A formula for the central value of certain Hecke L-functions

Let be the negative of a prime, and O_K its ring of integers. Let D be a prime ideal in O_K of prime norm congruent to . Under these assumptions, there exists Hecke characters ψ_D of K with conductor (D) and infinite type (1,0). Their L-series L(ψ_D,s) are associated to a CM elliptic curve A(N,D) defined over the Hilbert class field of K. We will prove a Waldspurger-type formula for L(ψ_D,s) of the form L(ψ_D,1)=Ω∑_[A],Ir(D,[A],I)m_[A],I([D]) where the sum is over class ideal representatives I of a maximal order in the quaternion algebra ramified at |N| and infinity and [A] are class group representatives of K. An application of this formula for the case N=-7 will allow us to prove the non-vanishing of a family of L-series of level 7|D| over K.     

Standing waves for supercritical nonlinear Schrödinger equations

Let V(x) be a non-negative, bounded potential in RN, N?3 and p supercritical, . We look for positive solutions of the standing-wave nonlinear Schrödinger equation Δu−V(x)u+up=0 in RN, with u(x)→0 as |x|→+∞. We prove that if V(x)=o(⁻²|x|) as |x|→+∞, then for N?4 and this problem admits a continuum of solutions. If in addition we have, for instance, V(x)=O(|x|^−μ) with μ>N, then this result still holds provided that N?3 and . Other conditions for solvability, involving behavior of V at ∞, are also provided.     

Integers with dense divisors 2

Let 1=d₁(n)<d₂(n)<?<d_τ(n)=n be the sequence of all positive divisors of the integer n in increasing order. We say that the divisors of n are t-dense iff max_1?i<τ(n)d_i+1(n)/d_i(n)?t. Let D(x,t) be the number of positive integers not exceeding x whose divisors are t-dense. We show that for x?3, and , we have , where , and d(w) is a continuous function which satisfies d(w)?1/w for w?1. We also consider other counting functions closely related to D(x,t).     

Symmetry and specializability in the continued fraction expansions of some infinite products

Let f(x)∈Z[x]. Set f₀(x)=x and, for n?1, define fn(x)=f(fn−1(x)). We describe several infinite families of polynomials for which the infinite product

     

Removable singularity of the polyharmonic equation

Let x₀∈Ω⊂Rⁿ, n≥2, be a domain and let m≥2. We will prove that a solution u of the polyharmonic equation Δ^mu=0 in Ω?{x₀} has a removable singularity at x₀ if and only if as |x−x₀|→0 for n≥3 and as |x−x₀|→0 for n=2. For m≥2 we will also prove that u has a removable singularity at x₀ if |u(x)|=o(|x−x₀|^2m−n) as |x−x₀|→0 for n≥3 and |u(x)|=o(|x−x₀|^2m−2log(|x−x₀|⁻¹)) as |x−x₀|→0 for n=2.     

Distance-two labelings of digraphs

For positive integers j?k, an L(j,k)-labeling of a digraph D is a function f from V(D) into the set of nonnegative integers such that |f(x)-f(y)|?j if x is adjacent to y in D and |f(x)-f(y)|?k if x is of distance two to y in D. Elements of the image of f are called labels. The L(j,k)-labeling problem is to determine the -number of a digraph D, which is the minimum of the maximum label used in an L(j,k)-labeling of D. This paper studies -numbers of digraphs. In particular, we determine -numbers of digraphs whose longest dipath is of length at most 2, and -numbers of ditrees having dipaths of length 4. We also give bounds for -numbers of bipartite digraphs whose longest dipath is of length 3. Finally, we present a linear-time algorithm for determining -numbers of ditrees whose longest dipath is of length 3.     

Carleson embeddings for Sobolev spaces via heat equation

Let u(t,x) be the solution of the heat equation (∂_t-Δ_x)u(t,x)=0 on subject to u(0,x)=f(x) on Rⁿ. The main goal of this paper is to characterize such a nonnegative measure μ on that f(x)?u(t²,x) induces a bounded embedding from the Sobolev space , p∈[1,n) into the Lebesgue space , q∈(0,∞).     

On an additive representation function

Let A be an infinite subset of natural numbers, and X a positive real number. Let r(n) denotes the number of solution of the equation n=a₁+a₂ where a₁?a₂ and a₁, a₂∈A. Also let |A(X)| denotes the number of natural numbers which are less than or equal to X and belong to A. For those A which satisfy the condition that for all sufficiently large natural numbers n we have r(n)≠1, we improve the lower bound of |A(X)| given by Nicolas et. al. [NRS98]. The bound which we obtain is essentially best possible.     

Approximation by smooth functions with no critical points on separable Banach spaces

We characterize the class of separable Banach spaces X such that for every continuous function and for every continuous function there exists a C¹ smooth function for which |f(x)−g(x)|?ε(x) and g^′(x)≠0 for all x∈X (that is, g has no critical points), as those infinite-dimensional Banach spaces X with separable dual X^∗. We also state sufficient conditions on a separable Banach space so that the function g can be taken to be of class Cp, for p=1,2,…,+∞. In particular, we obtain the optimal order of smoothness of the approximating functions with no critical points on the classical spaces ?p(N) and Lp(Rn). Some important consequences of the above results are (1) the existence of a non-linear Hahn-Banach theorem and the smooth approximation of closed sets, on the classes of spaces considered above; and (2) versions of all these results for a wide class of infinite-dimensional Banach manifolds.     

