All Questions
Tagged with nt.number-theory fa.functional-analysis
78 questions
2
votes
0
answers
95
views
Uncertainty principle: minimize $\int_{-\infty}^\infty |t| |\widehat{f}(t)|^2 dt$ for $f$ of compact support
This is a question of uncertainty-principle type stemming from Eigenvalue of a convolution and a restriction?
Let $f:\mathbb{R}\to \mathbb{R}$ be even, absolutely continuous and supported in $[-\frac{...
- 20.2k
2
votes
0
answers
76
views
Function that is (essentially) a self-convolution but not a multiple of a self-convolution
Call a function $F:\mathbb{R}\to C$ nice if it is of the form $F = f\ast \tilde{f}$, where $\tilde{f}(x) = \overline{f(-x)}$. (Of course nice functions are precisely those whose Fourier transform is ...
- 20.2k
2
votes
0
answers
194
views
Functions such that the *integral* of the Fourier transform is non-negative?
Let $f:\mathbb{R}\to \mathbb{R}$ be in $L^1$, with its Fourier transform $\widehat{f}$ also in $L^1$. What is a necessary and sufficient condition on $f$ so that
$$\int_{-\infty}^x \widehat{f}(t) dt \...
- 20.2k
0
votes
1
answer
117
views
Validity of approximation method for von Mangoldt function
I'm working on a problem involving the pointwise almost everywhere convergence of multilinear ergodic averages with the von Mangoldt function inspired by this paper. Specifically, I'm looking at ...
0
votes
0
answers
54
views
Functional equations with coupled arguments and additive sructure
Let $G$ be a locally compact abelian group and let $f: G \to \mathbb{R}^+$ be a continuous function satisfying the functional equation
$$f(x + \phi(y)) + f(y + \phi(x)) = 1 + f(x+y)$$
for all $x, y \...
1
vote
0
answers
175
views
Solution of recurrence relation with summation
I have the following recurrence relation:
$$b(n,k)=\sum _{\text{i}=0}^{2 n-1} \left(b(n-1,k-\text{i})+\frac{\text{i} (2 n-\text{i}) \binom{2 n-1}{\text{i}} \binom{(n-2)^2}{k-\text{i}}}{2 n-1} \right)$$...
- 47
1
vote
1
answer
170
views
Mean of probability distribution
I have a probability distribution defined by the following density function:
$f(k,j,n,m)=\frac{(m n)! \mathcal{S}_k^{(j)}}{(m n)^k (m n-j)!}$ (With $\mathcal{S}_k^{(j)}$ being the Stirling number of ...
- 47
3
votes
1
answer
172
views
1-1 map on the $\{0,1\}^k$
Let integer $k>0$ and let $\{0,1\}^k$ denote the set of all $1\times k$-dim vectors whose every coordinate is eithor 0 or 1, for example, $(0,1,1,0,\dots,1,0,0,1)$. For any such vector $\alpha$, ...
- 349
0
votes
1
answer
91
views
Construct next polynomial from predecessor and resulting GCD
I have a sequence of polynomials built from an interpolation derived in a combinatorial problem. For each integer value of a parameter $n$ there is a different polynomial.
After trying to find a way ...
- 47
0
votes
0
answers
163
views
Generalization of polynomial coefficients
I'm dealing with a hard combinatorial problem where for every positive integer value of a variable $n$ I have to calculate a list of numbers, specifically $n^2$, that depend on $n$ and its list index ...
- 47
6
votes
1
answer
248
views
Syndetic sets and Banach limits: reference request
First of all, let us give a few definitions. Suppose that $A$ is a subset of natural numbers. We say that $A$ is syndetic if there is a constant $M$ such that every set of $M$ consecutive natural ...
- 7,193
1
vote
0
answers
109
views
PDE coupled with the pronic numbers (related to triangular numbers)
I am studying the linear PDE:
$$ t^2\frac{\partial^3}{\partial t^3}\sum_{n=1}^\infty \Psi_n(t,s)=s^2\frac{\partial}{\partial s}\sum_{n=1}^\infty \Psi_n(t,s)+\sum_{n=2}^\infty b(n)\frac{\partial}{\...
