All Questions
Tagged with ca.classical-analysis-and-odes nt.number-theory
146 questions
9
votes
0
answers
546
views
Modern treatment of Delange's Tauberian Theorem
Tauberian theorems abound in the literature. One of the most general, powerful, and versatile is due to Delange, and appears as Theorem I of the paper:
H. Delange - Généralisation du théorème de ...
- 21.3k
9
votes
0
answers
262
views
Semi-norms for Schwartz-Bruhat space over Q_p
I'm an analyst beginning to do some work over the p-adics, in particular work with spaces of functions from $\mathbb{Q}_p$ to $\mathbb{C}$. The Schwartz-Bruhat space in this case is given by the space ...
- 91
8
votes
1
answer
3k
views
The Guinand-Weil explicit formula without entire function theory
I'll admit from the outset that this question is slightly vague. The actual question appears at the end of the post.
The explicit formula of Guinand and Weil can be written in the following way:
For ...
- 2,151
8
votes
1
answer
1k
views
An elementary lower bound on the number of primes
Recall the second Chebyshev function: $$\psi(x) = \sum_{p \leq x} \lfloor \log_p x \rfloor \log p$$ where $x$ is a positive integer, and $p$ runs over all primes $\leq x$.
In a hunt for an "...
- 11.3k
8
votes
0
answers
221
views
Inertia group vs. differential equations
The tame quotient of the inertia group of $\mathbf Q_p$, say, is the profinite group generated by the Frobenius $\sigma$ and the monodromy $\tau$, subject to the relation $\tau^{p-1} [\tau, \sigma] = ...
- 2,040
7
votes
2
answers
464
views
Asymptotics of the $q$-harmonic series as $q\to1$
The following (very simply looking!) problem occurs in regularization
of the harmonic series
which can be formally thought of as the limit as $q\to1$, $|q|<1$, of
$$
h(q):=(1-q)\sum_{n=1}^\infty\...
- 13.5k
7
votes
1
answer
409
views
P-adic functions on annuli
It is known that a complex analytic function defined on an annulus, say, takes its maximum on the boundary. Does an analogue hold for $p$-adic analytic functions?
More precisely suppose we have a ...
- 303
7
votes
2
answers
521
views
How large (small) can be the measure of a set where a polynomial takes small values ?
A $n$-th degree polynomial has precisely $n$ roots. So it is natural to ask the question
how large ( and small) can be the measure of a set where a polynomial takes small values ?
This, and other ...
- 1,795
7
votes
1
answer
488
views
Ref. request: Additive probability measure on $\mathcal P({\bf N})$ supplies subset of $\mathbf R$ without Baire property
ZFC proves, among the other things, the existence of a (finitely) additive probability measure $\theta: \mathcal P(\mathbf N) \to \mathbf R$ on the power set of $\mathbf N$ such that $\theta(X) = 0$ ...
- 10.5k
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 ...
- 11.2k
7
votes
1
answer
446
views
at which rational points does the Hypergeometric function take rational values
A generic example is ${}_2 F_1(\frac{1}{3},\frac{2}{3},\frac{5}{6};\frac{27}{32})=\frac{8}{5}$. So my question: Is there any description of the set of rational points at which the hypergeometric ...
- 605
7
votes
2
answers
659
views
Chebyshev-like polynomials with integral roots
Chebyshev polynomials have real roots and satisfy a recurrence relation. I was wondering if one can find a sequence of polynomials with integral or rational roots with similar properties. More ...
- 23.5k
7
votes
1
answer
507
views
Is the mapping $f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}$ surjective?
Is the mapping
$$
f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}
$$
surjective?
If not, what is its image?
If yes, what can be said about ...
- 19.6k
6
votes
2
answers
637
views
Alternating sums of GCDs
The exciting question on alternating sums of binomial coefficients triggered me to ask the following much simpler question (sorry if it is too simple, but I'm not a number theorist, so I must be ...
- 28.6k
6
votes
1
answer
392
views
How to show that $\log 2(1/2\log 2\log 4 + 1/3\log 3\log 6 + \dotsb) + 1/2\log 2 - 1/3\log 3 + 1/4\log 4 - \dotsb = 1/\log 2$ [closed]
I've been studying Ramanujan's work and I stumbled upon this question in the book: Collected Papers of Srinivasa Ramanujan. In there I found question number 769 which is about an infinite sum with ...
