All Questions
3,560 questions
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186
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Upper bound on the modulus of a power series and concentration inequalities for empirical processes
This is a research question I encountered when I as studying solutions of
Lebesgue-Stieltjes integral equations. It is related to a new statistical
method I am developing (which I cannot expose now) ...
- 984
0
votes
0
answers
248
views
Upper bound of $n$th derivative of reciprocal of $\zeta$
Is there any bound or asymptotic available for:
$$\sum\limits_{n=1}^{\infty} \frac{\mu(n)}{n^s} \log^k{n}$$
when $\Re(s) > 1$ and $k \to \infty$ ?
References are welcome.
- 129
0
votes
0
answers
158
views
On reasonable asymptotic estimates for some integral involving the logarithm of the Riemann zeta function
Let
$$I(T) = \int_{-T}^{T} \frac{\log|\zeta(\frac{1}{2} + it|)|}{\frac{1}{4}+t^2}\mathrm{d}t$$
where $\zeta$ denotes the Riemann zeta function.
What are the reasonable asymptotic estimates for $I(T)...
user123416
0
votes
0
answers
195
views
Section of a holomorphic line bundle with given differential at a zero
Let $X$ be a compact Kähler manifold of dimension $n$ with a given Kähler metric $\omega$. Let $L$ be a hermitian holomorphic line bundle on $X$ whose metric is positive. Let $x_0\in X$.
I would ...
- 404
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votes
0
answers
57
views
Questions on the behaviour of functions of exponential type 1
I am interested in understanding the properties of entire functions of exponential type 1. I have few questions about their growth.
How many sectors can a function of exponential type have, in which ...
- 503
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0
answers
83
views
Countable dense subset of functions of exponential type 1 that decay along the positive real axis
I am interested in the space of all holomorphic function of exponential type one, that decay exponentially along the positive real axis. I tried to define it as follows.
Let
$$\|f\|_n = \sup_{z\in\...
- 503
0
votes
0
answers
243
views
Sequence of analytic functions
Let $f_k$ be a sequence of rational functions analytic in the discs $\{ |z| < 1 + \epsilon_k\}$ (with some $\epsilon_k > 0$), which converge to an analytic function $f$ in every point $|z| < ...
- 163
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0
answers
297
views
Union of varieties
Let $Q_1, \ldots, Q_k$ and $P_1,\ldots, P_m$ be irredicable homogenous polynomials in $\mathbb{C}[x_0,\ldots, x_n]$ such that $V(Q_1, \ldots, Q_k) \subseteq \cup_i V(P_i)$. Here $V$ is projective ...
- 2,576
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votes
0
answers
87
views
Coefficients of a special meromorphic function
The problem described below appears elementary, but I can't figure out the answer or find it in the literature. I apologize if I have missed something very basic.
Let me begin with considering a ...
- 686
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votes
0
answers
194
views
Simple proof for Riemann/Hurwitz ζ functional equation
For the purpose of formalisation, I am looking for a simple proof of the function equation of the Riemann ζ function or the generalisation thereof, the Hurwitz's formula for the Hurwitz ζ function.
...
- 1,241
0
votes
1
answer
112
views
Certificate/Criterion for the existence of zero of a complex multivariate polynomial in a bounded region
I have a polynomial $f\in\mathbb{C}[X_1,...X_n]$ and a bounded (non-empty) compact region $\Omega \subset \mathbb{C}^n$. Let's say additionally that $f$ is not zero on the boundary of $\Omega$.
Does ...
0
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0
answers
127
views
Is $\lim_{z\to0} \exp_{\sqrt{2}}^{\circ z}(\xi)$ continuous?
This question has arisen in a bunch of my research, to the side of my research actually, I keep on getting curious about how it should be answered. I'll frame it in an anachronistic sense, but the ...
user78249
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230
views
Factorial : Gamma :: Primorial :?
Is there a unique function with the following properties:
f is meromorphic on the complex plane;
f is log-convex for n ≥ 1
$f(n) = n\#$ for n prime and ≥ 2, where # is the primorial function, and $f(...
