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Closed form of $\frac{1}{{\pi}^n}\sum_{k\in \mathbb{Z}^n}\prod_{j=1}^n\frac{1}{1+(m_{j}^{T}k)^2}$

Given vectors $m_{j}\in\mathbb{Z}^{n},M=(m_{1} \ldots m_{n}),\det(M)\not =0$. Is it possible to find a closed form of: $$S=\frac{1}{{\pi}^n}\sum_{k\in \mathbb{Z}^n}\prod_{j=1}^n\frac{1}{1+(m_{j}^{T}k)^...
Quý Nhân's user avatar
12 votes
0 answers
120 views

When could a diligent calculus student compute all Picard iterates algebraically?

As is well known, in the typical proof of the Picard–Lindelöf theorem, one shows the existence of a solution of the initial value problem $y'(t) = f(t,y(t))$, $y(t_0) = y_0$ by considering the Picard ...
James E Hanson's user avatar
5 votes
1 answer
732 views

Can the Pythagorean theorem be proved using imaginary numbers?

Can the Pythagorean theorem be proved using imaginary numbers? The proof must avoid circular reasoning, of course. I asked essentially the same question at MSE, but did not receive a definitive answer,...
Dan's user avatar
  • 3,577
-5 votes
0 answers
86 views

Every smooth function contains a bijection [closed]

Let $f:\mathbb{D}\rightarrow \mathbb{R}$ be a continuous non-constant over $\mathbb{D}$. Is there always a subset $\mathbb{A}\subseteq \mathbb{D}$ such that $f:\mathbb{A}\rightarrow \mathbb{R}$ is a ...
John Wayne's user avatar
-3 votes
1 answer
105 views

when does $h$ exist?

Let $\zeta(s)$ denote the Reimann zeta function in the critical strip. It is easy to see that $$ \zeta(s) = 0 \Longleftrightarrow \Re(\zeta(s))+\Im(\zeta(s)) = 0 ~~~~ \text{and} ~~~~~~ \Re(\zeta(s)) \...
Roy Burson's user avatar
-4 votes
0 answers
64 views

Is the real and imaginary part of the Dirichlet eta function closest to its partial sums when trigonometric function changes signs? [closed]

To grasp the question we are concerned with three Theorems 1,2, and 3 in bold font below. First let us consider the Dirichlet eta function $\eta: \mathbb{C}\rightarrow \mathbb{R}$ $$ \eta(s) = \sum_{n=...
Roy Burson's user avatar
14 votes
4 answers
2k views

The ten most fundamental topics in geometric group theory

What are the ten most fundamental topics in geometric group theory? This is a pedagogical question prompted by the fact that I am teaching geometric group theory to undergraduates. They are expected ...
2 votes
0 answers
50 views

Convergence of finite-difference method for Cauchy-Riemann equations

Let $I\subseteq \mathbb{R}$ an open interval. Let $f:I\rightarrow \mathbb{C}$ real analytic. Suppose we want to numerically compute an analytic extension of $f$. We will assume the following: we are ...
Plemath's user avatar
  • 312
1 vote
0 answers
53 views

Description of all biholomorphic maps from annulus [duplicate]

Consider the collection $\mathcal{C}$ of all maps $f \colon B_1 -\overline{B}_\beta \to \mathbb{C}$ such that $f$ is biholomorphic onto its image. Is this collection $\mathcal{C}$ path-connected? In ...
Jinyang wu's user avatar
0 votes
0 answers
83 views

Singular behavior of zeros of incomplete zeta function

I've been looking at the zeros of the incomplete zeta function $\zeta_{lower}(s, z)$ recently. $$ \zeta_{\mathrm{lower}}(s,z)=-\frac{{\Gamma(1-s)}}{2\pi i}\int_{z}^{\infty}\frac{{(-t)^{s-1}}}{e^{t}-1}...
Zhobbyist's user avatar
1 vote
0 answers
39 views

Currents with logarithmic poles compared with those with no poles

I am learning Deligne homology via U. Jannsen, "Deligne homology, Hodge-$\mathscr{D}$-conjecture, and motives." There, the currents with logarithmic poles are given in Definition 1.4 by $$ '\...
neander's user avatar
  • 161
3 votes
2 answers
147 views

Vector bundles over a Stein space are projective

It is a "well known" fact that locally free sheaves over a Stein space $X$ are projective as $\mathcal{O}_X$-modules (see e.g. just after Lemma 1.6 in O'Brian-Toledo-Tong's "The trace ...
Tim's user avatar
  • 1,109
0 votes
1 answer
127 views

