Kwok-Kwong Stephen Choi

Binary sequences with merit factor greater than 6.34

The maximum known asymptotic merit factor for binary sequences has been stuck at a value of 6 since the 1980s. Several authors have suggested that this value cannot be improved. In this paper, we construct an infinite family of binary sequences whose asymptotic merit factor we conjecture to be greater than 6.34. We present what we believe to be compelling evidence in support of this conjecture. The numerical experimentation that led to this construction is a significant part of the story.

On the Representations of xy + yz + ZX

We show that there are at most 19 integers that are not of the form xy + yz + xz with x, y, z ≥ 1. Eighteen of them are small and easily found. The remaining possibility must be greater than 1011 and cannot occur if we assume the Generalized Riemann Hypothesis.

Explicit merit factor formulae for Fekete and Turyn polynomials

We give explicit formulas for the L 4 norm (or equivalently for the merit factors) of various sequences of polynomials related to the Fekete polynomials.

On Dirichlet Series for Sums of Squares

AbstractHardy and Wright (An Introduction to the Theory of Numbers, 5th edn., Oxford, 1979) recorded elegant closed forms for the generating functions of the divisor functions σk(n) and σk2(n) in the terms of Riemann Zeta function ζ(s) only. In this paper, we explore other arithmetical functions enjoying this remarkable property. In Theorem 2.1 below, we are able to generalize the above result and prove that if fi and gi are completely multiplicative, then we have

Merit Factors of Polynomials Formed by Jacobi Symbols

\sum\limits_{n = 1}^\infty {\frac{{(f_1 * g_1 )(n) \cdot (f_2 * g_2 )(n)}}{{n^s }} = } \frac{{L_{f_1 f_2 } (s)L_{g_1 g_2 } (s)L_{f_1 g_2 } (s)L_{g_1 f_2 } (s)}}{{L_{f_1 f_2 g_1 g_2 } (2s)}}

Small Prime Solutions of Quadratic Equations

where Lf(s) := ∑n = 1∞f(n)n−s is the Dirichlet series corresponding to f. Let rN(n) be the number of solutions of x12 + ··· + xN2 = n and r2,P(n) be the number of solutions of x2 + Py2 = n. One of the applications of Theorem 2.1 is to obtain closed forms, in terms of ζ(s) and Dirichlet L-functions, for the generating functions of rN(n), rN2(n), r2,P(n) and r2,P(n)2 for certain N and P. We also use these generating functions to obtain asymptotic estimates of the average values for each function for which we obtain a Dirichlet series.

ON THE DISTRIBUTION OF POINTS IN PROJECTIVE SPACE OF BOUNDED HEIGHT

We characterize all cyclotomic polynomials of even degree with coefficients restricted to the set {+1, −1}. In this context a cyclotomic polynomial is any monic polynomial with integer coefficients and all roots of modulus 1. Inter alia we characterize all cyclotomic polynomials with odd coefficients. The characterization is as follows. A polynomial P(x) with coefficients ±1 of even degree N–l is cyclotomic if and only if where N = P1P1 … Pr and the Pi are primes, not necessarily distinct, and where ϕp(x) := (xp – 1)/ (x – 1) isthe p-th cyclotomic polynomial. We conjecture that this characterization also holds for polynomials of odd degree with ±1 coefficients. This conjecture is based on substantial computation plus a number of special cases. Central to this paper is a careful analysis of the effect of Graeffes root squaring algorithm on cyclotomic polynomials.

Conditional bounds for small prime solutions of linear equations

We give explicit formulas for the L4 norm (or equivalently for the merit factors) of various sequences of polynomials related to the polynomials