Michael J. Mossinghoff

Sign changes in sums of the Liouville function

The Liouville function A(n) is the completely multiplicative function whose value is -1 at each prime. We develop some algorithms for computing the sum T(n) = Σ n k=1 λ(k)/k, and use these methods to determine the smallest positive integer n where T(n) < 0. This answers a question originating in some work of Turan, who linked the behavior of T(n) to questions about the Riemann zeta function. We also study the problem of evaluating Polyas sum L(n) = Σ n k=1 λ(k), and we determine some new local extrema for this function, including some new positive values.

A

Since the proof of [3] is based on the maximum principle, which is available in its full generality only for (cooperative systems of) second order equations, one would tend to believe that such type of result cannot hold for (1) or (2). Trying to get a negative answer to the question whether or not positive solutions are radially symmetric will necessarily lead to a strict p.d.e. approach and will hence be hard to obtain. The radially decreasing part of the claim however allows an o.d.e.-counterexample as we will show shortly. Let us fix this part in a conjecture:

1 Problem

We determine the minimal Mahler measure of a primitive irreducible noncyclotomic polynomial with integer coefficients and fixed degree D, for each even degree D ≤ 54. We also compute all primitive irreducible noncyclotomic polynomials with measure less than 1.3 and degree at most 44.

Minimal Mahler Measures

The maximal area of a polygon with n = 2m edges and unit diameter is not known when m ≥ 5, nor is the maximal perimeter of a convex polygon with n = 2m edges and unit diameter known when m ≥ 4. We construct improved polygons in both problems, and show that the values we obtain cannot be improved for large n by more than c1/n3 in the area problem and c2/n5 in the perimeter problem, for certain constants c1 and c2.

Isodiametric Problems for Polygons

The minimum value of the Mahler measure of a nonreciprocal polynomial whose coefficients are all odd integers is proved here to be the golden ratio. The smallest measures of reciprocal polynomials with ±1 coefficients and degree at most 72 are also determined.

THE MAHLER MEASURE OF POLYNOMIALS WITH ODD COEFFICIENTS

We examine sequences of polynomials with {+1, -1} coefficients constructed using the iterations p(x) → p(x) ± x d+1 p * (-x), where d is the degree of p and p * is the reciprocal polynomial of p. If po = 1 these generate the Rudin-Shapiro polynomials, We show that the L4 norm of these polynomials is explicitly computable. We are particularly interested in the case where the iteration produces sequences with smallest possible asymptotic L4 norm (or, equivalently, with largest possible asymptotic merit factor). The Rudin-Shapiro polynomials form one such sequence. We determine all p 0 of degree less than 40 that generate sequences under the iteration with this property. These sequences have asymptotic merit factor 3. The first really distinct example has a p 0 of degree 19.

Rudin-Shapiro-like polynomials in L 4

A Barker sequence is a finite sequence of integers, each ±1, whose aperiodic autocorrelations are all as small as possible. It is widely conjectured that only finitely many Barker sequences exist. We describe connections between Barker sequences and several problems in analysis regarding the existence of polynomials with ±1 coefficients that remain flat over the unit circle according to some criterion. First, we amend an argument of Saffari to show that a polynomial constructed from a Barker sequence remains within a constant factor of its L2 norm over the unit circle, in connection with a problem of Littlewood. Second, we show that a Barker sequence produces a polynomial with very large Mahler’s measure, in connection with a question of Mahler. Third, we optimize an argument of Newman to prove that any polynomial with ±1 coefficients and positive degree n−1 has L1 norm less than √ n− .09, and note that a slightly stronger statement would imply that long Barker sequences do not exist. We also record polynomials with ±1 coefficients having maximal L1 norm or maximal Mahler’s measure for each fixed degree up to 24. Finally, we show that if one could establish that the polynomials in a particular sequence are all irreducible over Q, then an alternative proof that there are no long Barker sequences with odd length would follow.

Number Theory and Polynomials: Barker sequences and flat polynomials

Let M(P(z1, . . . , zn)) denote Mahlers measure of the polynomial P(z1, . . . , zn). Measures of polynomials in n variables arise naturally as limiting values of measures of polynomials in fewer variables. We describe several methods for searching for polynomials in two variables with integer coefficients having small measure, demonstrate effective methods for computing these measures, and identify 48 polynomials P(x, y) with integer coefficients, irreducible over ℚ, for which 1 < M(P(x, y)) < 1.37.

Small Limit Points of Mahler's Measure

We study the minimal degree d(m) of a polynomial with all coefficients in {—1, 0, 1} and a zero of order m at 1. We determine dlm) for m ≤ 10 and compute all the extremal polynomials. We also determine the minimal degree for m = 11 and m = 12 among certain symmetric polynomials, and we find explicit examples with small degree for m ≤ 21. Each of the extremal examples is a pure product polynomial. The method uses algebraic number theory and combinatorial computations and relies on showing that a, polynomial with bounded degree, restricted coefficients, and a zero of high order at 1 automatically vanishes at several roots of unity.

Polynomials with height 1 and prescribed vanishing at 1

A Newman polynomial has all its coefficients in {0, 1} and constant term 1. It is known that every root of a Newman polynomial lies in the slit annulus {z ∈ C : τ−1 < |z| < τ} \ R+, where τ denotes the golden ratio, but not every polynomial having all of its conjugates in this set divides a Newman polynomial. We show that every negative Pisot number in (−τ,−1) with no positive conjugates, and every negative Salem number in the same range obtained by using Salem’s construction on small negative Pisot numbers, is satisfied by a Newman polynomial. We also construct a number of polynomials having all their conjugates in this slit annulus, but that do not divide any Newman polynomial. Finally, we determine all negative Salem numbers in (−τ,−1) with degree at most 20, and verify that every one of these is satisfied by a Newman polynomial.