All Questions
3,560 questions
-1
votes
1
answer
122
views
On $\Re\frac{\zeta'}{\zeta}(s) \geq -A \log(|t|+4)$ for some $A>0$
On page 438 of "Multiplicative Number Theory I" by Montgomery-Vaughan one finds the following statement and its verification :
Assuming RH, there exists an absolute constant $A>0$ such ...
- 59
2
votes
1
answer
136
views
Is there a scalar product which makes orthonormal the family of complex functions $ (f_n)_{ n \geq 1 } $?
Let $ (f_n)_{ n \geq 1 } $ be a family of complex functions defined as follow,
$ \forall n \geq 1 $,
$$ f_n (z) = \dfrac{1}{n^{z}} $$
I would like to ask you if it is possible to construct a ( non-...
- 595
11
votes
3
answers
1k
views
"Simple" integral equation
Let $H(z)$ be a continuous solution of the problem
$$
H(z)=\frac1{1-z}\int_z^1 \frac{2\zeta}{1+\zeta} H(\zeta^2)\,d\zeta,\ \ \ z\in[0,1);\ \ \ H(1)=1.
$$
Is it true that $H(0)=1-\ln2$? The question ...
- 283
3
votes
1
answer
111
views
When entire or meromorphic map of finite type restricts to a Galois covering?
Suppose that $f \colon \mathbb{C} \to \mathbb{C}$ is an entire map of finite type, i.e., with finitely many singular values. Then we can consider the restriction $f| \mathbb{C} \setminus f^{-1}(S_f) \...
- 41
4
votes
1
answer
183
views
Notation for weak derivatives
I remember that, as a student, I felt a bit uncomfortable because I had to use the same notation (say $f'$, $D^\alpha f$, $\frac{\partial f}{\partial x^j}$, $\nabla \cdot f$ etc...) for classical and ...
- 4,459
7
votes
0
answers
167
views
Example of closed non-exact torsion differential form on variety
I asked this question some time ago on MSE and received close to no interest. I feel it is appropriate for this site:
I am interested in finding a particular example. I would like to find a variety (...
- 513
12
votes
4
answers
929
views
Interesting examples of systems of linear differential equations with constant coefficients
In this paper, Gian-Carlo Rota wrote:
A lot of interesting systems with constant coefficients have been discovered in the last thirty years: in control, in economics, in signal
processing, even in ...
4
votes
2
answers
360
views
Functions with asymmetrically decreasing Fourier transform?
$\def\ii{{\rm i}}\def\bbR{\mathbb R}\def\bbC{\mathbb C}\def\bbNo{\mathbb N_0}\def\Fou{\mathscr F}$Specifically, I would like to have a compactly supported continuous function $f=u+\ii\,v:\bbR\to\bbC$ ...
- 3,584
16
votes
0
answers
519
views
Gabriel's theorem for complex analytic spaces
Let $X,Y$ be noetherian schemes over $\mathbb{C}$.
Then, it is known that
$$
\text{Coh}(X) \simeq \text{Coh}(Y) \Rightarrow X \simeq Y,
$$
by P. Gabriel(1962).
Are there some results in the case of ...
- 889
1
vote
1
answer
152
views
complex optimization in the plane
I am trying to get the following condition:
$\forall\,\alpha,\beta,\gamma\in\mathbb{C}\,\,\exists\,u,v\in\mathbb{T}\colon\quad|\alpha^2uv-(\beta u-\gamma v)^2|-|\alpha^2+4\beta\gamma|=|\beta u+\gamma ...
- 375
4
votes
2
answers
364
views
Nontrivial invariant transformations for heat equations
It is well known that if $u$ is a harmonic function on $\mathbb R^2$ then its Kelvin transform defined by
$$ v(r,\theta) = u(\frac{1}{r},\theta)$$
is also harmonic for $r>0$. Note that the Kelvin ...
- 4,115
3
votes
1
answer
218
views
Subset of a complex manifold whose intersection with every holomorphic curve is analytic
The Osgood's lemma in complex analysis says that if $f \colon \mathbb{C}^n \to \mathbb{C}$ a continuous function such that $f|_{L}$ is holomorphic for every complex line $L \subset \mathbb{C}^n$, ...