- 383
1
vote
0
answers
103
views
Validity of analysis of summation of function of primes using Abel–Plana summation:
Consider the analytic function $g(x)$
Define
$$f(x)=g(x)\frac{\sin^2\left(\frac{π\Gamma(x)}{2x}\right)}{\cos^2\left(\frac{π}{2x}\right)}$$
Note that:
$$f(p)=g(p) \text{ for prime } p$$
And $f(n)=0$ ...
- 782
1
vote
0
answers
153
views
A transformation game for natural numbers?
Consider the completely additive function $\eta(n) := \sum_{p\mid n} v_p(n)p$ defined on natural numbers, with values in natural numbers. For literature, on this function, see the corresponding OEIS ...
- 3,064
1
vote
1
answer
355
views
Hilbert–Pólya conjecture with Grommer inequalities?
The Grommer inequalities are equivalent to RH and formulated on page 20 of Conrey - Riemann's hypothesis:
Let
$$\Xi(t) := \xi(1/2+it).$$
Then RH is equivalent to : All zeros of $\Xi(t)$ are real. The ...
- 3,064
0
votes
0
answers
346
views
On a Duality between Riemann-weil explicit formula and Abel- Plana summation of trigonometric prime counting function:
Consider the analytic function $g(x)$
Now define
$f(x)=g(x)\frac{\sin^2\left(\frac{π\Gamma(x)}{2x}\right)}{\cos^2\left(\frac{π}{2x}\right)}$
Such that
$|f(x+it)|=o(e^{2πt})$
uniformly for every $x$...
- 782
7
votes
0
answers
162
views
Relation between the additive Haar measure on $(K,+)$ and the multiplicative Haar measure on $K^{*}$ for a global field $K$
The following question comes from my studying of Alain Connes's paper Trace Formula in Noncommutative Geometry and the Zeros of the Riemann Zeta Function. In it, on p. 11, Connes notes that if $K$ is ...
- 1,805
7
votes
1
answer
334
views
Extremal problem for 2-dimensional lattices
Given a lattice $L$ in a Banach space $(B,\|\;\|)$, one denotes by $\lambda_1(L)$ the least norm of a nonzero element in $L$, and by $\lambda_k$ the least $\lambda$ such that there is a linearly ...
- 16.6k
1
vote
0
answers
180
views
Applications of hyperbolic polynomials?
The recently posted MO-Q "Positivity of the coefficients of Taylor series associated to the Riemann hypothesis" (see also this MO-Q) has re-kindled my interest in hyperbolic polynomials--...
- 10.5k
1
vote
0
answers
87
views
what is the relationship betwen $L(s,sym^mf\times sym^mg)$ symmetric L function of $f$ and $g$ and $\lambda_{f}(n^m)$, $\lambda_{g}(n^m)$?
what is the relationship betwen $L(s,sym^mf\times sym^mg)$ symmetric L function of $f$ and $g$ and $\lambda_{f}(n^m)$, $\lambda_{g}(n^m)$ ?
- 11
4
votes
0
answers
189
views
About the structure of smooth automorphic forms
Recently I read Prof. Cogdell's notes: Lectures on L-functions, Converse Theorems, and Functoriality for $GL_n$. (Co)
In chap.2.3, the conception of smooth automorphic forms is introdued. Explicitly, ...
- 111
7
votes
0
answers
198
views
The spectrum of the Banach algebra of certain arithmetic functions under Dirichlet convolution
I was thinking about using the tools of functional analysis to study some subring of arithmetic functions under Dirichlet convolution. If I let $D_s$ be the ring of arithmetic functions with finite ...
- 190
2
votes
0
answers
136
views
Eigenfunction of $h\mapsto H(h')|_{[-1,1]}$?
Let $H$ be the Hilbert transform. Is there a continuous, even function $h:\mathbb{R}\to \mathbb{R}$ with support on $[-1,1]$ such that, for some $\lambda\in \mathbb{R}$,
$$H(h')(t) = \lambda h(t)$$
...
- 20.2k
3
votes
0
answers
117
views
Arithmetic progressions and removal lemmas for graphs in arithmetic combinatorics
As it is well known, one can gets a proof of Roth's Theorem concerning arithmetic progressions of length 3 (APs for short) by using the celebrated Ruzsa-Szemerédi triangle removal lemma for graphs.
In ...
