On the finiteness of certain Rabinowitsch polynomials II

Let m be a positive integer and f_m(x) be a polynomial of the form f_m(x)=x²+x−m. We call a polynomial f_m(x) a Rabinowitsch polynomial if for and consecutive integers is either 1 or prime. In Byeon (J. Number Theory 94 (2002) 177), we showed that there are only finitely many Rabinowitsch polynomials f_m(x) such that 1+4m is square free. In this note, we shall remove the condition that 1+4m is square free.     

A complete determination of Rabinowitsch polynomials

Let m be a positive integer and fm(x) be a polynomial of the form fm(x)=x²+x−m. We call a polynomial fm(x) a Rabinowitsch polynomial if for and consecutive integers x=x₀,x₀+1,…,x₀+s−1, |fm(x)| is either 1 or prime. In this paper, we show that there are exactly 14 Rabinowitsch polynomials fm(x).     

Expansions in non-integer bases: Lower, middle and top orders

Let q∈(1,2); it is known that each x∈[0,1/(q−1)] has an expansion of the form with an∈{0,1}. It was shown in [P. Erd?s, I. Joó, V. Komornik, Characterization of the unique expansions and related problems, Bull. Soc. Math. France 118 (1990) 377-390] that if , then each x∈(0,1/(q−1)) has a continuum of such expansions; however, if , then there exist infinitely many x having a unique expansion [P. Glendinning, N. Sidorov, Unique representations of real numbers in non-integer bases, Math. Res. Lett. 8 (2001) 535-543]. In the present paper we begin the study of parameters q for which there exists x having a fixed finite number m>1 of expansions in base q. In particular, we show that if q<q₂=1.71…, then each x has either 1 or infinitely many expansions, i.e., there are no such q in . On the other hand, for each m>1 there exists γm>0 such that for any q∈(2−γm,2), there exists x which has exactly m expansions in base q.     

On the oscillations of multiplicative functions taking values ±1

For multiplicative functions f(n), which take on the values ±1, we show that under certain conditions on f(n), for all x sufficiently large, there are at least values of n?x for which f(n(n+1))=−1.     

Quadratic addition rules for quantum integers

For every positive integer n, the quantum integer [n]_q is the polynomial [n]_q=1+q+q²+?+q^n-1. A quadratic addition rule for quantum integers consists of sequences of polynomials , , and such that for all m and n. This paper gives a complete classification of quadratic addition rules, and also considers sequences of polynomials that satisfy the associated functional equation .     

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

     

On certain exponential sums over primes

Let f(x) be a real valued polynomial in x of degree k?4 with leading coefficient α. In this paper, we prove a non-trivial upper bound for the quantity

     

Kuznietsov trace formula and weighted distribution of Hecke eigenvalues

Let (X,μ) be a measurable topological space. Let S₁,S₂,… be a family of finite subsets of X. Suppose each x∈S_i has a weight w_ix∈R⁺ assigned to it. We say {S_i} is {w_i}-distributed with respect to the measure μ if for any continuous function f on X, we have .Let S(N,k) be the space of modular cusp forms over Γ₀(N) of weight k and let be a basis which consists of Hecke eigenforms. Let a_r(h) be the rth Fourier coefficient of h. Let x_p^h be the eigenvalue of h relative to the normalized Hecke operator T′_p. Let ||·|| be the Petersson norm on S(N,k). In this paper we will show that for any even integer k?3, is -distributed with respect to a polynomial times the Sato-Tate measure when N→∞.     

Class numbers of quadratic fields, Hasse invariants of elliptic curves, and the supersingular polynomial

Using the theory of elliptic curves, we show that the class number h(−p) of the field appears in the count of certain factors of the Legendre polynomials , where p is a prime >3 and m has the form (p−e)/k, with k=2,3 or 4 and . As part of the proof we explicitly compute the Hasse invariant of the Hessian curve y²+αxy+y=x³ and find an elementary expression for the supersingular polynomial ss_p(x) whose roots are the supersingular j-invariants of elliptic curves in characteristic p. As a corollary we show that the class number h(−p) also shows up in the factorization of certain Jacobi polynomials.     

On the prime power factorization of n!

In this paper, we prove two results. The first theorem uses a paper of Kim (J. Number Theory 74 (1999) 307) to show that for fixed primes p₁,…,p_k, and for fixed integers m₁,…,m_k, with , the numbers (e_p1(n),…,e_pk(n)) are uniformly distributed modulo (m₁,…,m_k), where e_p(n) is the order of the prime p in the factorization of n!. That implies one of Sander's conjectures from Sander (J. Number Theory 90 (2001) 316) for any set of odd primes. Berend (J. Number Theory 64 (1997) 13) asks to find the fastest growing function f(x) so that for large x and any given finite sequence , there exists n<x such that the congruences hold for all i?f(x). Here, p_i is the ith prime number. In our second result, we are able to show that f(x) can be taken to be at least , with some absolute constant c₁, provided that only the first odd prime numbers are involved.