit summation formulae and general symmetry identities are derived using different analytical means and applying generating functions.     

Fast Algorithm of Square Rooting in Some Finite Fields of Odd Characteristic

It was proved that the complexity of square root computation in the Galois field GF(3^s), s = 2^kr, is equal to O(M(2^k)M(r)k + M(r) log₂r) + 2^kkr^1+o(1), where M (n) is the complexity of multiplication of polynomials of degree n over fields of characteristics 3. The complexity of multiplication and division in the field GF(3^s) is equal to O(M(2^k)M(r)) and O(M(2^k)M(r)) + r^1+o(1), respectively. If the basis in the field GF(3^r) is determined by an irreducible binomial over GF(3) or is an optimal normal basis, then the summands 2^kkr^1+o(1) and r^1+o(1) can be omitted. For M(n) one may take n log₂nψ(n) where ψ(n) grows slower than any iteration of the logarithm. If k grow and r is fixed, than all the estimates presented here have the form O_r (M (s) log ₂s) = s (log ₂s)²ψ(s).