- 1,170
6
votes
1
answer
407
views
Conjectured closed form of $\int_0^1 \frac{\text{Li}_2\left(\frac{x}{4}\right)}{4-x}\,\log\left(\frac{1+\sqrt{1-x}}{1-\sqrt{1-x}}\right)\,dx$
After reading some meta posts, I've decided to post this question on MathOverflow since I didn't receive any comments or answers on MSE
Certainly, I apologize for any oversight. Here's a more refined ...
- 224
8
votes
1
answer
278
views
A real-valued analogue of the Weierstrass $\wp$ Function
I am interested in the following function:
$$\mathcal{Q}(z) = \sum_{w \in L^*} \frac{1}{|z-w|^2} - \frac{1}{|w|^2} \, . $$
This function is analogous to the Weierstrass $\wp$ function, the only ...
- 181
2
votes
0
answers
259
views
Least number of circles required to cover a continuous function on $[a,b]$
I asked this question on MSE here.
Given a continuous function $f :[a,b]\to\mathbb{R}$, what is the least number of closed circles with fixed radius $r$ required to cover the graph of $f$?
It is ...
- 541
-1
votes
1
answer
116
views
Riemann xi function strictly increasing along a half-plane
Matiyasevich, Saidak, Zvengrowsk proved the following result:
Let $σ_0$ be greater than or equal to the real part of any zero of ξ. Then $|ξ(s)|$ is strictly increasing in the half-plane $σ > σ_0$.
...
- 59
4
votes
2
answers
193
views
Is $\frac{|t|}{e^{a|t|}-e^{-b|t|}}$ the Fourier transform of a positive function
Consider the function $$\phi_{a,b}(t)=\frac{|t|}{e^{a|t|}-e^{-b|t|}}, \ \ t\in\mathbb{R},$$ where $0<a<b$. Can $\phi_{a,b}$ be the Fourier transform of a positive function for some $a<b$?
- 407
3
votes
1
answer
250
views
Is Toeplitz operator on the Bergman space bounded iff its symbol is bounded?
It is fairly well known that if $T_\varphi$ is a Toeplitz operator on the Hardy-Hilbert space, then $\lVert T_\varphi \rVert = \lVert \varphi \rVert _{\infty}$.
Now, if $\varphi \in L^\infty (\mathbb ...
- 151
7
votes
1
answer
521
views
Mellin-Barnes integral representation of the exponential function with a non-real argument
I have been studying a definite integral that I found out to be a particular (and possibly one of the simplest) case(s) of the arcane Mellin-Barnes integral. Solving this problem would lead to a ...
- 71
29
votes
2
answers
2k
views
Contractibility of the space of Jordan curves
Is the space of Jordan curves in $\textbf{R}^2$ contractible? In other words, is there a canonical or continuous way to deform each Jordan curve to the unit circle $\textbf{S}^1$.
If the curves are ...
- 7,179
2
votes
0
answers
68
views
Entire functions and Bergman spaces
Given an open set $D \subset \mathbb{C}$ with compact closure, let us consider the Bergman space $A^2(D)$ of all holomorphic functions on $D$ that are square integrable on $D$ with respect to the ...
- 13.5k
48
votes
8
answers
5k
views
Ideas for introducing Galois theory to advanced high school students
Briefly, I was wondering if someone can suggest an angle for introducing the gist of Galois groups of polynomials to (advanced) high school students who are already familiar with polynomials (...
-2
votes
2
answers
322
views
Bounds for analytic circles
It is known that for certain particular entire functions $f(s)$ of first order, in the circle $|s| = p$, if $\epsilon$ is a positive number as small as desired, the following bound holds:
$$|f(s)| = O(...
- 27
4
votes
1
answer
245
views
Conditional convergence of Artin $L$-functions
Let $k$ be a number field and $V$ a non-trivial irreducible Artin representation over $k$. Consider the associated Artin $L$-function with corresponding Euler product decomposition $L(V,s)= \prod_v ...
- 21.4k
0
votes
1
answer
279
views
If there are infinite poles inside an infinite integral contour, and the sum of residues of these poles converges. Can the residue theorem hold?
The residue theorem is commonly used to calculate situations where the number of isolated singularities is limited. However, I am very curious whether the residue theorem can be extended to cases with ...
