Differentiating Polynomials, and Ζ(2)
نویسنده
چکیده
Polynomials are fascinating because they have many facets to their personalities. By definition, a polynomial f ∈ C[x] is an expression of the form (1.1) f(x) = a0 + a1x+ · · ·+ anx , where the aj are complex numbers. By the fundamental theorem of algebra, f also has a representation as a product (1.2) f(x) = an(x− x1) · · · (x− xn), where xj are the roots of f . The fact that there are two ways of looking at polynomials provides possibilities that are hidden by their apparent simplicity. In this paper we look at two different polynomials which have equally spaced zeros, and we study the zeros of their derivatives. We will see that Euler’s famous result
منابع مشابه
Moments of the Derivative of the Riemann Zeta-function and of Characteristic Polynomials
Characteristic polynomials of unitary matrices are extremely useful models for the Riemann zeta-function ζ(s). The distribution of their eigenvalues give insight into the distribution of zeros of the Riemann zeta-function and the values of these characteristic polynomials give a model for the value distribution of ζ(s). See the works [KS] and [CFKRS] for detailed descriptions of how these model...
متن کاملAsymptotics of Zeros of Polynomials Arising from Rational Integrals
We prove that the zeros of the polynomials Pm(a) of degree m, defined by Borosh and Moll via Pm(a) = 2m+3/2 π (a + 1) Z ∞ 0 dx (x4 + 2ax2 + 1)m+1 , approach the lemniscate {ζ ∈ C : |ζ − 1| = 1, Rζ < 0}, as m diverges.
متن کاملOrthogonal polynomials for modified Gegenbauer weight and corresponding quadratures
In this paper we consider polynomials orthogonal with respect to the linear functional L : P → C, defined by L[p] = ∫ 1 −1 p(x)(1 − x 2)λ−1/2 exp(iζ x) dx, where P is a linear space of all algebraic polynomials, λ > −1/2 and ζ ∈ R. We prove the existence of such polynomials for some pairs of λ and ζ , give some their properties, and finally give an application to numerical integration of highly...
متن کاملIrrationality proof of a q-extension of ζ(2) using little q-Jacobi polynomials
We show how one can use Hermite-Padé approximation and little q-Jacobi polynomials to construct rational approximants for ζq(2). These numbers are qanalogues of the well known ζ(2). Here q = 1 p , with p an integer greater than one. These approximants are good enough to show the irrationality of ζq(2) and they allow us to calculate an upper bound for its measure of irrationality: μ (ζq(2)) ≤ 10...
متن کاملThe zeta function, L-functions, and irreducible polynomials
= 1 1− q1−s , where we have convergence for all s with <(s) > 1. This automatically gives us analytic continuation of ζ to all of C. We note the following simple observations: 1. The Riemann hypothesis: All zeroes of ζ lie on the line <(s) = 1/2, simply because there are no zeroes! 2. ζ has poles at s = 1 + 2πin log q . 3. Locally around s = 1, ζ has the Laurent series expansion: 1 (s− 1) log q...
متن کامل