Questions tagged [zeroes]

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What's the meaning of the nontrivial zeros of Selberg zeta function?

In the case of arithmetic variety over finite field, the zero points of the Hasse-Weil zeta function reflect the pure weights (i.e. dimension). On the other hand, in the case of the Selberg zeta ...
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1answer
60 views

Dominant root of a family of odd degree polynomials

Let $q$ be an odd positive integer and denote by $f_q(x)=x^q-x^{q-2}-1$. This family of polynomials appears in some problems related to recurrence sequences. In order to try to use some standard ...
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0answers
137 views

Chebyshev polynomials from roots [closed]

Given a polynomial $P_n(x)$ with roots at $\{a_i\}$: $P_n(x)=\prod\limits_{i=1}^{i=n} (x-a_i)$ one can obtain the Chebyshev coefficients from the roots. It is implemented eg in python. Can we ...
2
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1answer
151 views

Homogeneous polynomial in 4 variable with non degenerate zero

I've got a very simple question about a homogenous polynomial, for which I cannot see neatly how to proceed (probably due to my limitations in algebraic geometry though). Any help would be greatly ...
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0answers
60 views

Example of an integer $n_0$ such that $1+\sum_{k=2}^{n_0} \zeta(k)^s=0$ has repeated roots

After I was studying the exercise Problem 4.20 from [1] I was inspired to ask about next problem, where $\zeta(k)$ denotes, for integers $k>1$, particular values of the Riemann zeta function. And $...
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0answers
96 views

Solution of $s\zeta'(s)-\zeta(s)=0$ for some infinite strip $\subset\{0<\Re s<1\}$

I did the following calculation, first I take the $k$-th derivative with respect $s$ of the Mellin transform of the fractional part $\{\frac{1}{t}\}$ defined for $0<\Re s<1$ as it is showed in ...
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0answers
67 views

An elliptic function built from a log-theta-function integral?

I'm studying the apparent ellipticity of $$\Theta(z,a):=\frac{\exp(\tfrac2a\int_0^a\ln\vartheta(x+z)\,dx)}{\vartheta(z)\vartheta(z+a)},\tag1$$with $a$ is a free parameter and \begin{align}\vartheta(z)&...
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0answers
51 views

Root of the expectation of a random rational function

I am trying to figure out a formula for the unique $\lambda>1$ such that $$ \mathbb{E}\bigg[\frac{X}{\lambda -X}\bigg]=1 $$ where $X$ is a discrete random variable taking values in $\{\frac{1}{n},....
4
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1answer
99 views

Existence of Laurent series with zeroes at $𝑒^2𝑛$ (𝑛∈ℕ0 ) and even faster coefficient decay

This is an extension of an earlier question of mine which corresponds to the case $A = 1$. Precisely, my question is as follows: Given $A > 0$ fixed but arbitrary, is there a non-trivial ...
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1answer
200 views

Existence of Laurent series with zeroes at $e^{2n}$ ($n \in \Bbb{N}_0$) and extremely fast coefficient decay

I am working on a problem in harmonic analysis, which I converted into the following existence problem concerning Laurent series. I am a bit at a loss concerning this problem, since my knowledge of ...
33
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4answers
2k views

A family of polynomials whose zeros all lie on the unit circle

I had posted the following problem on stack exchange before. Suppose $\lambda$ is a real number in $\left( 0,1\right)$, and let $n$ be a positive integer. Prove that all the roots of the ...
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1answer
145 views

Is there always a polynomial with real zeroes between two polynomials with real zeroes?

Suppose that we have two complex polynomials $p(z)=\sum_{k=0}^n p_kz^k$ and $q(z)=\sum_{k=0}^n q_kz^k$ and also that we have $|p_k|<|q_k|$ for $k=0,1,...,n$. We say that a polynomial $r$ is ...
1
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1answer
128 views

Locus of roots of all convex combinations of two monic polynomials, II

This post contains a revised conjecture to a conjecture I posed previously which was shown to be false. Let $p, q \in \mathbb{C}[t]$ be two monic polynomials of degree $n \ge 1$. For $\alpha \in [0,1]...
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0answers
71 views

Deriving “quasi-theta” functions from theta functions' zeros

I've been trying to sift the zeros of \begin{align}\vartheta(z)&=e^{-\pi z^2}\prod_{k\ge1}(1+e^{-(2k-1)\pi+2\pi z})(1+e^{-(2k-1)\pi-2\pi z})\\ &=\prod_{k\ge1}(1+e^{-(2k-1)\pi+2i\pi z})(1+e^{-(...
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2answers
415 views

The $n$-th derivative has $n$ zeros. Can such a function be unbounded?

I asked this on Math.SE some days ago, but without any success. For some application I need a formal definition of bell-shaped function. So I had the following idea: Definition. A $C^\infty$-...
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1answer
138 views

A simple question about the zeros of an Entire Function in LP-class

We know that functions in $\mathcal{LP}$-class, and only these, are uniform limits, on compact subsets of $\mathbb{C}$, of polynomials with only real zeros. Question: Does it means that these ...
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2answers
529 views

Locus of roots of all convex combinations of two monic polynomials

Let $p,q$ be monic polynomials in $\mathbb{C}[t]$ and for $\alpha \in [0,1]$, let $c_\alpha := \alpha p + (1-\alpha)q \in \mathbb{C}[t]$. Since the roots of a polynomial vary continuously with respect ...
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2answers
424 views

Are trivial zeros of the zeta function important?

Non-trivial zeros play an important (main) role in the distribution of prime numbers. Are there theorems in which trivial zeros play an important (main) role?
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2answers
328 views

How to obtain an asymptotic formula for the zeros of the Airy function ($a_i$ for large $i$)?

Let $a_i$ be the zeros of the Airy function, which is the solution top the ODE $y''-xy=0$, such that Ai(a_i)=0. According to WolframMathWorld e.g., $a_{1..4}= -2.33811, -4.08795, -5.52056, -6....
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1answer
495 views

Zeroes of a not quite holomorphic (but random if helpful) function

I’m interested in the zeroes of the complex function $f(z,\bar{z}) = p(z) + \frac{1}{log(|z|)} q(z)$ where both $p$ and $q$ are polynomials of the complex variable $z$ (and are therefore holomorphic)...
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3answers
518 views

smooth functional to detect whether a function has a zero

Does there exist a function $F : C^\infty(\mathbb{R}, [0, \infty)) \to \mathbb{R}$ with the following properties: $F(f) = 0$ if and only if there exists an $x \in [0,1]$ such that $f(x) = 0$. $F$ is ...
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2answers
194 views

Zeroes of trigonometric-like function

Consider a function $f(z)=\cos(z)\cosh(az)+\sin(z)\sinh(bz)$ for $z\in \mathbb{C}, a,b \in \mathbb{R}$. Denote $D\subseteq \mathbb{R}^2$ being the set of such pairs $(a,b)$ of parameters so that NOT ...