- 1,561
4
votes
0
answers
197
views
Bailey's lemma in number theory
A pair of sequences $(α_n,β_n)$ is called a Bailey pair if they are related by
$$\beta_n=\sum_{r=0}^n\frac{\alpha_r}{(q;q)_{n-r}(aq;q)_{n+r}}$$
or equivalently
$$\alpha_n = (1-aq^{2n})\sum_{j=0}^n\...
- 41
3
votes
0
answers
174
views
On continuous seminorms on Fréchet-Stein algebras
Let $K$ be a discretely valued complete non-archimedean field and $U$ be a left Fréchet-Stein algebra as defined in Algebras of p-adic distributions and admissible representations, with a Fréchet-...
- 541
5
votes
1
answer
190
views
Describing the Gamma-transform explicitly in terms of power series
The Gamma transform of a measure is defined as follows. If $\alpha$ is a $\mathbf{Z}_p$-valued measure on $\mathbf{Z}_p$, then the Gamma transform of $\alpha$ is:
$$\Gamma_{\alpha}(s) = \int_{\mathbf{...
- 1,949
3
votes
0
answers
77
views
Unitary with entries $(i,j)$ only on equidistant lattice points $\|i-j\|^2 = c^2 \in \mathbb{N}$
My research needs help in finding examples of unitary matrices $U$ which have entries
\begin{align}
U_{ij} = \begin{cases} \alpha_{ij}, \ \text{ if } \|i-j\|^2 = c^2 \\ 0 , \text{ otherwise} \end{...
- 41
2
votes
1
answer
278
views
Diophantine equations and ergodic theorems
In the paper by Akos Magyar, Diophantine Equations and Ergodic Theorems, one states in page 923 the following theorem:
Theorem 1: Let $Q(m)$ be a nondegenerate polynomial and $\Lambda$ is ...
5
votes
0
answers
118
views
Good (Sidon) Approximation of "Bumps"
Given a rational point $p\in S^1$ and a continuous function $f:S^1\rightarrow \mathbb C$, we say that $f$ is an $\epsilon$-bump around $p$ (for some $\epsilon>0$) if $f(p)=1,|f|_{\infty}\leq 1+\...
- 113
9
votes
1
answer
346
views
Is there a uniform solution of the Ruziewicz problem?
For any integer $n\geq 2$ there is one and only one (up to rescaling) rotation-invariant, finitely-additive measure on the Lebesgue $\sigma$-algebra of $S^n$.
The proof of this statement I'm aware of ...
- 91
0
votes
1
answer
158
views
Encoding numbers with relationship into one and back
Given a set of many variables $S=\{x_1,x_2, ...., x_i\}$, and any subset $S'$ of $S$, I need a function $f$ which maps $S'$ to a value $x$ and a function $f'$ which maps $x$ back to set $S'$.
I know ...
6
votes
0
answers
348
views
Recent work on Pseudo-Laplacian and Pseudo-cuspform in the spirit of Riemann Hypothesis after the work of Bombieri and Garrett
( This is my first MO question . I'm totally inexperienced on MO so, forgive me for my mistakes .)
Paul Garrett and Enrico Bombieri were (are?) Secretly Working on Pseudo-Laplacians and Pseudo-...
user153636
6
votes
1
answer
340
views
The abc-conjecture over the positive rationals and Levy-Schoenberg kernels?
I am continuing the "abc-adventure" and have a specific question, which needs some explanation:
In this paper by Gangolli, the term "Levy-Schoenberg" kernel is defined (Definition 2.3).
Consider the ...
user6671
7
votes
0
answers
373
views
What is known about "almost orthogonal vectors"?
Motivation:
Suppose we have a kernel $k(a,b)$ defined over the natural numbers.
Then by the Moore–Aronszajn theorem, we can embedd the natural number $a$ in some Hilbert space $\mathbb{H}$, which we ...
user6671
2
votes
1
answer
920
views
Fourier transform of the von Mangoldt function?
Wikipedia states under the entry for the von Mangoldt function:
The Fourier transform of the von Mangoldt function gives a spectrum with spikes at ordinates equal to imaginary part of the Riemann ...
- 10.5k
13
votes
1
answer
528
views
Are the logarithms of the integer polynomials discrete in $L^1$ of the unit circle?
Tautologically, the integer polynomials form a discrete set in $L^1$ of the unit circle. On the other hand, a set of logarithms ordered by norm becomes generally rather denser than the original set.
