On the order of the error function of the (k,r)-integers

Let k and r be fixed integers such that 1 < r < k. Any positive integer n of the form n = a^kb, where b is r-free, is called a (k, r)-integer. In this paper we prove that if Q_k,r(x) denotes the number of (k, r)-integers ≤ x, then $\text{Q}{}_{\mathrm{k,r}}\text{(x) =}\text{xζ(k)}\text{ζ(r)}\text{+ Δ}{}_{\mathrm{k,r}}\text{(x)}$, where $\text{Δ}{}_{\mathrm{k,r}}\text{(x) = O(x}{}^{\mathrm{<mtext>1</mtext><mtext>r</mtext>}}\text{exp}\text{[?B}\text{log}{}^{\mathrm{<mtext>3</mtext><mtext>5</mtext>}}\text{x (}\text{log log}\text{x)}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>5</mtext>}}\text{])}$, B being a positive constant depending on r and the O-estimate is uniform in k. On the assumption of the Riemann hypothesis, we improve the above order estimate of Δ_k,r(x) and prove that

$\text{1}\text{x}\text{∫}\text{1}\text{α}\text{δ}{}_{\mathrm{k,r}}\text{(t)dt=0(x}\text{1}\text{k}\text{ω(x))}\text{or}\text{0(x}{}^{\mathrm{3/(4r+1)}}\text{ω(x))}$

, according as $\text{k ≤}\text{(4r + 1)}\text{3}$ or $\text{k >}\text{(4r + 1)}\text{3}$, where ω(x) = exp [B log x(log log x)^?1].     

Pairs of additive equations II. Large odd degree

Let k be an odd positive integer. Davenport and Lewis have shown that the equations

$\text{a}{}_1\text{x}{}_1{}^{\mathrm{k}}\text{+…+a}{}_{\mathrm{n}}\text{x}{}_{\mathrm{n}}{}^{\mathrm{k}}\text{=0}$

with integer coefficients, have a nontrivial solution in integers x₁,…, x_N provided that

$\text{N?[36klog6k]}$

Here it is shown that for any ? > 0 and k > k₀(?) the equations have a nontrivial solution provided that

$\text{N?}\text{8}\text{log 2}\text{+?}\text{k log k.}$

     

A decomposition theorem for an analytic function

Let f(z), an analytic function with radius of convergence R (0 < R < ∞) be represented by the gap series ∑_{k = 0}^∞c_kz^λ_k. Set and define the growth constants ?, λ, T, t by

$\text{?}\text{λ}\text{=}\text{lim sup}\text{r→R}\text{inf}\text{{}\text{log}\text{[Rr /(R?r)]}{}^{\mathrm{?1}}\text{log}{}^{\mathrm{+}}\text{log}{}^{\mathrm{+}}\text{M(r)}}$

, and if 0 < ? < ∞,

$\text{T}\text{t}\text{=}\text{lim sup}\text{r→R}\text{inf}\text{{[Rr /(R?r)]}{}^{\mathrm{??}}\text{log}{}^{\mathrm{+}}\text{M(r)}}$

. Then, assuming 0 < t < T < ∞, we obtain a decomposition theorem for f(z).     

An integral operator connected with the Helmholtz equation

Let Lu be the integral operator defined by $\text{(L}{}_{\mathrm{k?}}\text{)(x, y) = ∝ s ∝ ?(x′, y′)(}\text{e}{}^{\mathrm{ik?}}\text{?}\text{) dx′ dy′, (x, y) ? S}$ where S is the interior of a smooth, closed Jordan curve in the plane, k is a complex number with Re k ? 0, Im k ? 0, and ?² = (x ?x′)² + (y ? y′)². We define $\text{q(x, y) = [}\text{dist}\text{((x, y), ?S)]}\text{1}\text{2}\text{, (x, y) ? S; L}{}_2\text{(q, S) = {? : ∝ s ∝ ¦ ?(x, y)¦}{}^2\text{q(x, y) dx dy < ∞}; W}{}_2{}^1\text{(q, S) = {? : ? ? L}{}_2\text{(q, S),}\text{??}\text{?x}\text{,}\text{?f}\text{?y}\text{? L}{}_2\text{(q, S)}}$, where in the definition of W₂¹(q, S) the derivatives are taken in the sense of distributions. We prove that L_k is a continuous 1-l mapping of L₂(q, S) onto W₂¹(q, S).     