Integers with dense divisors

Let 1=d₁(n)<d₂(n)<?<d_τ(n)=n be the sequence of all positive divisors of the integer n in increasing order. We say that the divisors of n are y-dense iff max_1?i<τ(n)d_i+1(n)/d_i(n)?y. Let D(x,y,z) be the number of positive integers not exceeding x whose divisors are y-dense and whose prime divisors are bigger than z, and let , and . We show that is equivalent, in a large region, to a function d(u,v) which satisfies a difference-differential equation. Using that equation we find that d(u,v)?(1−u/v)/(u+1) for v?3+ε. Finally, we show that d(u,v)=e^−γd(u)+O(1/v), where γ is Euler's constant and d(u)∼x⁻¹D(x,y,1), for fixed u. This leads to a new estimate for d(u).     

Beta-expansion and continued fraction expansion

For any real number β>1, let ε(1,β)=(ε₁(1),ε₂(1),…,εn(1),…) be the infinite β-expansion of 1. Define . Let x∈[0,1) be an irrational number. We denote by kn(x) the exact number of partial quotients in the continued fraction expansion of x given by the first n digits in the β-expansion of x. If is bounded, we obtain that for all x∈[0,1)?Q,

     

On the prime power factorization of n!, II

Let p, q be primes and m be a positive integer. For a positive integer n, let ep(n) be the nonnegative integer with pep(n)|n and pep(n)+1?n. The following results are proved: (1) For any positive integer m, any prime p and any ε∈Zm, there are infinitely many positive integers n such that ; (2) For any positive integer m, there exists a constant D(m) such that if ε,δ∈Zm and p, q are two distinct primes with max{p,q}?D(m), then there exist infinitely many positive integers n such that , . Finally we pose four open problems.     

Fast Growing Solutions of Linear Differential Equations in the Unit Disc

For $n \in \mathbb{N}$ , the n-order of an analytic function f in the unit disc D is defined by $$\sigma _{{{M,n}}} (f) = {\mathop {\lim \sup }\limits_{r \to 1^{ - } } }\frac{{\log ^{ + }_{{n + 1}} M(r,f)}} {{ - \log (1 - r)}},$$ where log⁺ x  =  max{log x, 0}, log ⁺ ₁ x  =  log ⁺ x, log ⁺ _n+1 x  =  log ⁺ log ⁺ _n x, and M(r, f) is the maximum modulus of f on the circle of radius r centered at the origin. It is shown, for example, that the solutions f of the complex linear differential equation $$f^{{(k)}} + a_{{k - 1}} (z)f^{{(k - 1)}} + \cdots + a_{1} (z)f^{\prime} + a_{0} (z)f = 0,\quad \quad \quad (\dag)$$ where the coefficients are analytic in D, satisfy σ _M,n+1(f)  ≤  α if and only if σ _M,n (a _j )  ≤  α for all j  =  0, ..., k ? 1. Moreover, if q ∈{0, ..., k ? 1} is the largest index for which $\sigma _{M,n} ( a_{q}) = {\mathop {\max }\limits_{0 \leq j \leq k - 1} }{\left\{ {\sigma _{{M,n}} {\left( {a_{j} } \right)}} \right\}}$ , then there are at least k ? q linearly independent solutions f of ( $\dag$ ) such that σ _M,n+1(f) = σ _M,n (a _q ). Some refinements of these results in terms of the n-type of an analytic function in D are also given.     

Positive solutions for some Schrödinger equations having partially periodic potentials

This paper is concerned with the problem of finding positive solutions of the equation −Δu+(a_∞+a(x))u=|u|^q−2u, where q is subcritical, Ω is either RN or an unbounded domain which is periodic in the first p coordinates and whose complement is contained in a cylinder , a_∞>0, a∈C(RN,R) is periodic in the first p coordinates, infx∈RN(a_∞+a(x))>0 and a(x^′,x^″)→0 as |x^″|→∞ uniformly in x^′. The cases a?0 and a?0 are considered and it is shown that, under appropriate assumptions on a, the problem has one solution in the first case and p+1 solutions in the second case when p?N−2.     

Upper bounds on n-dimensional Kloosterman sums

Let p^m be any prime power and K_n(a,p^m) be the Kloosterman sum , where the x_i are restricted to values not divisible by p. Let m,n be positive integers with m?2 and suppose that p^γ||(n+1). We obtain the upper bound , for odd p. For p=2 we obtain the same bound, with an extra factor of 2 inserted.     

A general estimation for partitions with large difference: For even case

Let xN,i(n) denote the number of partitions of n with difference at least N and minimal component at least i, and yM,j(n) the number of partitions of n into parts which are . If N is even and i is co-prime with N+2i+1, we prove that

xN,i(n)?yN+2i+1,i(n)