Fractional parts of the function <Emphasis Type="Italic">x/n</Emphasis>

Asymptotic formulas for sums of values of some class of smooth functions of fractional parts of numbers of the form x/n, where the parameter x increases unboundedly and the integer n ranges over various subsets of the interval [1, x], are obtained.     

Model of <Emphasis Type="Italic">p</Emphasis>-Adic Random Walk in a Potential

We consider the p-adic random walk model in a potential which can be viewed as a generalization of p-adic random walk models used for describing protein conformational dynamics. This model is based on the Kolmogorov-Feller equations for the distribution function defined on the field of p-adic numbers in which the transition rate depends on ultrametric distance between the transition points as well as on function of potential violating the spatial homogeneity of a random process. This equation which will be called the equation of p-adic random walk in a potential, is equivalent to the equation of p-adic random walk with modified measure and reaction source. With a special choice of a power-law potential the last equation is shown to have an exact analytic solution. We find the analytic solution of the Cauchy problem for such equation with an initial condition, whose support lies in the ring of integer p-adic numbers.We also examine the asymptotic behaviour of the distribution function for large times. It is shown that in the limit t→∞ the distribution function tends to the equilibrium solution according to the law, which is bounded from above and below by power laws with the same exponent. Our principal conclusion is that the introduction of a potential in the model of p-adic random walk conserves the power-law behaviour of relaxation curves for large times.     

The finite Littlewood problem in $${{\mathbb {F}}}_p$$

Let p be a large prime number and f(x) be an integer-valued function defined in \({\mathbb F}_p\). The Littlewood problem in \({{\mathbb {F}}}_p\) is to establish non-trivial lower bounds for the \(\ell _1\) norm of exponential sums involving f(x). In the present paper, we establish new lower bounds for exponential sums including polynomials, powers of any primitive root and subgroups of \(\mathbb {F}_p^*.\)     

Simultaneous approximation problems of <Emphasis Type="Italic">p</Emphasis>-adic numbers and <Emphasis Type="Italic">p</Emphasis>-adic knapsack cryptosystems - Alice in <Emphasis Type="Italic">p</Emphasis>-adic numberland

In this paper we construct the multi-dimensional p-adic approximation lattices by using simultaneous approximation problems (SAP) of p-adic numbers and we estimate the l _∞ norm of the p-adic SAP solutions theoretically by applying Dirichlet’s principle and numerically by using the LLL algorithm. By using the SAP solutions as private keys, the security of which depends on NP-hardness of SAP or the shortest vector problems (SVP) of p-adic lattices, we propose a p-adic knapsack cryptosystem with commitment schemes, in which the sender Alice prepares ciphertexts and the verification keys in her p-adic numberland.     

Maximal functions and ergodic averages related to Waring’s problem

We study the arithmetic analogue of maximal functions on diagonal hypersurfaces. This paper is a natural step following the papers of [13], [14] and [16]. We combine more precise knowledge of oscillatory integrals and exponential sums to generalize the asymptotic formula in Waring’s problem to an approximation formula for the Fourier transform of the solution set of lattice points on hypersurfaces arising in Waring’s problem and apply this result to arithmetic maximal functions and ergodic averages. In sufficiently large dimensions, the approximation formula, ? ²-maximal theorems and ergodic theorems were previously known. Our contribution is in reducing the dimensional constraint in the approximation formula using recent bounds of Wooley, and improving the range of ? ^p spaces in the maximal and ergodic theorems. We also conjecture the expected range of spaces.     

The <Emphasis Type="Italic">p</Emphasis>-adic weighted Hardy-Cesàro operators on weighted Morrey-Herz space

In this article we give necessary and sufficient conditions for the boundedness of the weighted Hardy-Cesà ro operators which is associated to the parameter curve γ(t, x) = γ(t)x defined by \({U_{\psi ,\gamma }}f\left( x \right) = \int {\left( {\gamma \left( t \right)x} \right)} \psi \left( t \right)dt\) on the weighted Morrey-Herz space over the p-adic field. Especially, the corresponding operator norms are established in each case. These results actually extend those of K. S. Rim and J. Lee [27] and of the authors [9]. Moreover, the sufficient conditions of boundedness of commutators of p-adic weighted Hardy-Cesàro operator with symbols in the Lipschitz space on the weighted Morrey-Herz space are also established.     