On the van der Waerden conjecture and zeros of polynomials

Let

$\text{F(x) =}\text{∑}\text{k=o}\text{n}\text{n}\text{k}\text{A}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}$

$\text{A}{}_{\mathrm{n}}\text{≠ 0,}$

and

$\text{G(x) =}\text{∑}\text{k=o}\text{n}\text{n}\text{k}\text{B}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}$

$\text{B}{}_{\mathrm{n}}\text{≠ 0,}$

be polynomials with real zeros satisfying A_n?1 = B_n?1 = 0, and let

$\text{H(x) =}\text{∑}\text{k=o}\text{n-2}\text{n}\text{k}\text{A}{}_{\mathrm{k}}\text{B}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}\text{.}$

Using the recently proved validity of the van der Waerden conjecture on permanents, some results on the real zeros of H(x) are obtained. These results are related to classical results on composite polynomials.     

Some aspects of the finite moment problem

The approximate solution of the finite moment problem $\text{μ}{}_{\mathrm{k}}\text{= ∫}{}_0{}^1\text{x}{}^{\mathrm{k?1}}\text{?(x)}\text{d}\text{x}$, k = 1, 2, 3, …, is considered. This problem is related to the problem of finding a best polynomial least squares approximation to a given function $\text{?(x)}$ in [0,1]. The connection with Laplace transform inversion is emphasized, and a set of special square matrices with integral elements is introduced, which has an intimate relation to the above two problems. These matrices are the well-known inverses of finite segments of the infinite Hilbert matrix.     

Two-parameter Turan inequalities for ultraspherical and Laguerre polynomials

It is known that the classical orthogonal polynomials satisfy inequalities of the form U_n²(x) ? U_{n + 1}(x) U_{n ? 1}(x) > 0 when x lies in the spectral interval. These are called Turan inequalities. In this paper we will prove a generalized Turan inequality for ultraspherical and Laguerre polynomials. Specifically if P_n^λ(x) and L_n^α(x) are the ultraspherical and Laguerre polynomials and $\text{F}{}_{\mathrm{n}}{}^{\mathrm{\lambda }}\text{(x) =}\text{P}{}_{\mathrm{n}}{}^{\mathrm{\lambda }}\text{(x)}\text{P}{}_{\mathrm{n}}{}^{\mathrm{\lambda }}\text{(1)}\text{, G}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x) =}\text{L}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x)}\text{L}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(0)}\text{,}\text{then}\text{F}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x) F}{}_{\mathrm{n}}{}^{\mathrm{\beta }}\text{(x) ? F}{}_{\mathrm{n\; +\; 1}}{}^{\mathrm{\alpha }}\text{(x) F}{}_{\mathrm{n\; ?\; 1}}{}^{\mathrm{\beta }}\text{(x) > 0, ? 1 < x < 1, ?}\text{1}\text{2}\text{< α ? β ? α + 1}\text{and}\text{G}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x) G}{}_{\mathrm{n}}{}^{\mathrm{\beta }}\text{(x) ? G}{}_{\mathrm{n\; +\; 1}}{}^{\mathrm{\alpha }}\text{(x) G}{}_{\mathrm{n\; ?\; 1}}{}^{\mathrm{\beta }}\text{(x) > 0, x > 0, 0 < α ? β ? α + 1}$. We also prove the inequality $\text{(n + 1) F}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x) F}{}_{\mathrm{n}}{}^{\mathrm{\beta }}\text{(x) ? nF}{}_{\mathrm{n\; +\; 1}}{}^{\mathrm{\alpha }}\text{(x) F}{}_{\mathrm{n\; ?\; 1}}{}^{\mathrm{\beta }}\text{(x) > A}{}_{\mathrm{n}}\text{[F}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x)]}{}^2\text{, ?1 < x < 1, ?}\text{1}\text{2}\text{< α ? β < α + 1,}\text{where}\text{A}{}_{\mathrm{n}}$ is a positive constant depending on α and β.     

An asymptotic evaluation of the cycle index of a symmetric group

Let $\text{Z(S}{}_{\mathrm{n}}\text{;?(x))}$ denote the polynomial obtained from the cycle index of the symmetric group Z(S_n) by replacing each variable s_i by f(x¹). Let f(x) have a Taylor series with radius of convergence ? of the form f(x)=x^k + a_k+1x^k+1 + a_k+2x^k+2+? with every a₁?0. Finally, let 0<x<1 and let x??. We prove that

This limit is used to estimate the probability (for n and p both large) that a point chosen at random from a random p-point tree has degree n + 1. These limiting probabilities are independent of p and decrease geometrically in n, contrasting with the labeled limiting probabilities of $\text{1}\text{n!e}$.In order to prove the main theorem, an appealing generalization of the principle of inclusion and exclusion is presented.     