- 17
4
votes
0
answers
821
views
One of the numbers $\zeta(5), \zeta(7), \zeta(9), \zeta(11)$ is irrational
I am reading an interesting paper One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational by Zudilin. We fix odd numbers $q$ and $r$, $q\geq r+4$ and a tuple $\eta_0,\eta_1,...,\eta_q$ of positive ...
- 11
0
votes
0
answers
65
views
Rotations and bi-analytic functions
Are the bi-analytic functions $\partial^2_{\overline{z}} f=0$
invariant under rotations?
- 41
3
votes
1
answer
459
views
Limit of an infinite series with quadratic arguments
I have encountered a limiting process on some infinite series. So, I would like to ask:
QUESTION. Assume $n$ is an even positive integer. Is this true?
$$\lim_{r\rightarrow1^{-}}\sum_{j=1}^{\infty}\...
- 43.2k
4
votes
0
answers
227
views
Holomorphic non vanishing modular form
Let $\mathcal{O}(\mathcal{H})^\times$ be the multiplicativee group of holomorphic functions on the Poincaré half-plane $\mathcal{H}$ that do not vanish there.
Let $j(g,z)=(cz+d)$ and $gz=(az+b)/(cz+d)$...
1
vote
0
answers
42
views
Concerning the definition of a class of functions introduced by Nilsson
In the paper 'Some growth and ramification properties of certain integrals on algebraic manifolds' by Nilsson we find the following definitions:
My question is how does one prove the remark "It ...
- 360
2
votes
1
answer
628
views
Does some published texbook take a particular approach (described here) to the transition from discrete to continuous probability distributions?
(I posted this question at matheducators.stackexchange.com and it seems to be considered an inappropriate question for that site. I don't understand why.)
Imagine an introductory probability course ...
3
votes
1
answer
141
views
Is inverse image along finite group quotient $t$-exact for the perverse $t$-structure?
Let $q: \mathbb{C}\to \mathbb{C}$ be the quotient by $\mathbb{Z}/n\mathbb{Z},$ i.e. the map taking $z\mapsto z^n$. In the accepter answer to Operations on perverse sheaves on disk the inverse image of ...
- 1,071
1
vote
1
answer
155
views
Accessible points of a simply connected domain
We know that if $U$ is an open subset of $\mathbb{\widehat C}$ (extended complex plane), a point $v\in\partial U$ is called accessible from $U$ if there exists a curve $\gamma:[0,1)\to U$ such that $\...
3
votes
2
answers
646
views
Upper bound for complex integral
I am interested in obtaining a good upper bound for the absolute value of the following integral
$$
\left| \int_{0}^{\pi/3} e^{-itn} \left( 1-e^{it} \right)^{k} dt \right|,
$$
when $n>k>0$ are ...
- 43
4
votes
0
answers
72
views
Stability of analytic continuation
Let $(Q,\| \cdot \|)$ be a certain Banach space of entire functions, say certain functions of finite order that satisfy a growth condition of the form $|f(z)| \leq c e^{a|z|^\gamma}$ for some $c,a,\...
3
votes
0
answers
167
views
Suitability of formal type theory for mathematical thinking (vs. traditional set theory)
Type theory has advantages over set theory for the (computer) formalisation of mathematics, but has anybody who does mathematics with pen and paper found proof assistants or automated theorem provers, ...
0
votes
0
answers
59
views
Convergence of Liouville correlation functions
A key object in Liouville conformal field theory is the random Liouville measure $M$ defined heuristically as $M(d^2x) = :e^{2bX(x)}: d^2x$, where $X$ is a Gaussian free field and $:e^{2bX}:$ denotes ...
user528837
1
vote
0
answers
64
views
Extension of meromorphic distribution
Let $W$ be a topological vector space (e.g. Frechet) with a dense subspace $V$. Let $D_s$ be a distribution on $V$ that is meromorphic in $s\in\mathbb C$ and extends continuously to $W$ with respect ...
- 3,799
1
vote
0
answers
118
views
Poles/Residues of the Gamma function under action of Mobius transform $\Gamma(A(z))$
I am not sure whether this is rather an MO or MSE question but it results from my research, so I put it here.
In my effort to find (or to disprove the existence of) $k,l,h\in\mathbb{N}$ such that $2^{...