Holomorphic functions of certain blow up at origin

Suppose that $D=\{z\in \mathbb C\,:\, |z|\leq 1\}$ and let $f$ be holomorphic on $D\setminus\{0\}$ such that $|f(z)|\leq e^{\frac{1}{|z|}}$ for all $0<|z|\leq 1$ and assume additionally that $\lim\...
Ali's user avatar
  • 4,115
1 vote
0 answers
148 views

integral over the unit sphere of $\Bbb C^n$

Please, is there a way to calculate this integral $$\int_{S_{2n-1}} \frac{e^{a \langle z, \zeta \rangle}}{|z - \zeta|^{\beta}} \, d\sigma(\zeta)$$ where $ z $ is a fixed point in the complex unit ball ...
zoran  Vicovic's user avatar
2 votes
1 answer
201 views

Section 3 of Atiyah's "On analytic surfaces with double points" — some questions

I have some questions about section 3 of Atiyah's "On analytic surfaces with double points," a short 9 page paper. Section 3 is all dedicated to proving lemma 4. Near the end of section 3, ...
maxo's user avatar
  • 129
0 votes
0 answers
57 views

Complexity of evaluation of analytic functions

Given an analytic function $f(x)$ (say as combination of elementary functions and operators), is it possible to compute $n$ first bits of the value of the function on the whole interval $[a, b]$ ...
roignoirewg's user avatar
1 vote
0 answers
53 views

Can one explicitly define a right inverse for a convolution operator on the space of entire functions?

A result of Meise and Taylor in 1988 shows that every non-zero convolution operator on the Frechet space $H(\mathbb{C})$ of all entire functions on $\mathbb{C}$ has a continuous linear right inverse $...
David Walmsley's user avatar
5 votes
1 answer
274 views

Why "no wandering domain" fails in parabolic basin?

Theorem (Sullivan). Every Fatou component $U$ of $f$ rational map is eventually periodic, that is, there exist $n > m > 0$ such that $f^n(U) = f^m(U)$ I am familiar with the proof: spread around ...
Ricky Simanjuntak's user avatar
0 votes
0 answers
79 views

Alternative proof of parabolic implosion

I am working on an alternative proof of parabolic implosion from complex dynamics, but only allowing hyperbolic perturbation. Theorem (Parabolic Implosion) Let $f(z)=z^2+z$ and $U_f$ be parabolic ...
Ricky Simanjuntak's user avatar
-3 votes
1 answer
195 views

Bounding a number-theoretic integral

Find a good upper bound on $$\int_1^T\frac{\zeta'(s)}{\zeta(s)\zeta(1-s)}X^sdt,$$ where $s=c+it$ for a constant $c>1$ and $X>0$ is a parameter. If needed, we can assume RH. My attempt here is ...
charlie_beck's user avatar
2 votes
0 answers
70 views

Is the hypothesis "uniformly equivalent" needed?

I am reading S. Shimorin's paper titled Complete Nevanlinna-Pick property of Dirichlet-type spaces. My question concerns Lemma 2.3. which is as follows: Assume $\mathscr{H}$ is a Hilbert space of ...
ash's user avatar
  • 151
0 votes
1 answer
119 views

Nonstationary phase method for oscillatory integral

I want to approximate an integral of the form $$\int_a^bf(t)e^{ig(t)}dt,$$where $f(t)$ is smooth, $g(t)$ is real-valued and smooth. The stationary phase method says that if $t_0\in [a,b]$ is such that ...
charlie_beck's user avatar
1 vote
1 answer
63 views

Need bound for absolute value of complex-valued special functions (Taylor coefficients of Faddeeva's w(z))

To guarantee accuracy for code [1] that computes Faddeeva's w(z) [2] using Taylor expansions around different centers, I would need upper bounds for the absolute values $|w_n(z)|$ of the coefficients $...
Joachim W's user avatar
  • 111
1 vote
2 answers
225 views

Bounds of zeta function near $\Re(s)=1$

Richert proved in https://link.springer.com/article/10.1007/BF01399533 that $$ \zeta(s) =O\left( |\Im(s)|^{100(1-\Re(s))^{3/2}} (\log |\Im(s)|)^{2/3}\right)$$ uniformly in the region $\Re(s)\in [1/2,1]...
Dr. Pi's user avatar
  • 3,062
16 votes
1 answer
977 views

Pedagogically intuitive reformulation of Zorn's Lemma for functional analysis

While teaching an applied functional analysis class, I’ve noticed that students often struggle to develop an intuitive understanding of Zorn’s lemma. It’s relatively straightforward to explain why ...
Tobias Diez's user avatar
  • 5,824
2 votes
1 answer
215 views

How can one test whether a given analytic curve in the plane is algebraic or not?

