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Register a new userCollatz polynomials: an introduction with bounds on their zeros
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The Collatz Conjecture (also known as the 3x+1 Problem) proposes that the following algorithm will, after a certain number of iterations, always yield the number 1: given a natural number, multiply by three and add one if the number is odd, halve the resulting number, then repeat. In this article, for each $N$ for which the Collatz Conjecture holds we define the $N^{th}$ Collatz polynomial to be the monic polynomial with constant term $N$ and $k^{th}$ term (for $k > 1$) the $k^{th}$ iterate of $N$ under the Collatz function. In particular, we bound the moduli of the roots of these polynomials, prove theorems on when they have rational integer roots, and suggest further applications and avenues of research.
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In a previous work, we introduced the Collatz polynomials; these are the polynomials $left[P_N(z)right]_{Ninmathbb{N}}$ such that $left[z^0right]P_N = N$ and $left[z^{k+1}right]P_N = cleft(left[z^kright]P_Nright)$, where $c:mathbb{N}rightarrow mathbb{N}$ is the Collatz function $1rightarrow 0$, $2nrightarrow n$, $2n+1rightarrow 3n+2$ (for example, $P_5(z) = 5 + 8z + 4z^2 + 2z^3 + z^4$). In this article, we prove that all zeros of $P_N$ (which we call Collatz zeros) lie in an annulus centered at the origin, with outer radius 2 and inner radius a function of the largest odd iterate of $N$. Moreover, using an extension of the Enestrom-Kakeya Theorem, we prove that $|z| = 2$ for a root of $P_N$ if and only if the Collatz trajectory of $N$ has a certain form; as a corollary, the set of $N$ for which our upper bound is an equality is sparse in $mathbb{N}$. Inspired by these results, we close with some questions for further study.
Period polynomials have long been fruitful tools for the study of values of $L$-functions in the context of major outstanding conjectures. In this paper, we survey some facets of this study from the perspective of Eichler cohomology. We discuss ways to incorporate non-cuspidal modular forms and values of derivatives of $L$-functions into the same framework. We further review investigations of the location of zeros of the period polynomial as well as of its analogue for $L$-derivatives.
In recent years, a number of papers have been devoted to the study of roots of period polynomials of modular forms. Here, we study cohomological analogues of the Eichler-Shimura period polynomials corresponding to higher $L$-derivatives. We state general conjectures about the locations of the roots of the full and odd parts of the polynomials, in analogy with the existing literature on period polynomials, and we also give numerical evidence that similar results hold for our higher derivative period polynomials in the case of cusp forms. We prove a special case of this conjecture in the case of Eisenstein series.
Let $x_1$ and $x_k$ be the least and the largest zeros of the Laguerre or Jacobi polynomial of degree $k.$ We shall establish sharp inequalities of the form $x_1 <A, x_k >B,$ which are uniform in all the parameters involved. Together with inequalities in the opposite direction, recently obtained by the author, this locates the extreme zeros of classical orthogonal polynomials with the relative precision, roughly speaking, $O(k^{-2/3}).$
The Collatz conjecture is explored using polynomials based on a binary numeral system. It is shown that the degree of the polynomials, on average, decreases after a finite number of steps of the Collatz operation, which provides a weak proof of the conjecture by using induction with respect to the degree of the polynomials.
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