Questions tagged [cv.complex-variables]
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
3,447 questions
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How to construct elliptic functions with predescribled zeros and poles by means of Weierstrass ℘-function and its derivatives?
Given a lattice $\Lambda=\mathbb{Z}\omega_1+\mathbb{Z}\omega_2$ in $\mathbb{C}$. It's well known that given $n_i\in\mathbb{Z}, z_i\in \mathbb{C}$ satisfying $\sum n_i z\in\Lambda$ and $\sum n_i=0$, ...
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Zeros of the partial sums $\sum_{k=0}^n (-1)^k/(z-k)$
let
$$
D_n(z)=\sum_{k=0}^n \frac{(-1)^k}{z-k}
=\frac{P_n(z)}{Q_n(z)}, \qquad
Q_n(z) = \prod_{k=0}^n (z-k).
$$
We have $\deg P_n = n$ ($n$ even), and $\deg P_n = n-1$ ($n$ odd). Moreover, using the ...
- 703
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Picard-Fuchs equation, Schwarzian derivative and Bers embedding of Teichmuller Space
Let $X$ be the elliptic curve
$$y^2=x^3-g_2x+g_3$$
The $j$ invariant of $X$ is
$$j=\frac{g_2^3}{g_2^3-27g_3^2}$$
I came across the formula of Dedekind
$S(\tau)(j)=\frac{1-\frac{1}{2^2}}{(1-j)^2}+\frac{...
- 515
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Convexity principle in several complex variables
I would like a reference for an analogue of the Phragmen-Lindelof in several complex variables. Specifically, if f is analytic in a region $A \subset \mathbb C^n$ and, by Bochner's theorem it extends ...
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Upper bound for maximum of reciprocal of zeta
The following appears in this paper:
Lemma Let $H=T^{1/3}$. Then we have
$$\min_{T\le t\le T+H}\max_{1/2\le\sigma\le 2}\frac{1}{|\zeta(\sigma+it)|}<\exp(C(\log\log T)^2)$$
where $C$ is an absolute ...
user574181
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Explicit convexity bound for Dirichlet $L$-functions
The following is a well-known convexity bound for Dirichlet $L$-functions.
Theorem Let $\chi$ be a primitive character to the modulus $q$. Then, for any fixed $\varepsilon>0$ and any $k\in\mathbb{...
user574181
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No real roots of $\frac{1}{2^{1+it}} + \frac{1}{3^{1+it}} + \frac{1}{5^{1+it}}$
Prove or disprove that $$\frac{1}{2^{1+it}} + \frac{1}{3^{1+it}} + \frac{1}{5^{1+it}} \neq 0$$ for any real number $t$.
I find it surprising that so simple looking equations involving complex numbers ...
- 163
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Critical zeros and Levinson's method
In this paper it was shown that one can prove at least one third of the non-trivial zeros of the Riemann zeta function lie on the critical line. This method relies on the twisted second moment of $\...
user574181
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When does the positive part of an entire function being in $L^2$ imply that the entire function is in $L^2$?
Let $f$ be an entire function of exponential type, real on the real axis, and $f^+$ its positive part, i.e.
$$
f^+(x) = \begin{cases}
f(x) & \text{if} f(x) \geq 0,\\
0 & \text{otherwise}.
\end{...
- 167
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Is phase $S(T)$ of Riemann Zeta function jumping maximum by one for small increase of $T$?
The number of non-trivial zeros of the $\zeta$ function is strongly coupled to
the hypothetical number of zeros outside of the critical line that are
counter-examples for the Riemann Hypothesis. Hence,...
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Necessary condition for a partial sum in the limit
For $s\in\mathbb{C}$, let $S(s)$ be a series that checks for the Hurwitz Theorem,
If $S(s)=0$, then by Hurwitz theorem, there exists a sequence $s_n$ so that $s_n \xrightarrow[n\to \infty]{} s$ ...
- 17
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Is the Minkowski sum of two strictly pseudo-convex bounded domains pseudo-convex?
Let me just recall that the Minkowski sum of two sets is defined by
$$A+B=\{a+b|\, a\in A, b\in B\}.$$
- 23.1k
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Computing Pic with the exponential exact sequence for singular Varieties
For a smooth complex projective variety $X$, the exponential exact sequence is $0\to \mathbb{Z} \to O \to O^*\to 0$, and gives rise to a LES of cohomology. Here, $O$ is the sheaf of holomorphic ...
- 731
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Composition of two functions is holomorphic and second is holomorphic then first is holomorphic
Let $f, g: \mathbb{C}^n \rightarrow \mathbb{C}^n$, $g$ is surjective, $f \circ g$ is holomorphic and $g$ is holomorphic. Is $f$ holomorphic? I found this is true for 1-dimensional case but is it such ...
- 121
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Best formulation of Riemann hypothesis for a general audience
There are many equivalent ways to state the Riemann Hypothesis. I'm looking for a statement that is mathematically precise and yet at the same time as accessible as possible to a general audience. The ...
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