Integral averaging technique for oscillation of damped half-linear oscillators

This paper is concerned with the oscillatory behavior of the damped half-linear oscillator (a(t)?_p(x′))′ + b(t)?_p(x′) + c(t)?_p(x) = 0, where ?p(x) = |x|^p?1 sgn x for x ∈ ? and p > 1. A sufficient condition is established for oscillation of all nontrivial solutions of the damped half-linear oscillator under the integral averaging conditions. The main result can be given by using a generalized Young’s inequality and the Riccati type technique. Some examples are included to illustrate the result. Especially, an example which asserts that all nontrivial solutions are oscillatory if and only if p ≠ 2 is presented.     

On the rank of compact <Emphasis Type="Italic">p</Emphasis>-adic Lie groups

The rank of a profinite group G is the basic invariant \({{\rm rk}(G):={\rm sup}\{d(H) \mid H \leq G\}}\), where H ranges over all closed subgroups of G and d(H) denotes the minimal cardinality of a topological generating set for H. A compact topological group G admits the structure of a p-adic Lie group if and only if it contains an open pro-p subgroup of finite rank. For every compact p-adic Lie group G one has rk(G) ≥ dim(G), where dim(G) denotes the dimension of G as a p-adic manifold. In this paper we consider the converse problem, bounding rk(G) in terms of dim(G). Every profinite group G of finite rank admits a maximal finite normal subgroup, its periodic radical π(G). One of our main results is the following. Let G be a compact p-adic Lie group such that π(G) = 1, and suppose that p is odd. If \(\{g \in G \mid g^{p-1}=1 \}\) is equal to {1}, then rk(G) = dim(G).     

Commutator of Riesz potential in <Emphasis Type="Italic">p</Emphasis>-adic generalized Morrey spaces

Suppose that I _p ^α is the p-adic Riesz potential. In this paper, we established the boundedness of I _p ^α on the p-adic generalized Morrey spaces, as well as the boundedness of the commutators generated by the p-adic Riesz potential I _p ^α and p-adic generalized Campanato functions.     

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.”     

Exact Solutions of a Second-Order Nonlinear Partial Differential Equation

The equation ?²u/?t?x + u^p?u/?x = u^q describing a nonstationary process in semiconductors, with parameters p and q that are a nonnegative integer and a positive integer, respectively, and satisfy p + q ≥ 2, is considered in the half-plane (x, t) ∈ ? × (0,∞). All in all, fourteen families of its exact solutions are constructed for various parameter values, and qualitative properties of these solutions are noted. One of these families is defined for all parameter values indicated above.     

Verbal width in anabelian groups

The class A of anabelian groups is defined as the collection of finite groups without abelian composition factors. We prove that the commutator word [x₁, x₂] and the power word x₁^p have bounded width in A when p is an odd integer. By contrast, the word x³⁰ does not have bounded width in A. On the other hand, any given word w has bounded width for those groups G ∈ A whose composition factors are sufficiently large as a function of w. In the course of the proof we establish that sufficiently large almost simple groups cannot satisfy w as a coset identity.     

On the Classification of <Emphasis Type="Italic">p</Emphasis>-Adic UHF Banach Algebras

For any prime number p let Ω_p be the p-adic counterpart of the complex numbers C. In this paper we investigate the class of p-adic UHF Banach algebras. A p-adic UHF Banach algebra is any unital p-adic Banach algebra A of the form \(A = \overline {U{M_n}} \), where (M_n) is an increasing sequence of p-adic Banach subalgebras of M such that each M_n is generated over Ω_p by an algebraic system of matrix units {e _ij ^{( n)} | 1 ≤ i, j ≤ p_n }. The main result is that the supernatural number associated to a p-adic TUHF Banach algebra is an invariant of the algebra.     

Derivative at s = 1 of the p-adic L-function of the symmetric square of a Hilbert modular form

Let p ≥ 3 be a prime and F a totally real number field. Let f be a Hilbert cuspidal eigenform of parallel weight 2, trivial Nebentypus and ordinary at p. It is possible to construct a p-adic L function which interpolates the complex L-function associated with the symmetric square representation of f. This p-adic L-function vanishes at s = 1 even if the complex L-function does not. Assuming p inert and f Steinberg at p, we give a formula for the p-adic derivative at s = 1 of this p-adic L-function, generalizing unpublished work of Greenberg and Tilouine. Under some hypotheses on the conductor of f we prove a particular case of a conjecture of Greenberg on trivial zeros.     