- 309
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votes
0
answers
102
views
Finite group action on germs of holomorphic functions
Let $f(x,y) \in \mathcal{O}^\ast_{\mathbb{C}^2,0}$, a germ of holomorphic function at the origin of $\mathbb{C}^2$ with $f(0,0)=1$. Let $$\varphi(x,y)=(ax+by,cx+dy)$$ be a linear germ of ...
- 502
0
votes
0
answers
244
views
Extension of a holomorphic function
Let $H^2$ be the polydisk given by the set of points $(z,w)$ in $\mathbb C^2$ with by $\Im(z) > 0$ and $\Im(w) > 0$.
Consider the surface $S$ defined by $z^2 + w^2 = -1$ within $H^2$. On this ...
- 131
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votes
0
answers
130
views
summability and analytic continuation
Let $d_n=LCM(1,\cdots,n)$. It is well-known that $d_n=e^{\Psi(n)}$ where $\Psi$ est the second Chebyshev function. One knows that $\Psi(x)=\sum_{k\le x}\Lambda(k)$ where $\Lambda$ is the Von Mangold ...
- 3,998
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0
answers
260
views
Concluding that the Poisson kernel is indeed the Cauchy distribution?
See here.
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. We see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, y)}\...
- 13
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answers
116
views
Dimension of the set of the polynomial growth harmonic function on the hyperbolic plane
We consider the hyperbolic plane and the harmonic function there. Pick any point $p$. Let $H_n, n \in\mathbb N$ be the set of the harmonic functions $f$ such that $|f(x)|\leq c(1+ d(x,p))^n$.
What is ...
- 5,063
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0
answers
109
views
solutions of elliptic linear pde depending analytically on a parameter
Fix $ \Omega$ a bounded smooth domain in $ R^N$ and suppose $0<w(x)$ is a smooth solution of $ -\Delta w(x)=w(x)^2$ in $ \Omega$ with $ w=0$ on $ \partial \Omega$ (were are assuming $2< \frac{N+...
- 1,385
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0
answers
49
views
Non interacting complex unit
How to work with two non interacting complex units say i and j. These two imaginary complex unit represent different quantities. For example i is for periodicity in theta and j is for frequency or ...
- 1
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votes
0
answers
53
views
On sequences of rational functions [duplicate]
Let $\{f_n\}_{n=0}^\infty$ be a sequence of rational functions of the following form: $$ f_n(z) = \sum_{m=1}^\infty \frac{C_{m,n}}{z-m}$$ with $C_{m,n} \in \mathbb{Z}$, $C_{1,n} = 1$, and for each $n \...
- 11.3k
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votes
0
answers
90
views
Derivative of a conjugation of matrices
Let $\mathcal{M}_n$ be the space of complex $n\times n$ matrices. Let $\Phi\colon \mathbb{D}\to \mathcal{M}_n$ and $\psi \colon \mathbb{D}\to \mathcal{M}_n$ be holomorphic functions. Consider the ...
- 758
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0
answers
256
views
Explicit formula for Bergman kernel on the unit ball
On page 173 in Krantz's book "Explorations in Harmonic analysis" in the proof of Lemma 7.1.21 there is a part that I really don't understand. What I don't understand is why is $$\sum_{\alpha}\frac{z^{\...
- 325
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votes
0
answers
65
views
what is the first non-constant term in the Kronecker Limit formula?
The Kronecker Limit formula gives the constant term in the Laurent expansion about s=1 of the Eisenstein series E(s,\tau). What is the next term? I.e., the coefficient of the first power of (s-1)? I ...
- 153
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votes
0
answers
161
views
Asymptotic analysis of a sum of complex summands using integral
I'm trying to find the exact asymptotics of a sum:
$$A = \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^{i} y^{2n-i} $$
as $n\rightarrow\infty$. Here $x,y$ are complex numbers, $|x|\leq1, |y|\...
