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Kolodziej's acta paper "the complex monge-ampere equation"——a detailed ploblem [closed]

Recently,I am reading kolodziej's acta paper,there are some ditails that i do not know clearly. In the top line of page 99,"it's no restriction to assume that for each s we have $\nu(\cup_{I\in{B_s}}...
Unknown's user avatar
  • 247
-1 votes
2 answers
352 views

Image of a complex disc by this function? [closed]

I am working with the function $f(z) = \frac{z+1}{z-1}$, for a complex variable $z$. I understood that for $z$ in the unit disc, i.e $\lvert z\rvert \le 1$, $\mathrm{Re}(f(z)) \le 0$. What if $z$ is ...
lrnv's user avatar
  • 686
-1 votes
1 answer
512 views

Does $\int_{2}^{\infty} (\pi(x)-Li(x))x^{-s-1} \mathrm{d}x$ converge on the real axis for $s>1/2$? [closed]

Consider the prime zeta function, defined for $\Re(s)>1$, by the infinite series $$\sum_{p} p^{-s} = \sum_{m=1}^{\infty}\frac{\mu(m)}{m}\log \zeta(ms)$$ where $p$ denotes a prime, $\mu$ the Mobius ...
user avatar
-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 ...
12321's user avatar
  • 59
-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 ...
swami's user avatar
  • 375
-1 votes
1 answer
471 views

Infinite sum and product associated with the Weierstrass elliptic function [closed]

Can anyone help me figure out how the identity below was obtained? $ \frac{1}{\sqrt{(e_1-e_3)(e_2-e_3)}} = R \prod \limits_{n=1}^{\infty} \left(1 - \frac{1}{R^{4n}} \right)^{-4}\left(1 + \frac{1}{R^{...
Fareeda's user avatar
  • 45
-1 votes
2 answers
129 views

Is it possible for all of the smooth/continuous curves in $R^3$ to form a Hilbert space? [closed]

Under which condition can it form a Hilbert space? Or what space can it form? You can write down certain condition to make it to be a Hilbert space, e.g., Let $$p(t)=[x(t),y(t),z(t)]^T\in \text{R}^3$$ ...
Nan Zhang's user avatar
-1 votes
1 answer
208 views

smooth holomorphic functions are CR on the boundary? [closed]

Is this true that any holomorphic functions in a domain with smooth boundary, and which is smooth on the boundary is a CR function ?
beni-oui's user avatar
-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 $σ > σ_0$. ...
12321's user avatar
  • 59
-1 votes
1 answer
250 views

Significance of $N_0(T+1)-N_0(T)\sim \frac{1}{2\pi}\log \frac{T}{2\pi}$

Let $N(T)$ be the number of zeros of Riemann zeta function upto height $T$ in the critical strip and $N_0(T)$ be the number of zeros on the critical line. What will be the significance of proving ...
user avatar
-1 votes
1 answer
243 views

On a certain representation of the Riemann zeta function

Let $\zeta$ denote the Riemann zeta function. In this answer: https://mathoverflow.net/a/314066/133634, @Paul Garret considers the representation $$\frac{\zeta(s)}{s} = \int_1^\infty (\sum_{1 \le n \...
OneTwoOne's user avatar
  • 105
-1 votes
1 answer
2k views

Real and imaginary part of an holomorphic function

I guess this could be a very elementary question. Anyway I can not find an answer in literature. Let $f:U\rightarrow\mathbb{C}$ be an holomorphic function on an upen subset $U\subseteq\mathbb{C}$. ...
user avatar
-1 votes
1 answer
2k views

Reducibility (or not) of algebraic curves [closed]

[ I am a bit clueless about why this question is getting downvotes!? I put it up with a genuine serious interest and I don't seem to be making any egregious error either - apart from those unsure ...
Anirbit's user avatar
  • 3,541
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1 answer
85 views