On a conjecture by D. Ž. D̄okovic

It is shown that if $\text{A?Ω}{}_{\mathrm{n}}\text{?{J}{}_{\mathrm{n}}\text{}}$ satisfies

$\text{nkσ}{}_{\mathrm{k}}\text{(A)?(n?k+1)}{}^2\text{σ}{}_{\mathrm{k?1}}\text{(A)}$

$\text{(k=1,2,…,n)}$

, where σ_k(A) denotes the sum of all kth order subpermanent of A, then Per[λJ_n+(1?λ)A] is strictly decreasing in the interval 0<λ<1.     

On the Cauchy problem for the generalized Benjamin–Ono equation with small initial data

We prove global well-posedness results for small initial data in $\text{H}{}^{\mathrm{s}}\text{(}\text{R}\text{),}\text{s>s}{}_{\mathrm{k}}$, and in , s_k=1/2?1/k, for the generalized Benjamin–Ono equation $\text{?}{}_{\mathrm{t}}\text{u+}\text{H}\text{?}{}^2{}_{\mathrm{x}}\text{u+?}{}_{\mathrm{x}}\text{(u}{}^{\mathrm{k+1}}\text{)=0,}\text{k?4}$. We also consider the cases k=2,3. To cite this article: L. Molinet, F. Ribaud, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Construction of a recurrence relation for modified moments

In this paper we are constructing a recurrence relation of the form

$\text{∑}\text{i=0}\text{r}\text{ω}{}_{\mathrm{i}}\text{(k)m}{}_{\mathrm{k+i}}{}^{\mathrm{\{\lambda \}}}\text{[f] = ω(k)}$

for integrals (called modified moments)

$\text{m}{}_{\mathrm{k}}{}^{\mathrm{\{\lambda \}}}\text{[f]df=}\text{∫}\text{?1}\text{1}\text{f(x)C}{}_{\mathrm{k}}{}^{\mathrm{(\lambda )}}\text{(x)dx (k = 0,1,…)}$

in which C_k^(λ) is the k-th Gegenbauer polynomial of order $\text{λ(λ > ?}\text{1}\text{2}\text{)}$, and f is a function satisfying the differential equation

$\text{∑}\text{i=0}\text{n}\text{P}{}_{\mathrm{i}}\text{(x)f}{}^{\mathrm{(i)}}\text{(x) = p(x) (?1?x?1)}$

of order n, where p₀, p₁, …, p_n ? 0 are polynomials, and m_k^〈λ〉[p] is known for every k. We give three methods of construction of such a recurrence relation. The first of them (called Method I) is optimum in a certain sense.     

Lp estimates for the wave equation

Let u(x, t) be the solution of u_tt ? Δ_xu = 0 with initial conditions $\text{u(x, 0) = g(x)}\text{and}\text{u}{}_{\mathrm{t}}\text{(x, 0) = ?;(x)}$. Consider the linear operator $\text{T: ?; → u(x, t)}$. (Here g = 0.) We prove for t fixed the following result. Theorem 1: T is bounded in L^p if and only if . Theorem 2: If the coefficients are variables in C and constant outside of some compact set we get: (a) If n = 2k the result holds for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ < (n ? 1)}{}^{\mathrm{?1}}$. (b) If n = 2k ? 1, the result is valid for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ ? (n ? 1)}$. This result are sharp in the sense that for p such that $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ > (n ? 1)}{}^{\mathrm{?1}}$ we prove the existence of $\text{?; ? L}{}^{\mathrm{P}}$ in such a way that $\text{T?; ? L}{}^{\mathrm{P}}$. Several applications are given, one of them is to the study of the Klein-Gordon equation, the other to the completion of the study of the family of multipliers $\text{m(ξ) = ψ(ξ) e}{}^{\mathrm{i¦\xi ¦}}\text{¦ ξ ¦}{}^{\mathrm{?b}}$ and finally we get that the convolution against the kernel $\text{K(x) = ?(x)(1 ? ¦ x ¦)}{}^{\mathrm{?1}}$ is bounded in H¹.     