- 93
0
votes
0
answers
309
views
Holder continuous analytic function
Assume that $0<\alpha<1/2$ and $f$ is analytic in the unit disk $D$ with $|f(z)-f(w)|\le M|z-w|^\alpha$. Can we state that in general $$\int_D |f'(z)|^2 dx dy <\infty?$$
- 33
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0
answers
332
views
Examples of functions with natural boundary that do not satisfy Fabry or Hadamard gap theorem condition
there are examples of lacunary functions with natural boundary that do not satisfy Fabry or Hadamard gap theorem condition.I want to know more examples of those functions,the more the better,...
- 3,104
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votes
0
answers
134
views
Mittag-Leffler function and Laplace Integral
Let $E_{\alpha}(z)\triangleq \sum_{n=0}^{\infty} \frac{z^n}{\Gamma(\alpha n + 1)}$ be the Mittag-Leffler function.
I am looking for a full proof of the following fact (a reference to a proof in the ...
- 424
0
votes
1
answer
552
views
Teaching profession:Differential Equations and Mean Value Theorems
Usually I teach Algebra,Algebra and Geometyry, Topology, at various University levels. This semester (Spring 2014) I have to teach Differential Equations to University second year students (4th ...
- 1,422
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votes
0
answers
143
views
Arakelian's approximation theorem
I have a difficulty in understanding how one gets relation (2) in the proof of the Theorem, in the nice paper
[Jean-Pierre Rosay and Walter Rudin, Arakelian's Approximation Theorem, The American ...
- 1
0
votes
0
answers
173
views
Derivative of a function related to Dedekind zeta function
Lef $K$ be an algebraic number field of degree $[K:\mathbb{Q}]=n$. For simplicity suppose $K$ is totally real. Define $f(s) = \zeta_K(s) \zeta(1-s)^{n-1}$ where $\zeta = \zeta_{\mathbb{Q}}$.
From the ...
- 101
0
votes
1
answer
106
views
What is the corresponding version in the complex space of this proposition got in the real space real
How can I transform the following proposition that is gotten in $real$ space into the corresponding one used in the $complex$ space,i.e.,$A\in C^{n\times n},x=(x_1,...,x_n)\in C^n$ ?
suppose that $\...
- 1
0
votes
0
answers
141
views
Boundary behavior of Harmonic functions
Assume that $f$ is harmonic in the unit disk $|z|<1$, with boundary function of bounded variation, such that $$\lim_{r\to 1}f(re^{it})= 0$$ for $t\in[0,\pi]\setminus \mathbf{Q}$, where $\mathbf{Q}$ ...
- 303
0
votes
0
answers
381
views
Help with an irregular integral
I am looking for help with doing the following integral :
$$\frac{1}{2\pi i}\int_{1}^{\infty}\ln\left(\frac{1-e^{-2\pi i x}}{1-e^{2\pi i x}} \right )\frac{dx}{x\left(\ln x+z\right)}\;\;\;\;z\in \...
- 181
0
votes
0
answers
191
views
Differentiability of minimax objective function with respect to a decision variable
I have the following optimization problem:
$$\text{find } x= \min_{a} \max_{\lambda\in\Lambda} |R(h\lambda)|$$
where $\Lambda$ is some finite, fixed set of complex numbers and $R(z)$ is a ...
- 1,253
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votes
0
answers
100
views
Is this set of curves discrete?
Let $\alpha_1, \dots, \alpha_n$ be complex numbers whose sum is zero and $u_1, \dots, u_{2g-2+n}$ be pariwise distinct nonzero complex numbers. Consider the the set of smooth genus g curves with n+1 ...
0
votes
1
answer
448
views
Uniform convergence of a series to exponent [closed]
I'm trying to prove that in the complex plane $\left(1+\frac{z}{n}\right)^n$ converges uniformly to $e^z$ in every closed disc $|z|\leq c$. I thought about showing the sequence as a logarithm of ...
- 3
0
votes
1
answer
446
views
About an integral transform or generalized Laurent series
We start with a little of context. I needed that a function from $\mathbb{R}^+$ to $\mathbb{R}$ could be represented in the following form, not necessarily uniquely:
$$
K(z)=\int_{-\infty}^{\infty}A(\...