Reference Request: Continuous extension of conformal maps

currently I am trying to find some references on the continuous extension of conformal maps between two simply connected domains of the Riemann sphere $\hat{\mathbb C}$. Let $\gamma_1,\gamma_2$ be two ...
A.s. graduate student's user avatar
-1 votes
1 answer
175 views

On the bound for $\int_{x}^{x+i\infty} (\cot(\pi z)+ i)z^{-s} \, \mathrm{d}z$

I'm reading Titchmarsh's "The theory of the Riemann zeta function", and on p.81 it is claimed that $$ \int_{x}^{x+i\infty} (\cot(\pi z)+i)z^{-s} \, \mathrm{d}z \ll \frac{x^{-\sigma}}{2(n+1)\...
Q_p's user avatar
  • 1,019
-1 votes
1 answer
87 views

Inferring polynomial rate of convergence from polynomial bound

Let $x_n$ be a non-negative valued sequence and suppose that the following hold: $\lim\limits_{n\to\infty} x_n =0$ There exists some polynomial function $p$ of degree at-least $1$ such that: $$ \|x_n\...
ABIM's user avatar
  • 5,405
-1 votes
1 answer
119 views

Existence of a function with slow growth on derivatives

Does there exist a smooth compactly supported function $$f \in C^{\infty}_c((0,1))$$ such that $$ \|D^k f\|_{L^{2}(0,1)} \leq \left\lfloor{\alpha\,k}\right \rfloor! \quad \forall\, k\in \mathbb N$$ ...
Ali's user avatar
  • 4,115
-1 votes
1 answer
211 views

Stone Cech compactification for exponential map

Recently I met with a problem related to Stone-Cech Compactification theorem in Furstenberg's famous paper "non-commuting product." I try my best to understand Stone-Cech compactification theorem by ...
yaoxiao's user avatar
  • 1,706
-1 votes
1 answer
327 views

Residue at an integration border in case of a limit? [closed]

I am dealing with an integral in a limit of the following shape: $$\lim_{\epsilon \to 0} \int_0^{\frac{\pi}{2}} dx \frac{2 \epsilon}{1-(1-\epsilon^2)\sin^2(x)}$$ Formally, assuming that $x=\arcsin(\...
Kagaratsch's user avatar
-1 votes
1 answer
215 views

Best approximation of the modulus function

While there is extensive study regarding the best approximation of function with polynomial functions in the real domain, the study of approximation of complex variables becomes much sparse. See this ...
ironmanaudi's user avatar
-1 votes
2 answers
87 views

Limits of integral series

Suppose we have the series of functions: \begin{equation} F(x)=\sum_{n=1}^{\infty} f_n(x) \end{equation} where convergence is uniform. Additionally, consider the partial functions of the series: \...
george andrade's user avatar
-1 votes
1 answer
124 views

Borel summation

If $f(z)=\sum_{n=0}^\infty a_n z^n$ is a formal power series with complex coefficients, then its Borel transform is defined by $$B(f)(z)=\sum_{n=0}^\infty a_n \tfrac{z^n}{n!}.$$ Suppose that $f$ and ...
Todor's user avatar
  • 139
-1 votes
1 answer
406 views

Topological properties of complex valued Riemann sum limit curve and a particular integral inequality

I am studying under what conditions the following integral inequality would hold ($a$ real, $a>0$): $$ \int_{-\infty} ^{\infty} \frac{f(ix)}{a\pm ix}dx\ = 0 \ \ \ \ \Rightarrow \ \ \ \int_{-\...
Luca's user avatar
  • 362
-2 votes
2 answers
322 views

Is there a term for a countour integral that disregards direction?

