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3 votes
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175 views

polynomial relations between modular functions

$\newcommand{\Qbar}{\overline{\mathbb{Q}}}$ We define a modular function to be a meromorphic modular form of weight 0 for some subgroup (not necessarily congruence) $\Gamma\le\text{SL}_2(\mathbb{Z})$ ...
Will Chen's user avatar
  • 10.7k
6 votes
1 answer
425 views

Uniform definition of $S(\mathbb{R})$ and $S(\mathbb{Q}_p)$

Let $\mathcal{P}=\{\infty, 2,3,5,7,11,\ldots\}$ be the set of primes of $\mathbb{Q}$ and let $\mathbb{Q}_p$ denote the corresponding completions, so in particular $\mathbb{Q}_{\infty}=\mathbb{R}$. Is ...
Abdelmalek Abdesselam's user avatar
1 vote
0 answers
182 views

Existence of Euler product on critical line for $L(\chi,s) L(\overline{\chi},1-s)$?

Generally there is no Euler product for Dirichlet L-functions $L(\chi,s)$ in the critical strip.(cf Is the Euler product formula always divergent for 0<Re(s)<1?) But I would like to know if ...
Bertrand's user avatar
  • 1,199
1 vote
1 answer
262 views

Uniform convergence of infinite sum with Dirichlet characters

I would like to prove uniform convergence of function series like : $$\sum\limits_{n=1}^{\infty} \chi(n) f(nx)$$ where $\chi$ is a primitive character and $f(x)$ a function decreasing to zero in ...
Bertrand's user avatar
  • 1,199
2 votes
0 answers
223 views

On uniform or simple convergence of Poisson Summation formula

Under good conditions on an even function $f(x)$ we have the Poisson Summation formula ($x>0$): $$f(0) + 2 \sum\limits_{n =1}^{\infty} f(nx)= \frac{1}{x} \left( \hat{f}(0) + 2 \sum\limits_{n =1}^{\...
Bertrand's user avatar
  • 1,199
3 votes
0 answers
443 views

Infinite sums with Mobius Inversion : can we have uniform convergence of inversion formula?

My question is on Mobius inversion formula convergence/properties when used with infinite sums of function. Lets consider (on $\mathbb{R}^{+}$): $$S(x)= \sum\limits_{n=1}^{\infty} f(nx)$$ We call $...
Bertrand's user avatar
  • 1,199
2 votes
1 answer
223 views

Infinite sum of asymptotic expansions

I have a question about an infinite sum of asymptotic expansions: Assume that $f_k(x)\sim a_{0k}+\dfrac{a_{1k}}{x}+\dfrac{a_{2k}}{x^2}+\cdots$ with $a_{0k}\leq \dfrac{1}{k^2}$, $a_{1k}\leq \dfrac{1}{k^...
Analysis's user avatar
1 vote
0 answers
348 views

On a property of Riemann Zeta function zeros

Lets consider the function : $$F(x) = \sum_{n=1} (xn)^{-s_0} e^{-nx} $$ with $s_0$ a zero of the Riemann Zeta function in the critical strip. This sum is well defined for $x \in \mathbb{R}^{+*}$. It ...
Bertrand's user avatar
  • 1,199
8 votes
3 answers
526 views

Lower bound for spectral radius on $\operatorname{GL}(n,\mathbb{Z})$

Consider the group of matrices $G =\operatorname{GL}(n,\mathbb{Z})$ with integer entries and determinant $\pm 1$. For each matrix $D \in G$, the product of the eigenvalues of $D$ is equal to $\det D =\...
Liam Baker's user avatar
10 votes
2 answers
925 views

Isomorphisms between spaces of test functions and sequence spaces

I am in the process of writing some self-contained notes on probability theory in spaces of distributions, for the purposes of statistical mechanics and quantum field theory. Perhaps the simplest ...
Abdelmalek Abdesselam's user avatar
0 votes
0 answers
79 views

Stable analytic manifold under simple action

For an integer $m > 1$, let us define the action $$ f: X_i \to (1+X_i)^{m} - 1 $$ on $C[[X_1,...,X_N]]$, where $C$ is the complex number field. Consider the analytic manifold $V(I)$ defined by the ...
Pierre's user avatar
  • 1
1 vote
2 answers
325 views

Riemann zeta function in the framework of Bergman spaces

My question is about connectedness of the Riemann zeta function and the theory of Bergman spaces. Because I'm working in this area of Bergman spaces, and I had Analytic number theory as a subject on ...
Alem's user avatar
  • 325
2 votes
0 answers
238 views

Fractional Derivatives [closed]

How far these Theories of "Fractional Derivatives" be rigorized ? I have few books and references on Fractional Differential Equations etc (mainly they stress on Applied Mathematics parts and similar ...
user avatar
3 votes
1 answer
273 views