Author index

A matrix T=(t_ik) is introduced, the coefficients of which are defined by $\text{k}{}_{\mathrm{ik}}\text{:= (}\text{i}{}^{\mathrm{k}}\text{(ik)!)}\text{Σ}{}_{\mathrm{x?S}}{}_{\mathrm{n}}\text{a}{}_{\mathrm{i}}\text{(x)}{}^{\mathrm{k}}\text{, i, k?N={1, 2, 3,…,}}$, where a_i(x) denotes the s the number of i cycles in the element x of the symmetric group S_n. It is shown that these numbers are natural numbers, that they are easy to evaluate, and that they serve very well in order to formulate an infinite number of characterizations of multiply transitive subgroups of symmetric groups in terms of the cycle structure of their elements.     

Proof of a conjecture of S. Chowla

Let θ(k, p) be the least s such that the congruence $\text{x}{}_1{}^{\mathrm{k}}\text{+ … + x}{}_{\mathrm{s}}{}^{\mathrm{k}}\text{≡ 0(}\text{mod}\text{p)}$ has a nontrivial solution. Let θ(k) = {max θ(k, p)| p > 1 + 2k}. The purpose of this note is to prove the following conjecture of S. Chowla: $\text{θ(k) = O(k}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext><mtext>+?</mtext>}}\text{)}$.     

Votes and a half-binomial

In two party elections with popular vote ratio $\text{p}\text{q}\text{,}\text{1}\text{2}\text{≤p=1 ?q}$, a theoretical model suggests replacing the so-called MacMahon cube law approximation $\text{(}\text{p}\text{q}\text{)}{}^3$, for the ratio $\text{P}\text{Q}$ of candidates elected, by the ratio $\text{?}{}_{\mathrm{k}}\text{(p)}\text{?}{}_{\mathrm{k}}\text{(q)}$ of the two half sums in the binomial expansion of (p+q)^2k+1 for some k. This ratio is nearly $\text{(}\text{p}\text{q}\text{)}{}^3$ when k = 6. The success probability $\text{g}{}_{\mathrm{k}}\text{(p)=(}\text{p}{}^{\mathrm{a}}\text{(p}{}^{\mathrm{a}}\text{+q}{}^{\mathrm{a}}\text{)}$ for the power law $\text{(}\text{p}\text{q}\text{)}{}^{\mathrm{a}}\text{?}\text{P}\text{Q}$ is shown to so closely approximate $\text{?}{}_{\mathrm{k}}\text{(p)=Σ}{}_0{}^{\mathrm{k}}\text{(}{}_{\mathrm{r}}{}^{\mathrm{2k+1}}\text{)p}{}^{\mathrm{2k+1?r}}\text{q}{}^{\mathrm{r}}$, if we choose $\text{a = a}{}_{\mathrm{k}}\text{=}\text{(2k+1)!}\text{4}{}^{\mathrm{k}}\text{k!k!}$, that $\text{1≤}\text{?}{}_{\mathrm{k}}\text{(p)}\text{g}{}_{\mathrm{k}}\text{(p)}\text{≤1.01884086}$ for $\text{k≥1}\text{if}\text{1}\text{2}\text{≤p≤1}$. Computationally, we avoid large binomial coefficients in computing $\text{?}{}_{\mathrm{k}}\text{(p)}$ for k>22 by expressing $\text{2?}{}_{\mathrm{k}}\text{(p)?1}$ as the sum $\text{(p?q) Σ}{}_0{}^{\mathrm{k}}\text{(4pq)}{}^{\mathrm{s}}\text{a}{}_{\mathrm{s}}\text{(2s+1)}$, whose terms decrease by the factors $\text{(4pq)(1?}\text{1}\text{2s}\text{)}$. Setting K = 4k+3, we compute a_k for the large k using a continued fraction $\text{πa}{}_{\mathrm{k}}{}^2\text{=}\text{K+1}{}^2\text{(}\text{2K+3}{}^2\text{(}\text{2K+5}{}^2\text{(2K+…)}\text{)}\text{)}$ derived from the ratio of π to the finite Wallis product approximation.     