Is there a name for integration of the form $\oint_\gamma f(z) |dz|$? In other words, the integral that only takes into account the length of the contour and the values of the function but not the ...
Anixx's user avatar
  • 10.1k
-2 votes
2 answers
393 views

Expression for infinite product

can anyone show me how $$\displaystyle\frac{4}{R}\displaystyle\Pi_{n=1}^{\infty} \left(\frac{1+R^{-4n}}{1+R^{-4n+2}}\right)^4= \frac{1}{R}\left(1+2 \sum_{n=1}^ {\infty} \frac{1}{R^{2n(n+1)}}\...
Fareeda's user avatar
  • 45
-2 votes
2 answers
2k views

Taylor series of a complex function that is not holomorphic

I want to create Taylor series of a complex function that has complex conjugate in it. Obviously I cannot do a total derivative but derivations over real and imag parts exist. Bonus question: Can I ...
Domagoj Peharda's user avatar
-2 votes
1 answer
139 views

Convergence of scrambled product for Dirichlet-$L$ function with modulo 4 character

A Dirichlet-$L$ function is typically defined by its series, and its Euler product is a consequence of the definition. Here my approach is the other way around. I define the function $$ L_4^*(s) = \...
Vincent Granville's user avatar
-2 votes
1 answer
203 views

Holomorphic maps on $\mathbb{R}^{n}$ (for n not necessarily even)

Edit according to the comment of user36931 I remove the "motivation" from the previous version and I add an statement to the first question We consider the following two classes of smooth maps on $...
Ali Taghavi's user avatar
-2 votes
1 answer
1k views

holomorphic extension of a function [closed]

hi, I have the following question: let $U \subset \mathbb{C}^{n}$ be some open set containing zero. let $\tilde{U} = U \cap \mathbb{R}^{n}$. assume we have a real-valued analytic function $f : \tilde{...
bruno's user avatar
  • 29
-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(...
Bo Jonsson's user avatar
-2 votes
1 answer
271 views

A curious relationship betwen $|\zeta(\sigma+it)|$ and $|\zeta(1-\sigma - it)|$

By use of the Riemann functional equation, it can be shown (see corollary 10.5 of Montgomery-Vaughan) that $$|\zeta(\sigma + it)| \asymp |t|^{\sigma-1/2}|\zeta(1-\sigma - it)|$$. where $\zeta$ ...
sigma's user avatar
  • 35
-2 votes
1 answer
151 views

Is there some lattice not rigid

I heard that in complex hyperbolic space setting for example CH2, there is some deformation of lattice nontrivial. What confused me is it seems contradicting Mostow Rigidity. Could someone explain ...
user42804's user avatar
  • 1,121
-2 votes
1 answer
578 views

Simply-Connected Regions and Phragmen-Lindelöf Theorem

It's easy to see that the Phargmen-Lindelöf theorem from complex analysis can be generalized to non-simply-connected regions. Namely to regions $G$ with the property that for each $z \in \partial_\...
Jason Mraz's user avatar
-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
-2 votes
1 answer
1k views

Degree of a rational function [closed]

I would like to have a simple proof for the following result: Let $f=\frac{p}{q}:\mathbb{C}\longrightarrow\mathbb{C}$ be a quotient of polynomials (of course, at some points it may be undefined). ...
Jjm's user avatar
  • 2,091
-2 votes
1 answer
219 views

Howto plot a specific complex function [closed]

We need to plot the real and imaginary parts of a complex function $k(\omega)$, and cannot find a good way to do this without using "ad hoc tricks." Definitions $k$ is a complex-valued function ...
Petern's user avatar
  • 33
-2 votes
1 answer
74 views

Behavior of "integer complex number" on computer [closed]

I want to provide software to compute with "integer complex numbers", that live in $\mathbb{Z}\times i \mathbb{Z}$, rather than the $\mathbb{C}$. Some operations are going to give results that are ...
user26415's user avatar
-2 votes
1 answer
185 views

Order of the zero of a meromorphic function under the action of $Gal(\mathbb{C},\mathbb{Q})$

