All Questions
3,560 questions
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On an oscillatory property of the Riemann Xi-function
In their paper "The Integral of the Riemann Xi-Function", Lagarias and Montague mention Wintner's 1947 proof that
$$
\Xi^{(-1)}(t) > 0 \quad \text{when} \quad t > 0.
$$
This result ...
- 61
1
vote
1
answer
155
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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 $\...
CommunityBot
- 1
27
votes
5
answers
6k
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The Matrix-Tree Theorem without the matrix
I'm teaching an introductory graph theory course in the Fall, which I'm excited about because it gives me the chance to improve my understanding of graphs (my work is in topology). A highlight for me ...
- 5,429
3
votes
2
answers
141
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Accessible literature on fractional dimensions of subsets of $\mathbb R^n$
I am currently wondering whether it is realistically possible to choose the topic "Fractals and fractal dimensions" for a seminar aimed at undergraduate students in the 2nd semester, with ...
- 81
38
votes
2
answers
13k
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What, exactly, has Louis de Branges proved about the Riemann Hypothesis?
I know this is a dangerous topic which could attract many cranks and nutters, but:
According to Wikipedia [and probably his own website, but I have a hard time seeing exactly what he's claiming] Louis ...
CommunityBot
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-2
votes
2
answers
322
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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 ...
- 91.8k
1
vote
1
answer
132
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Deriving a specific bound for functions in Hardy Space
Reading some article a while ago I read the following: (here $H^2$ represents the Hardy space)
Let $f\in H^2$ be such that $f(0)=1$, and let $0<\lvert\lambda\rvert<1$, then $$\lVert f(\lambda z)\...
- 24.2k
14
votes
1
answer
1k
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The location of the zeros of the "new" function $\Psi(s)=\sum _{n=1}^{\infty }{\frac{1}{n!^s}}$
Defining the Psi function as $$\Psi(s)=\sum _{n=1}^{\infty }{\frac{1}{n!^s}}$$ and by studying, from a numerical point of view, the location of the zeros in the complex plane up to $0<\Im(s)<...
- 161
3
votes
1
answer
104
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Characterization of bi-Hermitian structures with equal Lee forms
Let $(M,g,I_+,I_-)$ be a compact bi-Hermitian manifold, where $g$ is a Riemannian metric and $I_+$, $I_-$ are two complex structures that are both compatible with $g$. We assume that $I_+$ and $I_-$ ...
- 108k
6
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0
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209
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Partial fraction expansions of meromorphic functions
Sorry if this question (inspired by the recent flurry of activity around a "new" formula for $\pi$) is too naive.
Imitating what one does with Hadamard products, one can try to do the same ...
- 6,377
30
votes
1
answer
4k
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Proof of "Possible new series for $\pi$" without use of physics
Related post: The post Possible new series for $\pi$ is about whether the identity is new, so to avoid confusion I was advised to ask this question separately.
I am looking for a proof of the ...
- 82.7k
32
votes
9
answers
21k
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Interesting applications of the classical Stokes theorem?
When students learn multivariable calculus they're typically barraged with a collection of examples of the type "given surface X with boundary curve Y, evaluate the line integral of a vector field Y ...
- 2,754
7
votes
2
answers
389
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Systematic way to compute $\sum_{n=1}^\infty P(n) / Q(n)$ for polynomials $P$ and $Q$
This may be well known so feel free to downvote.
When $P = 1$ and $Q(x) = x^k$, this is of course the Riemann zeta function. But what about other cases?
For instance is it always possible to express $\...
- 231
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0
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338
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Recognizing when a $2\pi$-periodic function is a shifted sine
Consider a real analytic function $f(x)$ with a period of $2\pi.$ Let $t\in \operatorname{Range}(f(x))$, $$I(t)=\int_{0}^{2\pi}\log |t-f(x)|dx.$$ Prove that $I(t)$ is equal to a constant if and only ...
