# Questions tagged [zeroes]

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22
questions

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167 views

### 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 ...

**2**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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 $...

**1**

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**0**answers

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 ...

**3**

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**0**answers

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)&...

**0**

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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**

votes

**1**answer

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 ...

**9**

votes

**1**answer

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**

votes

**4**answers

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 ...

**1**

vote

**1**answer

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**

vote

**1**answer

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]...

**0**

votes

**0**answers

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^{-(...

**13**

votes

**2**answers

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 $C^\infty$-...

**-1**

votes

**1**answer

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 ...

**7**

votes

**2**answers

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 ...

**8**

votes

**2**answers

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?

**4**

votes

**2**answers

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....

**5**

votes

**1**answer

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)...

**11**

votes

**3**answers

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 ...

**2**

votes

**2**answers

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 ...