Real modular form, inverse transform

The real Eisenstein series $G_s^* = \frac{\Gamma(s)}{\pi^s} \sum'_{m,n}\frac{Im(\tau)}{|m+n \tau|^{2s}}$ admits the following integral representation (their Mellin transform): $G_s^* = \frac{1}{2}...
fernando's user avatar
  • 303
5 votes
3 answers
2k views

Extension of Poisson Summation formula

Under the condition f continuous, integrable and: $|f(t)| + |\hat{f}(t)| \le C (1+|t|)^{-1-a}$ (with a>0) we have the twisted Poisson formula that holds (where $\chi(n)$ is a primitive Dirichlet ...
Bertrand's user avatar
  • 1,199
3 votes
1 answer
497 views

Possible to have Poisson Summation formula with coefficient of modular forms? (for some functions)

Taking a modular form such that we have Fricke involution: $\sum_{n=1} a_n e^{-\pi nx^2} = \frac{A}{x^k} \sum_{n=1} a_n e^{-\pi \frac{n}{x^2}}$ [1] I would like to know if there exists results on ...
Bertrand's user avatar
  • 1,199
8 votes
1 answer
1k views

A reformulation of the Riemann Hypothesis

I am studying Sieve theory from Iwaniec's notes. I have come across a theorem which estimates $\varphi(x,N)=\#\{1\leq n \leq x:(n,N)=1\}$, where $N$ is product of distinct primes. Let's define $R(x,...
Subhajit Jana's user avatar
5 votes
2 answers
1k views

Bruhat-Schwartz functions and derivatives in p-adic numbers

First of all, I am not an expert in neither classical, nor $p$-adic functional analysis, but anyway, I stumbled over the following lately: Let $\varphi:\mathbb{Q}_p\rightarrow\mathbb{C}$. ...
MHMertens's user avatar
  • 105
3 votes
1 answer
445 views

Bohr sets, Coin-flip sets and Roth's theorem

I have been learning about Roth's theorem, trying to understand how Fourier series and dynamical systems (or even graph theory and binary sequences)are involved in counting arithmetic sequences in ...
john mangual's user avatar
  • 22.8k
1 vote
2 answers
437 views

Analytical predicate for integers over complex numbers

A complex number $z$ is an integer if and only if $\sin(\pi z)=0$. It follows that a complex number $z$ is an integer if and only $\sin^2(\pi z) = 0$. So for a real analytic function $f$ and any real ...
Stephan Wehner's user avatar
3 votes
2 answers
3k views

Uniqueness of power series

Is there two sequences of real numbers $a_i$ and $b_i\neq 8$, not depending on $x$, such that $x^8=\sum_{k=1}^{\infty}a_kx^{b_k}$ for all $x$? If $\displaystyle\sum_{k=1}^{\infty}a_kx^{b_k}=\sum_{k=1}...
Paco's user avatar
  • 39
0 votes
1 answer
426 views

Lower bounds for partial sums of multiplicative functions

The motivation for this enquiry is to understand something about the impact of multiplicativity for $f:\mathbb{N}\rightarrow\mathbb{C}$ on the conditional convergence of Dirichlet series $$F(s)=\...
Kevin Smith's user avatar
  • 2,480
6 votes
0 answers
733 views

$f(x) \ne g(x)$ but $f(f(x))=g(g(x))$ - is there a name/some discussion of this property?

In the context of iteration of functions I look at the eigenvalues of the associated matrixoperator/Carleman-matrix . If a function $\small f(x)$ has a negative eigenvalue in its associated ...
Gottfried Helms's user avatar
7 votes
4 answers
2k views

Invariant means on the integers

Let $A\subseteq\mathbb Z$, as usual we define the lower Beurling density $d^{-}(A)=\lim\inf_{n\rightarrow\infty}\frac{|A\cap[-n,n]|}{2n+1}$ and the upper Beurling density $d^+(A)=\lim\sup_{n\...
Valerio Capraro's user avatar
3 votes
1 answer
181 views

Reference request - spectral radius formula for linear transformations in char p

I am finishing up a paper and I would like to be able to quote a theorem that does what is said in the title. To be specific let me introduce some notations: ${\bf F}$ is a local field of ...
Valerio Talamanca's user avatar
7 votes
3 answers
1k views

A Question concerning the Fourier Transform of $\mathbb{R}$

Consider the classical Schwartz space $\mathcal{S}(\mathbb{R})$ together with the Fourier transform $\mathcal{F} : \mathcal{S}(\mathbb{R}) \rightarrow \mathcal{S}( \mathbb{R})$. Consider the subspace ...
Marc Palm's user avatar
  • 11.2k
38 votes
2 answers
13k views

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 ...
Zen Harper's user avatar
  • 1,990
7 votes
1 answer
286 views

a.e. convergence of the powers of an operator built from rotations

Consider two numbers $a,b\in R/Z$ and some integer $p\geq 1$. Let $T:L^p(R/Z)\rightarrow L^p(R/Z)$ be the operator given by $$T(f)(x)=1/2(f(x+a)+f(x+b))$$ For which values of $a,b$ do we have almost ...
coudy's user avatar
  • 18.7k

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