Let's take $X$ a Riemann surface as an algebraic curve in $\mathbb{P}^n$. The group $Gal(\mathbb{C},\mathbb{Q})$ (automorphisms of $\mathbb{C}$ which act as the identity on $\mathbb{Q}$) acts on $X$ ...
Andreas Kisser's user avatar
-2 votes
1 answer
314 views

holomorphic equation

hi, i am working for some time on a problem and at some point i cant go further. here the critical part: Let $U \subset \mathbb{C}^{n}$ be a open set and consider $c : U \rightarrow \mathbb{R}$ a ...
miriam's user avatar
  • 11
-2 votes
1 answer
318 views

Holder class of analytic functions

Assume that $\lim_{(nt) |z|\to 1}|f(z)|(1-|z|)^p=0$, where $f$ is analytic in the unit disk and $p>0$,where $(nt)|z|\to 1$ nontangentially. Does this implies that $\lim_{|z|\to 1}|f(z)|(1-|z|)^p=0$...
user36162's user avatar
  • 259
-3 votes
1 answer
251 views

Is the imaginary part of $\displaystyle\ \zeta(s)\zeta(1-s)=0$ for $\operatorname{Re}(s)=\frac{1}{2}$ [closed]

Some of my computations here showed to me that the imaginary part of $\displaystyle\ \zeta(s)\zeta(1-s)=0$ for $\operatorname{Re}(s)=\frac{1}{2}$, really i w'd like to know if there is any paper ...
zeraoulia rafik's user avatar
-3 votes
1 answer
208 views

Conformal map from a 7-sided polyhedron to a square pyramid

I have a right-angled square pyramid, $A$, whose height and base-length are $l$. By 'right-angled', I mean that the apex of $A$ lies vertically above one of the vertices in its base. Now supposed I ...
niran90's user avatar
  • 167
-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
-3 votes
2 answers
318 views

When is $\Re(\zeta(s)) - \Im(\zeta(s)) = 0 $ with $\Re(\zeta(s))\neq 0$ and $\Im(\zeta(s))\neq 0$? [closed]

When is $\Re(\zeta(s)) - \Im(\zeta(s)) = 0 $ for $0<\Re(s)<1$. Here $\zeta$ denotes the Reimann zeta function. Does the solution live on a vertical line? Or is this another coincidence when both ...
MrPie 's user avatar
  • 317
-3 votes
2 answers
225 views

Zeroes of linear combination of sines [closed]

Let us consider $$f(z):=\sum\limits_{j=1}^{j=n}a_j\sin(\lambda_jz) $$ where all $a_j$ and $\lambda_j$ (of course, $\lambda_j$ are distinct) are real numbers and $n \in \mathbb{Z},\, n \ge 3$. The ...
user64494's user avatar
  • 3,486
-3 votes
1 answer
245 views

An interesting phenomenon of the analytic continuation of Riemann zeta function [closed]

It is known that $$\Gamma (s) \zeta (s)=\int_0^{\infty} \frac{x^{s-1}}{e^x-1}dx$$ this function is valid only for $\Re{s}>1$. However, if we ignore this restriction, and integrate by using $$\frac{...
Milin's user avatar
  • 395
-3 votes
1 answer
411 views

Can Hartogs' extension theorem be used to prove there's no naked singularity?

Maybe for the first time my question doesn't deal with number theory. Tonight a friend of mine told me about Hartogs' extension theorem and said his work in analytic microlocal analysis was somehow ...
Sylvain JULIEN's user avatar
-3 votes
1 answer
158 views

Randomness about coefficients of series

$B\subset \mathbb{N}\bigcup \{0\}$ is finite and not empty, infinite series:$$f(x)=\sum_{i=1}^{\infty}a_i x^i,a_i \in B$$ Now $f(x)$ is rational or has a natural boundary. Now,the question :if $f(x)$...
XL _At_Here_There'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
-4 votes
2 answers
228 views

An elementary-looking integral inequality

This might seem a bit easy but I still like to ask it for pedagogical reasons. QUESTION. Is this inequality true for non-negative integers $n$? $$\frac{\pi}2\int_0^1x^n\sin\left(\frac{\pi}2x\right)dx\...
T. Amdeberhan's user avatar