- 12.9k
4
votes
2
answers
287
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Teaching suggestions for Kleene fixed point theorem
I will take over two lectures from a colleague in which we discuss fixed point theory in the context of complete partial orders, and culminates in showing the Kleene fixed point theorem (see f.e. ...
- 349
2
votes
1
answer
154
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Branched covering maps between Riemann surfaces
What is an example of a branched covering map between Riemann surfaces of infinite degree? i.e. something like a branched version of the exponential map $exp: \mathbb{C} \to \mathbb{C}^*$.
Thanks!
- 91.8k
3
votes
1
answer
344
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On the upper bound for $|\zeta(s)|$ near the zeta zeros
Let $T \in \mathbb{R}$ be large and $\rho$ be a non-trivial zero of the Riemann zeta function. Assume that $|\rho|=|\rho_T| \approx T$ and let $\varepsilon_T \approx \frac{\log \log T}{\log T}$. Is it ...
CommunityBot
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3
votes
1
answer
232
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Prescribe the type of an entire function which inverse zeros are summable
According to Lindelöf's theorem, given points $z_i\in \mathbb C\setminus \{0\}$ ordered by increasing modulus with possible repetitions, we can define a function
$$
f(z)=\prod_{n=1}^\infty (1-z/z_n)e^{...
- 91.8k
1
vote
0
answers
42
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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
407
views
Finite generation of certain graded sequences of ideals
Let $U\subset\mathbb{C}^n$ be an open set containing the origin $o$ and $Y\subset U$ a complex analytic subvariety of pure codimension $c$ with ideal sheaf $\mathcal{I}_Y$. Let $\frak{a}_{\bullet}=\{{\...
CommunityBot
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votes
3
answers
853
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The smallest nontrivial zero of the Riemann zeta function
Consider the Riemann zeta function $$\zeta(s)=\dfrac{1}{1-2^{1-s}}\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^s},\operatorname {R}(s)>0.$$ Riemann suggested that all nontrivial zeroes lie on the line $\...
- 14.7k
0
votes
0
answers
145
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Function of several complex variables with prescribed zeros
I accidentally stumbled upon a problem of complex analysis in several variables, and I have a hard time understanding what I read, it might be related to the Cousin II problem but I cannot say for ...
- 1,352
6
votes
0
answers
342
views
Does the Poincaré lemma (Dolbeault–Grothendieck lemma) still hold on singular complex space?
Let $X$ be a complex manifold, then we have the Poincaré lemma (or say, Dolbeault-Grothendieck lemma) (locally) on $X$, whose formulation is as follows:
( $\bar{\partial}$-Poincaré lemma) If $\...
- 327
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
7
votes
0
answers
167
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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 (...
- 12.9k
3
votes
0
answers
219
views
Schwartz's theorem without English language reference
I'm reading the paper "Spectral Synthesis And The Pompeiu Problem" by Leon Brown, Bertram M. Schreiber and B. Alan Taylor,
Annales de l’Institut Fourier 23, No. 3, 125-154 (1973), MR352492, ...
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1
answer
376
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Bringing a Heun equation into canonical form
It is a well known fact that any second order Fuchsian differential equation on the complex plane $$u''(x) + p(x)u'(x) + q(x)u(x)=0$$ with exactly $4$ regular singular points may be suitably ...
CommunityBot
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votes
0
answers
39
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Contraction of an inclusion with respect to Kobayshi hyperbolic metric
Suppose that $X = \mathbb{C}^n - \Delta_X$ and $Y = \mathbb{C}^n - \Delta_Y$, where $\Delta_X$ and $\Delta_Y$ are unions of hyperplanes in $\mathbb{C}^n$ such that $\Delta_Y \subset \Delta_X$, $\...
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1
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1
answer
175
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Motivation for defining polar derivative
The polar derivative of a polynomial $p(z)$ is defined as
$$ D_\alpha p(z):=np(z)+(\alpha -z)p^{\prime}(z) \qquad \alpha\in\mathbb{C}. $$ It is a polynomial of degree $n-1.$ I am new to complex ...