On some arithmetic properties of Mahler functions

Mahler functions are power series f(x) with complex coefficients for which there exist a natural number n and an integer ? ≥ 2 such that f(x), f(x^?),..., \(f({x^{{\ell ^{n - 1}}}}),f({x^{{\ell ^n}}})\) are linearly dependent over ?(x). The study of the transcendence of their values at algebraic points was initiated by Mahler around the’ 30s and then developed by many authors. This paper is concerned with some arithmetic aspects of these functions. In particular, if f(x) satisfies f(x) = p(x)f(x^?) with p(x) a polynomial with integer coefficients, we show how the behaviour of f(x) mirrors on the polynomial p(x). We also prove some general results on Mahler functions in analogy with G-functions and E-functions.     

A <Emphasis Type="Italic">p</Emphasis>-adic hard-core model with three states on a Cayley tree

We examine the p-adic hard-core model with three states on a Cayley tree. Translationinvariant and periodic p-adic Gibbs measures are studied for the hard-core model for k = 2. We prove that every p-adic Gibbs measure is bounded for p ≠ 2. We show in particular that there is no strong phased transition for a hard-core model on a Cayley tree of order k.     

The unramified computation of Rankin–Selberg integrals expressed in terms of Bessel models for split orthogonal groups: Part I

In this paper, we compute the local integrals, with normalized unramified data, over a p-adic field F, arising from general Rankin–Selberg integrals for SO _m × GL_r+k+1, where the orthogonal group is split over F, \(k \leqslant \left[ {\frac{{m - 1}}{2}} \right]\), and the irreducible representation of SO _m (F) has a Bessel model with respect to an irreducible representation of the split orthogonal group SO_m?2k?1(F). Our proof is by “analytic continuation from the unramified computation in the generic case”. We let the unramified parameters of the representations involved vary, and express the local integrals in terms of the Whittaker models of the representations, which exist at points in general position. Then we apply analytic continuation and the known unramified computation in the generic case. We discuss some applications to poles of partial L-functions and functorial lifting.     

Répartitio n simulta née de S( n) et $$ S( n+1)$$ S( n+1) da ns les progressio ns arithmétiques

If \(q\ge 2\) is an integer, we denote by \(S_q(n)\) the sum of the digits in base q of the positive integer n and by \(v_q(n)\) its q-adic valuation. The goal of this work is to study exponential sums of the form \(\displaystyle \sum \nolimits _{n\le x}\exp \big (2i\pi \big (\frac{l}{m} S_q(n)+\frac{k}{m'}S_q(n+1)+\theta n\big )\big )\) in order to prove some statistical properties of integers n for which \(S_q(n)\) and \(S_q(n+1)\) belong to given arithmetic progressions. This extends the results obtained by Gelfond in 1968 and those obtained by Mauduit–Sárközy in 1996.     

Sets with even partition functions and 2-adic integers

For P ? \(\mathbb{F}_2 \)[z] with _P(0) = 1 and deg(_P) ≥ 1, let \(\mathcal{A}\) = \(\mathcal{A}\)(P) (cf. [4], [5], [13]) be the unique subset of ? such that Σ _n≥0 p(\(\mathcal{A}\), n)z ⁿ ≡ P(z) (mod 2), where p(\(\mathcal{A}\), n) is the number of partitions of n with parts in \(\mathcal{A}\). Let p be an odd prime and P ? \(\mathbb{F}_2 \)[z] be some irreducible polynomial of order p, i.e., p is the smallest positive integer such that P(z) divides 1 + z ^p in \(\mathbb{F}_2 \)[z]. In this paper, we prove that if m is an odd positive integer, the elements of \(\mathcal{A}\) = \(\mathcal{A}\)(P) of the form 2 ^k m are determined by the 2-adic expansion of some root of a polynomial with integer coefficients. This extends a result of F. Ben Saïd and J.-L. Nicolas [6] to all primes p.     

On Lehmer’s problem and Dedekind sums

Let p be an odd prime and c a fixed integer with (c, p) = 1. For each integer a with 1 ≤ a ≤ p ? 1, it is clear that there exists one and only one b with 0 ? b ? p ? 1 such that ab ≡ c (mod p). Let N(c, p) denote the number of all solutions of the congruence equation ab ≡ c (mod p) for 1 ? a, b ? p?1 in which a and \(\overline b \) are of opposite parity, where \(\overline b \) is defined by the congruence equation b\(\overline b \) ≡ 1 (mod p). The main purpose of this paper is to use the properties of Dedekind sums and the mean value theorem for Dirichlet L-functions to study the hybrid mean value problem involving N(c, p)?½φ(p) and the Dedekind sums S(c, p), and to establish a sharp asymptotic formula for it.