- 157
0
votes
0
answers
191
views
Asymtotic Complexity Analysis using logarithms and binomial coefficients
On page 11 of "Smaller decoding exponents: ball-collision decoding" by Berstein et.al. they have the formula \begin{equation}\lim_{n \rightarrow \infty} \frac{1}{n}\log_{2}\left(\dbinom{k_{1}}{p_{1}}\...
- 101
0
votes
0
answers
173
views
Implications of complex solutions of Matiyasevich / Chaitin diophantine polynomials.
This is a shot in the dark: In twf:202, an isomorphism $T\cong T^{7}$ between binary trees $T$ and seven tuples of binary trees T^{7} is mentioned. The argument for this isomorphism starts with the ...
- 975
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0
answers
169
views
Is degree a "strict -transform" birational invariant for surfaces in the complex projective 3-space?
(Edit) My question is as follows: My previous question was about a kind of "strict-transform birational" invariant not birational invariants as usual. So I just delete the question since it ...
0
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0
answers
213
views
unbounded plurisubharmonic function
Reading Demailly's Algebraic and Complex geometry in particular chapter 3 about positive currents, tha author defines the unbounded locus $L(u)$ of a plurisubharmonic function $u$ to be the set of ...
- 75
0
votes
0
answers
624
views
zeros of a holomorphic function in several variable
Let $f$ be a holomorphic function in $n$ complex variables on a domain $D\subset \mathbb C^n$. Let $S$ be a subset of $D$ such that for a polynomial $P$ in $n$ variable, $P(S)=(a,b)$ for an interval $...
- 415
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votes
0
answers
268
views
Is the absolute value of the j-invariant bounded from below on an annulus
Let $j:\mathbf{H}\to \mathbf{C}$ be the $j$-invariant. It's a modular function for $\Gamma(1) = \textrm{PSL}_2(\mathbf{Z})$.
For $\epsilon>0$ small, let $B(\epsilon)$ be the image of the strip $$\{...
- 225
0
votes
0
answers
819
views
Possible application of Rouche's theorem to aproblem of complex roots of polynomials
The following holds:
Let $P(x)$ be a polynomial in one variable $x$ of degree $3$ with complex coefficients
such that
a)
$$
P(-1)=P(1)=0
$$
Then
b)
the formal derivative $P^{'}(x)$ has a root in ...
- 1,641
0
votes
0
answers
841
views
covariant derivative complex manifold
Assume we have $X$ a complex manifold and $Y = Y^{\alpha} \frac{\partial}{\partial z^{\alpha}}$ and $Z = Z^{\alpha} \frac{\partial}{\partial z^{\alpha}}$ two vector fields on $X$. Let $\nabla$ be the ...
- 23
0
votes
0
answers
520
views
Motivation of proof of Riemann-Roch for elliptic curve and generalizations
Given a lattice $L \subseteq \mathbb{C}$, Alain Robert defines a theta function as a meromorphic function such that $\theta(z+\omega)=a(\omega) e^{\pi h(\omega)(z+\frac{\omega}{2})} \theta(z)$ for all ...
- 15.4k
0
votes
1
answer
122
views
Existence of an eigenpair for d-bar operator in the unit disck
Let $\overline{\partial}=\frac{1}{2}(\partial_{x}+\textrm{i} \,\partial_y)$ and let $D$ be the unit disc in the complex plane. For each $\lambda \in \mathbb C$, consider the problem:
$$ \overline{\...
- 4,115
0
votes
1
answer
226
views
Subspaces of $H^{\infty}(\mathbb{D})$ which contains a nontrivial weak* closed subalgebra
Let $H^{\infty}(\mathbb{D})$ denotes the Banach space of bounded holomorphic functions in the unit disc. Consider the weak* topology on $L^{\infty}(\mathbb{T})$
that it inherits as the dual of $L^{1}(\...
- 175
-1
votes
1
answer
1k
views
Non trivial zeros of Riemann zeta function [closed]
Question Define $f(z)=(s-1)\zeta(s)$ where $s=\frac{1}{1+z^2}$ and $\zeta$ denotes the Riemann zeta function. Prove that if $\rho$ denotes the non trivial zeros of $\zeta(s)$ then, $$\sum_{|\alpha|&...
user292590