- 1,538
0
votes
0
answers
188
views
Behavior of subtree of $\mathbb{Z}^2$ embedded in $\mathbb{C}$ under compactification of the latter to the riemann sphere
I consider a countable subtree $T$ of the integer lattice isomorphic to $\mathbb{Z}^2$ with directed edges. It shall be embedded in $\mathbb{C}$ where the edge $(u,v)$ points from $u$ to $v$ if and ...
- 19.6k
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
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,\...
0
votes
1
answer
191
views
Estimates for the first coefficient of the Taylor expansion of $\zeta$ around a zero
Let's suppose that $s_0=\frac{1}{2}-\Delta+it$ with $0<\Delta<\frac{1}{2}$ is a simple zeta zero (i.e a zero not on the critical line). Then $1-\overline{s_0}$ is also a zero.
If we take the ...
- 105k
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 ...
CommunityBot
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votes
1
answer
122
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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 ...
- 105k
0
votes
1
answer
211
views
Sign of $\Re\frac{\xi'(s)}{\xi(s)}$ locally around a zeta zero
In the article ”On a positivity property of the real part of logarithmic derivative of the Riemann $\xi$–function” the authors check the positivity of $\Re \frac{\xi'}{\xi}(s)$ for $\frac{1}{2}<\...
- 105k
2
votes
1
answer
261
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Uniqueness results for holomorphic functions bounded in a strip with a certain decay to zero
Let $A=\{ z \in \mathbb C : |\mathrm{Im}(z)| < a \}$ for some $a >0$. Further let $F:A \to \mathbb C$ be a holomorphic function on $A$ with the following properties:
$F$ is bounded on $A$
For ...
- 91.8k
-6
votes
1
answer
139
views
Is the real part of the Eta function bounded by $2 \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}}{n^{\alpha}}} $ [closed]
Consider the series defined by
\begin{equation}
f(\alpha,\beta) := \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}}{n^{\alpha}}\cos(\beta\ln(n))}
\end{equation}
is it true that $$f(\alpha,\beta) \le 2\sum_{n=1}...
- 188k
1
vote
0
answers
64
views
Extension of meromorphic distribution
Let $W$ be a topological vector space (e.g. Frechet) with a dense subspace $V$. Let $D_s$ be a distribution on $V$ that is meromorphic in $s\in\mathbb C$ and extends continuously to $W$ with respect ...
- 3,799
11
votes
1
answer
961
views
Can the topologist's sine curve be realized as a Julia set?
Does there exist a rational function $f\in\Bbb{C}(z)$ whose Julia set coincides with
$$
T:=\left\{\left(x,\sin\left(\frac{1}{x}\right)\right)\,\Big|\,x\in\left(0,\frac{1}{\pi}\right]\right\}\cup\big(\{...
- 91.8k
1
vote
0
answers
133
views
Fundamental set for families of abelian varieties
I'm considering the universal family of principally polarized g-dimensional abelian varieties with level N-structure. Let's briefly recall the usual construction as a quotient of $\mathbb{C}^g \times \...
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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 $σ > σ_0$.
...
- 105k
41
votes
3
answers
3k
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Can the unsolvability of quintics be seen in the geometry of the icosahedron?
Q1. Is it possible to somehow "see" the unsolvability of quintic polynomials
in the $A_5$ symmetries of the icosahedron (or dodecahedron)?
Perhaps this is too vague a question.
Q2. Are there ...
CommunityBot
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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
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 ...
CommunityBot
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0
votes
1
answer
281
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 ...
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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 ...
- 360
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) \...
- 91.8k
1
vote
0
answers
118
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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
2
votes
1
answer
589
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Bounds for Dirichlet L-functions
Let $L$ denote a Dirichlet L-function attached to the primitive character $\chi$. What are the best known bounds for $L(\sigma+it, \chi)$?
PS: For $L=\zeta$ and $0\leq\sigma\leq 1$, i'm aware of a ...
- 105k