...
- 13.8k
63
votes
5
answers
10k
views
Jean Bourgain's relatively lesser known significant contributions
Jean Bourgain passed away on December 22, 2018.
A great mathematician is no longer with us.
Terry Tao has blogged about Bourgain's death and mentioned some of his more recent significant contributions,...
0
votes
0
answers
169
views
Is there an option to handle Neumann-series when it diverges? (using infinite-sized Carleman matrices)
(I asked this in MSE but did not find resonance, there is also a relation to an older discussion here on summability see here and a followup formulating an $\text{ais}()$ already here)
...
- 5,342
4
votes
1
answer
237
views
Meromorphic continuation of local zeta integrals
Let $f$ be a Maass cusp form for $\text{SL}_2(\mathbb{Z})$ on the upper half plane. Let $\varphi_0$ be its lift to an automorphic form on $G = \text{PGL}_2(\mathbb{R})$ and let $\pi = \pi_{f} =\langle ...
- 163
1
vote
0
answers
137
views
Is there an analysis theorem analogous to Kuznetsov/Petersson trace formula?
I am thinking about general differential operator acts on a compact manifold. Is there something similar to Kuznetsov trace formula?
For example, let $f_i $ be the eigenfunctions of an operator $D$, ...
- 3,804
0
votes
1
answer
110
views
Number theory for operator bound
Let $\gamma_i$ be such that for even $i$ $\gamma_i=1$ and for odd $i$ $\gamma_i$ shall have absolute value $1$ and the product of all of the odd ones is also on the complex unit circle but not 1 or -1....
- 501
0
votes
2
answers
193
views
Space of functions f such that the number of primes in $ [x, x+f(x)] $ remains bounded
Given a positive integer $ n $ , let $ S_{b}(n) $ the set of functions $ f $ fulfilling the following conditions :
1) $ f $ is continuous, positive and increasing on $(n,+\infty) $
2) for ...
- 6,974
3
votes
1
answer
541
views
Adelic Schwartz class
I am not a specialist in automorphic forms, can someone explain to me typical elements of adelic Schwartz class, $\mathcal{S}(\mathbb{A})$. Over the real numbers there are obviously elements like:
$$ ...
- 22.8k
8
votes
0
answers
265
views
$L^2$ norms of Whittaker vectors and zeros of Intertwining operators
For $\mu,\nu\in \mathbb{C}^2$ we denote $I(\mu,\nu)$ to be the principal series of $\mathrm{GL}_2(\mathbb{Q}_p)$ induced from $|.|^\mu\otimes |.|^\nu$. For $s=\mu-\nu$ one defines the standard ...
- 1,692
2
votes
0
answers
132
views
Wiener-Ikehara Theorem and Signal Processing
I am trying to understand the Wiener-Ikehara Tauberian theorem which can be a step to understanding the prime number theorem. Let
$$ \hat{a}(s) = \int_0^\infty e^{-us}\, da(u) $$
with $a(u)$ some ...
- 22.8k
9
votes
1
answer
299
views
Sequence of nested sets in $[0, 1]$ with bound on gaps
What is the best possible $\epsilon$ and sequence $(a_n)_{n = 1}^\infty \subset [0, 1]$ we can find such that
$$
d_{N}:=\sup_{x\in [0,1]}\inf_{n=1}^N |x-a_n|\leq \frac{1+\epsilon}{N}
$$
for all $N\in ...
user78817
1
vote
1
answer
129
views
Linear functions
Let $(f_1, f_2, \ldots, f_n)$ be an $n$-tuple of functions mapping non-negative integers to non-negative integers. Let $m$ be a positive integer.Suppose there exists a function $f$ apping non-negative ...
- 1,129
5
votes
0
answers
168
views
Functional equations about Conway's box function
Conway's box function is the inverse of Minkowski's question mark function. It maps the dyadic rationals on the unit interval to the rationals using the Stern-Brocot tree (Farey sequence).
The ...
- 213
4
votes
1
answer
633
views
Quantum Mechanics derivation of Wallis' Formula?
Recently there was a proof of the Wallis Product using quantum mechanics on the arXiv. However, there are many proofs of the result, Wikipedia has 4.
Fine Print the first proof has on Wikipedia, the ...
- 22.8k