Some asymptotic boundary behavior of solutions of non-linear parabolic initial-boundary value problems

For parabolic initial boundary value problems various results such as $\text{lim}{}_{\mathrm{t\; \downarrow \; 0}}\text{{}\text{(}\text{?u}\text{t6x}\text{)(0, t)}\text{(}\text{?u}{}_{\mathrm{\alpha }}\text{?x}\text{)(0, t)}}\text{= 1}$, where u satisfies $\text{?u}\text{?t}\text{= a(u)(}\text{?}{}^2\text{u}\text{?x}{}^2\text{), 0 < x < 1, 0 < t ? T, u(x, 0) = 0, u(0, t) = |}{}_1\text{(t), 0 < t ? T, u(1, t) = |}{}_2\text{(t), 0 < t ? T, u}{}_{\mathrm{\alpha }}\text{satisfies}\text{(}\text{?u}{}_{\mathrm{\alpha }}\text{?t}\text{) = α(}\text{?}{}^2\text{u}{}_{\mathrm{\alpha }}\text{?x}{}^2\text{), 0 < x < 1, 0 < t ? T, u}{}_{\mathrm{\alpha }}\text{(x, 0) = 0, u}{}_{\mathrm{\alpha }}\text{(0, t) = |}{}_1\text{(t), 0 < t ? T, u}{}_{\mathrm{\alpha }}\text{(1, t) = |}{}_2\text{(t), 0 < t ? T,}\text{and}\text{α = a(0)}$, are demonstrated via the maximum principle and potential theoretic estimates.     

On the probability that k positive integers are relatively prime II

Given a set S of positive integers let $\text{Z}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(t)}$ denote the number of k-tuples 〈m₁, …, m_k〉 for which $\text{m}{}_{\mathrm{i}}\text{∈ S ? [1, t]}$ and (m₁, …, m_k) = 1. Also let $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n)}$ denote the probability that k integers, chosen at random from $\text{S ? [1, n]}$, are relatively prime. It is shown that if P = {p₁, …, p_r} is a finite set of primes and S = {m : (m, p₁ … p_r) = 1}, then $\text{Z}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(t) = (td(S))}{}^{\mathrm{k}}\text{Π}{}_{\mathrm{\nu ?P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{) + O(t}{}^{\mathrm{k?1}}\text{)}$ if k ≥ 3 and $\text{Z}{}_2{}^{\mathrm{S}}\text{(t) = (td(S))}{}^2\text{Π}{}_{\mathrm{p?P}}\text{(1 ?}\text{1}\text{p}{}^2\text{) + O(t}\text{log}\text{t)}$ where d(S) denotes the natural density of S. From this result it follows immediately that $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n) → Π}{}_{\mathrm{p?P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{) = (ζ(k))}{}^{\mathrm{?1}}\text{Π}{}_{\mathrm{p\in P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{)}{}^{\mathrm{?1}}$ as n → ∞. This result generalizes an earlier result of the author's where $\text{P = ?}$ and S is then the whole set of positive integers. It is also shown that if S = {p₁^x₁ … p_r^x_r : x_i = 0, 1, 2,…}, then $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n) → 0}$ as n → ∞.     

Asymptotic properties of general symmetric hyperbolic systems

Results on partition of energy and on energy decay are derived for solutions of the Cauchy problem $\text{?u}\text{?t}\text{+ ∑}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{n}}\text{A}{}_{\mathrm{j}}\text{?u}\text{?x}{}_{\mathrm{j}}\text{= 0, u(0, x) = ?(x)}$. Here the A_j's are constant, k × k Hermitian matrices, x = (x₁,…, x_n), t represents time, and u = u(t, x) is a k-vector. It is shown that the energy of Mu approaches a limit $\text{E}{}_{\mathrm{M}}\text{(?)}\text{as}\text{¦ t ¦ → ∞}$, where M is an arbitrary matrix; that there exists a sufficiently large subspace of data ?, which is invariant under the solution group U₀(t) and such that $\text{U}{}_0\text{(t)? = 0}\text{for}\text{¦ x ¦ ? a ¦ t ¦ ? R, a}\text{and}\text{R}$ depending on ? and that the local energy of nonstatic solutions decays as $\text{¦ t ¦ → ∞}$. More refined results on energy decay are also given and the existence of wave operators is established, considering a perturbed equation $\text{E(x)}\text{?u}\text{?t}\text{+ ∑}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{n}}\text{A}{}_{\mathrm{j}}\text{?u}\text{?x}{}_{\mathrm{j}}\text{= 0,}\text{where}\text{¦ E(x) ? I ¦ = O(¦ x ¦}{}^{\mathrm{?1\; ?\; ?}}\text{)}$ at infinity.