All Questions
3,560 questions
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1
answer
1k
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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}}...
-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 ...
-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 ...
-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 ...
-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 ...
-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^{...
-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 $$p(t)=[x(t),y(t),z(t)]^T\in \text{R}^3$$ ...
-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 ?
-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 $σ > σ_0$.
...
-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 ...
-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
$$\frac{\zeta(s)}{s} = \int_1^\infty (\sum_{1 \le n \...
-1
votes
1
answer
2k
views
Real and imaginary part of an holomorphic function
I guess this could be a very elementary question. Anyway I can not find an answer in literature.
Let $f:U\rightarrow\mathbb{C}$ be an holomorphic function on an upen subset $U\subseteq\mathbb{C}$. ...
-1
votes
1
answer
2k
views
Reducibility (or not) of algebraic curves [closed]
[ I am a bit clueless about why this question is getting downvotes!? I put it up with a genuine serious interest and I don't seem to be making any egregious error either - apart from those unsure ...
-1
votes
1
answer
85
views
Reference Request: Continuous extension of conformal maps
currently I am trying to find some references on the continuous extension of conformal maps between two simply connected domains of the Riemann sphere $\hat{\mathbb C}$. Let $\gamma_1,\gamma_2$ be two ...
-1
votes
1
answer
175
views
On the bound for $\int_{x}^{x+i\infty} (\cot(\pi z)+ i)z^{-s} \, \mathrm{d}z$
I'm reading Titchmarsh's "The theory of the Riemann zeta function", and on p.81 it is claimed that
$$ \int_{x}^{x+i\infty} (\cot(\pi z)+i)z^{-s} \, \mathrm{d}z \ll \frac{x^{-\sigma}}{2(n+1)\...
-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:
$$
\|x_n\...
-1
votes
1
answer
119
views
Existence of a function with slow growth on derivatives
Does there exist a smooth compactly supported function $$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$$
...
-1
votes
1
answer
211
views
Stone Cech compactification for exponential map
Recently I met with a problem related to Stone-Cech Compactification theorem
in Furstenberg's famous paper "non-commuting product."
I try my best to understand Stone-Cech compactification theorem by ...
-1
votes
1
answer
327
views
Residue at an integration border in case of a limit? [closed]
I am dealing with an integral in a limit of the following shape:
$$\lim_{\epsilon \to 0} \int_0^{\frac{\pi}{2}} dx \frac{2 \epsilon}{1-(1-\epsilon^2)\sin^2(x)}$$
Formally, assuming that $x=\arcsin(\...
-1
votes
1
answer
215
views
Best approximation of the modulus function
While there is extensive study regarding the best approximation of function with polynomial functions in the real domain, the study of approximation of complex variables becomes much sparse. See this ...
-1
votes
2
answers
87
views
Limits of integral series
Suppose we have the series of functions:
\begin{equation}
F(x)=\sum_{n=1}^{\infty} f_n(x)
\end{equation}
where convergence is uniform.
Additionally, consider the partial functions of the series:
\...
-1
votes
1
answer
124
views
Borel summation
If $f(z)=\sum_{n=0}^\infty a_n z^n$ is a formal power series with complex coefficients, then its Borel transform is defined by
$$B(f)(z)=\sum_{n=0}^\infty a_n \tfrac{z^n}{n!}.$$
Suppose that $f$ and ...
-1
votes
1
answer
406
views
Topological properties of complex valued Riemann sum limit curve and a particular integral inequality
I am studying under what conditions the following integral inequality would hold ($a$ real, $a>0$):
$$ \int_{-\infty} ^{\infty} \frac{f(ix)}{a\pm ix}dx\ = 0 \ \ \ \ \Rightarrow \ \ \ \int_{-\...
-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 ...
-2
votes
2
answers
393
views
Expression for infinite product
can anyone show me how
$$\displaystyle\frac{4}{R}\displaystyle\Pi_{n=1}^{\infty} \left(\frac{1+R^{-4n}}{1+R^{-4n+2}}\right)^4= \frac{1}{R}\left(1+2 \sum_{n=1}^ {\infty} \frac{1}{R^{2n(n+1)}}\...
-2
votes
2
answers
2k
views
Taylor series of a complex function that is not holomorphic
I want to create Taylor series of a complex function that has complex conjugate in it. Obviously I cannot do a total derivative but derivations over real and imag parts exist.
Bonus question: Can I ...
-2
votes
1
answer
139
views
Convergence of scrambled product for Dirichlet-$L$ function with modulo 4 character
A Dirichlet-$L$ function is typically defined by its series, and its Euler product is a consequence of the definition. Here my approach is the other way around.
I define the function
$$
L_4^*(s) = \...
-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 $...
-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{...
-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:
$$|f(s)| = O(...
-2
votes
1
answer
271
views
A curious relationship betwen $|\zeta(\sigma+it)|$ and $|\zeta(1-\sigma - it)|$
By use of the Riemann functional equation, it can be shown (see corollary 10.5 of Montgomery-Vaughan) that
$$|\zeta(\sigma + it)| \asymp |t|^{\sigma-1/2}|\zeta(1-\sigma - it)|$$.
where $\zeta$ ...
-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 ...
-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_\...
-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 ...
-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). ...
-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 ...
-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 ...
-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$ ...
-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 ...
-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$...
-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 ...
-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 ...
-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)) \...
-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 ...
-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 ...
-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
$$\frac{...
-3
votes
1
answer
411
views
Can Hartogs' extension theorem be used to prove there's no naked singularity?
Maybe for the first time my question doesn't deal with number theory. Tonight a friend of mine told me about Hartogs' extension theorem and said his work in analytic microlocal analysis was somehow ...
-3
votes
1
answer
158
views
Randomness about coefficients of series
$B\subset \mathbb{N}\bigcup \{0\}$ is finite and not empty, infinite series:$$f(x)=\sum_{i=1}^{\infty}a_i x^i,a_i \in B$$ Now $f(x)$ is rational or has a natural boundary.
Now,the question :if $f(x)$...
-3
votes
1
answer
195
views
Bounding a number-theoretic integral
Find a good upper bound on $$\int_1^T\frac{\zeta'(s)}{\zeta(s)\zeta(1-s)}X^sdt,$$ where $s=c+it$ for a constant $c>1$ and $X>0$ is a parameter. If needed, we can assume RH.
My attempt here is ...
-4
votes
2
answers
228
views
An elementary-looking integral inequality
This might seem a bit easy but I still like to ask it for pedagogical reasons.
QUESTION. Is this inequality true for non-negative integers $n$?
$$\frac{\pi}2\int_0^1x^n\sin\left(\frac{\pi}2x\right)dx\...