Suppose $\Gamma$ is an analytic Jordan curve in the complex plane $\mathbb{C}$. What are ways to test whether $\Gamma$ is (contained in) an algebraic curve, i.e., whether there exists a real ...
Malik Younsi's user avatar
  • 2,154
3 votes
1 answer
177 views

Mellin transform at $0$

Let $f$ be a smooth complex valued function with support on $[0,2]$ and with Mellin transform $\tilde{f}$. Recall that the Mellin transform is defined as $$\tilde{f}(w)=\int_0^{\infty}f(x)x^{w-1}dx.$$ ...
user avatar
0 votes
0 answers
146 views

On the pointwise limit of a sequence of analytic functions

I have been confused with this problem for weeks now. Suppose I have Banach spaces $E$ and $F$ and a sequence of functions $f_{n}: U \subset E \to F$, where $U$ is open and nonempty. Let $x \in U$ be ...
InMathweTrust's user avatar
1 vote
1 answer
193 views

How to evaluate the following integral?

How to (analytically) calculate the following integral, $$I = \int_{S_{2n-1}} \left( \langle z, \zeta \rangle \right)^a e^{ b \langle z, \zeta \rangle} \, d\sigma(\zeta),$$ where $\langle z, \zeta \...
zoran  Vicovic's user avatar
20 votes
1 answer
620 views

Conjecture on the number of roots of $z^n + P(z)$ within the unit disk

Some other people and I have noticed that the following seems to be true. Fix an integer polynomial $P \in \mathbb{Z}[x]$. Let $a_n$ be the number of roots of $z^n + P(z) = 0$ that lie in the unit ...
Incompleteusern's user avatar
1 vote
0 answers
71 views

Integral formula of quantum dilogarithm

In the paper"Level N Teichmüller TQFT and Complex Chern-Simons Theory" arXiv:1612.06986, the authors study the quantum dilogarithm function: \begin{equation} \mathrm{D}_{\rm b}(x,n)=\prod_{...
color's user avatar
  • 109
0 votes
0 answers
77 views

Constant mean curvature hypersurface

Assume that $f:\mathbb{B}^2\to \mathbb{C}$ is a holomorphic function defined in the unit ball in $\mathbb{C}^2$. Let $u(z)=|f(z)|(1-|z|^2)$ and consider $\Sigma =\{z: u(z)=c\}$. It seems to me that if ...
user67184's user avatar
1 vote
1 answer
151 views

Does this sequence of Blaschke Product have rescaling limit $z-1$?

Background: The conformal conjugacy class of parabolic isometry of upper half plane $\mathbb{H}$ consists of $f(z) = z+1$ and $g(z)=z-1$. Consider surjective proper holomorphic $F_n: \mathbb{H} \...
Ricky Simanjuntak's user avatar
0 votes
0 answers
57 views

Double-periodic functions with (possible) poles

Consider the set of double-periodic function $f:\mathbb C/(\mathbb Z+i \mathbb Z) \setminus \{z_0\} \to \mathbb C$, where $z_0$ is a fixed point inside $\mathbb C/(\mathbb Z+i \mathbb Z),$ that have a ...
António Borges Santos's user avatar
1 vote
0 answers
39 views

About Carleson measures on the Hardy space on the bidisc

I have been reading the paper "Carleson Measures in Hardy and Weighted Bergman Spaces of Polydiscs" by F. Jafari and there are a few things that going on that I am not entirely convinced of. ...
an_ordinary_mathematician's user avatar
2 votes
0 answers
165 views

Bounds of modular functions on the Ford circles

Assume a holomorphic function from a product of two upper half planes $Z: \mathbb{H}_+\times \mathbb{H}_+\rightarrow \mathbb{C}$ with an expansion of the form $$ Z(\tau,\tau') = \sum_{(h,h')\in S} a_{...
Yiannis T.'s user avatar
0 votes
0 answers
27 views

Conformal mapping which is $\alpha$-holder continuous for each $\alpha<1$

Recall that the Jordan curve $J\subset \mathbb{C}$ is called asymptotically conformal if $$\text{$\max_{z\in J(a,b)}\frac{|a-z|+|z-b|}{|a-b|}\to 1$ as $a,b\in J, |a-b|\to 0$}$$ where $J(a,b)$ is the ...
ljjpfx's user avatar
  • 207
0 votes
0 answers
66 views