- 91
6
votes
2
answers
606
views
Infinite sum of even Bessel functions - Identities
Recently, I came across the following identities among first-kind Bessel functions, namely
$$
2\sum_{k=1}^{\infty}(-1)^k\,k^5\,J_{2k}(x) = \frac{x^2}{4}\left[x\,J_1(x)-J_0(x)\right] \label{1}\tag{1}
$$...
5
votes
1
answer
167
views
Upper bound for the $n$-th derivative of a rational function $\frac{f}{f+g}$
Let $f$ and $g$ be real polynomials with nonnegative coefficients. Let
$$
h = \frac{f}{f+g}.
$$
I want to prove that the $n$-th derivative of $h$ satisfies:
There exists $C > 0$ such that
$$
|h^{(...
- 187
1
vote
1
answer
121
views
An asymptotic integral with complex phase
Suppose that $D\subset \mathbb R^2$ is the closed unit disk and that $f\in C^{\infty}(D)$. Assume that for all $\lambda \in (1,\infty)$ there holds
$$ \left|\int_D f(x^1,x^2)\, e^{\lambda (x^1+ix^2)}\,...
- 4,115
12
votes
3
answers
784
views
Can the equation $1+z+z^2=z^n$ for natural $n$ have multiple complex roots $z$?
The question is stated in the title of this post.
It is easy to see that, if $z$ is a multiple root of $p_n(z):=1+z+z^2-z^n$, then $(n-2)z^2+(n-1)z+n=0$, so that we can successively express $z^2,\dots,...
- 128k
3
votes
1
answer
344
views
On the upper bound for $|\zeta(s)|$ near the zeta zeros
Let $T \in \mathbb{R}$ be large and $\rho$ be a non-trivial zero of the Riemann zeta function. Assume that $|\rho|=|\rho_T| \approx T$ and let $\varepsilon_T \approx \frac{\log \log T}{\log T}$. Is it ...
- 1,019
25
votes
1
answer
2k
views
Can we just use the linear term of exponential sums to sum divergent series
Suppose you want to compute the sum $\sum_{n=0}^{\infty} a_n $
You could consider the expression $f(x) = \sum_{n=0}^{\infty} e^{a_n x}$ and try to compute the coefficient of an $x^1$ term in the ...
- 2,689
0
votes
1
answer
102
views
On weighted Fourier transforms
Suppose that $f\in L^{\infty}((0,1))$ and that there exists $c_1,c_2>0$ such that
$$ \left|\int_0^1 e^{i \xi x} e^{-|\xi|^{-1}x}f(x)\,dx \right| \leq c_1 e^{-c_2|\xi|} \quad \forall\, |\xi|>1.$$
...
- 4,115
3
votes
1
answer
136
views
Growth of preimages of singular values of finite type entire map
Let $f\colon \mathbb{C} \to \mathbb{C}$ be an entire map having precisely two distinct singular values $w^1$ and $w^2$. If $w^i$ has infinitely many preimages under $f$, we write $(z_n^i)_{n \in \...
- 41
3
votes
1
answer
107
views
Does the reproducing property of the unitary group Poisson kernel require a multiple of the identity?
The Poisson kernel of the unitary group is
$$ P(Z,U)=\frac{\det(1-ZZ^\dagger)^N}{\det(1-ZU^\dagger)^N\det(1-UZ^\dagger)^N}.$$
It has a reproducing property, $\int dU P(Z,U)f(U)=f(Z)$, where $dU$ is ...
- 2,552
-1
votes
1
answer
109
views
Analyzing a Dirichlet series with log-oscillating terms via Fourier methods
I am investigating the series $S(z)$ defined as follows:
$$
S(z) = \sum_{n=1}^{\infty} n^{-a}\cos(b\ln(n)),
$$
where $z = a + bi \in \mathbb{C}$, with $0 < a < 1$, and $b \in \mathbb{R}$.
I want ...
- 375
0
votes
0
answers
46
views
Function samplable from the past and Hardy spaces
What I am ultimately looking for is a $L^2$ function $f$ on the real line that can be sampled from the past, i.e. for each $x<0$ there are $L^2$ coefficients $c_n(x)$, $n\in \mathbb{N}$ such that, ...
- 1,352