Uniformization and constructive analytic continuation of Taylor-Maclaurin series

Context. In their paper, "Uniformization and Constructive Analytic Continuation of Taylor Series", Costin and Dunne present a constructive method to greatly increase the accuracy of a ...
butsurigakusha's user avatar
4 votes
0 answers
160 views

An unusual uniqueness property for entire functions

For given $q\in (0,1),$ coefficients $|c_k|\leq Cq^{k^2/3},$ and non-negative non-decreasing convergent sequences $\{a_k\}_{k=0}^\infty$ and $\{b_k\}_{k=0}^\infty$ satisfying $a_k\geqslant b_k,\;k=0,...
Deepti's user avatar
  • 783
1 vote
0 answers
113 views

Are there any known statistics on the sign of the Stieltjes Constants?

The Stieltjes Constants $\gamma_n$ arise from considering the laurent series of the Riemann Zeta function at $s=1$ $$ \zeta(s) = \frac{1}{s-1} + \sum_{n=0}^{\infty} (-1)^n \frac{\gamma_n}{n!} (s-1)^n $...
Sidharth Ghoshal's user avatar
2 votes
0 answers
37 views

Theta series of well-rounded lattices

I've started looking into well-rounded Euclidean lattices and I was interested in learning whether their theta series have any interesting properties, but haven't found much in terms of bibliography ...
JBuck's user avatar
  • 223
2 votes
0 answers
179 views

Analytic continuation of $\int_V (f(x_1,\cdots,x_n))^s dx_i$

Let $V$ be an $n$-dimensional simplex, let $f(\boldsymbol{x}) = f(x_1,\cdots,x_n)\in \mathbb{C}[x_1,\cdots,x_n]$ be a product of linear polynomials that is non-zero in interior of $V$. Also let $E(\...
pisco's user avatar
  • 528
13 votes
2 answers
801 views

For which rationals is this exponential sum bounded?

Given $x \in [0, 1]$, we denote by $e(x)$ the complex number $e^{2 \pi i x}$. Can we characterise the set of rationals $x$ for which the sum $$A_N(x)\, :=\, \sum_{n = 0}^N e(2^n x)$$ remains bounded ...
Nate River's user avatar
  • 6,321
0 votes
0 answers
82 views

A far reaching generalization of the WPT?

I encountered the the Nullstellensatz for Germs of Holomorphic Functions in Daniel Huybrechts' Complex Geometry: An Introduction, specifically Proposition 1.1.29 If $I\subset \mathcal{O}_{\mathbb{C}^...
LuckyJollyMoments's user avatar
7 votes
2 answers
186 views

Non-locally connected polynomial Julia sets

What are some examples of complex polynomials whose Julia sets are connected, but not locally? In the book Complex Dynamics by Carleson and Gamelin, I found: They seem to reference: But what is a ...
D.S. Lipham's user avatar
  • 3,317
13 votes
1 answer
291 views

Descriptive complexity of analytic continuation

Consider the set of complex power series $$ f(z)=\sum_{n=0}^\infty a_nz^n $$ that have radius of convergence $1$ and can be analytically continued to the neighborhood of some point on the unit circle. ...
183orbco3's user avatar
  • 623
4 votes
1 answer
173 views

Is every Cantor set $C\subseteq\mathbb R^{\infty}$ the limit set of a Fuchsian group?

Is every Cantor set $C\subseteq\mathbb R^{\infty}$ the limit set of a Fuchsian group? Let's recall the definitions. A Fuchsian group $G$ is a discrete subgroup of $\operatorname{PSL}(2,\mathbb R)=\...
Christian Remling's user avatar
4 votes
1 answer
328 views

Holomorphic homotopy conjecture

Let $X$ be a smooth projective variety over the complex numbers, and let $\text{Coh}(X)$ be the category of coherent sheaves on $X$. Consider the dg-category $\text{Perf}(X)$ of perfect complexes on $...
Nhan Le's user avatar
  • 41
-2 votes
1 answer
121 views

Infinite sum related to Hurwitz Zeta

I want to evaluate the following sum: \begin{equation} \sum_{-\infty}^{\infty}\frac{(-1)^n}{(n+a)^2} \end{equation} Where $a\in\mathbb{R}$ is not an integer. Such is similar to $\zeta(2,a)$, but it ...
Johann Wagner's user avatar
1 vote
0 answers
106 views

The proposition associated with a set

Given a set $U$ and a set $A \subseteq U$, is there an accepted symbol for the proposition $p$ over the universe $U$ such that for each $x \in U$, $p(x)$ is the assertion that $x \in A$? (The symbol $...
James Propp's user avatar
  • 19.7k

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