6
votes
2
answers
389
views
asymptotic for li(x)-Ri(x)
Is it true that $$\operatorname{li}(x)-\operatorname{Ri}(x) \sim \frac{1}{2}\operatorname{li}(x^{1/2}) \ (x \to \infty),$$
where
$$\operatorname{Ri}(x) = \sum_{n = 1}^\infty \frac{\mu(n)}{n} \...
- 4,985
6
votes
2
answers
1k
views
A (likely) positivity property of the Lerch zeta-function
The problem is to show that $\Re L(b/2,1/2,p+1)>0$ for all real $b\ne0$ and all real $p>-1$, where
$$L(\lambda,c,s):=\sum_{k=0}^\infty\frac{\exp(2\pi i\lambda k)}{(k+c)^s}$$
is the Lerch zeta-...
- 128k
6
votes
1
answer
426
views
Asymptotic expansion of the Mordell integral
my question concerns the Mordell integral
$$h(z;\tau):=\int_{-\infty}^\infty \frac{e^{\pi i\tau w^2-2\pi zw}}{\cosh(\pi w)}dw,\qquad \Im(\tau)>0,\quad z\in\mathbb{C},$$
which frequently occurs in ...
- 189
6
votes
1
answer
2k
views
Do the tails of the decimal expansion of pi form a dense set in [0,1]?
Let $a_n=10^n \cdot \pi$. Is the set of numbers $\{a_n-\lfloor a_n \rfloor : n \in \mathbb{N}\}$ dense in [0,1]?
What is the best known result near this question?
Apparently John Nash asked this on ...
- 1,068
6
votes
0
answers
431
views
How to prove these identities for $\log(2)$ based on $_3F_2$ integrals?
In this MO post I have placed 4 Ramanujan-type hypergeometric series found using the LLL algorithm for fast computing of some logarithms. I could prove 3 of them by means of classical methods based on ...
- 2,836
5
votes
3
answers
1k
views
Product of sine
For which $n\in \mathbb{N}$, can we find (reps. find explicitly) $n+1$ integers $0 < k_1 < k_2 <\cdots < k_n < q<2^{2n}$
such that
$$\prod_{i=1}^{n} \sin\left(\frac{k_i \pi}{q} \...
- 2,829
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}$.
...
- 105
5
votes
1
answer
728
views
Linear independence of exponentials
Let $X$ be the set of functions $e^{p(x)}$ of the real vector $x$, where $p$ is a multivariate polynomial with $p(0)=0$.
Is any finite subset of $X$ linearly independent? If yes, why? If no, is the ...
- 3,873
5
votes
1
answer
227
views
Unicity of additive, $(-1)$-homogeneous, and shift invariant probability measures on $\mathbf N^+$
Let $\mathcal D$ be the set of all (finitely) additive probability measures $\mu^\ast: \mathcal P(\mathbf N^+) \to [0,\infty[$ such that $\mu^\ast(k \cdot X + h) = \frac{1}{k} \mu^\ast(X)$ for all $X \...
- 10.5k
5
votes
2
answers
484
views
Optimizing a smoothing function with the Prime Number Theorem in mind
Let $f:[0,\infty)\to \mathbb{R}$ be a function with $f(x)=1$ for $0\leq x\leq 1$. Write $Mf$ for the Mellin transform of $f$. Let $c>0$, $T>10^6$ be constants. We are interested in minimizing ...
- 20.2k
5
votes
0
answers
194
views
Proximity of zeroes of Bessel functions
I have been running into a question for which I found no reference in the litterature. I do not have a strong background in number theory ; for me this is motivated by a question in PDEs (how close ...
- 51
5
votes
0
answers
79
views
Some questions about the Lévy monoid of certain densities
Let $\bf H$ be a set, $f: \mathcal P({\bf H}) \rightharpoonup \bf R$ a partial function, and $\mathcal{D}$ the domain of $f$.
Next, denote by $\mathcal M(f)$ the set of all (total) functions $\theta: ...
- 10.5k
5
votes
0
answers
498
views
What are the consequences of allowing the ABC-conjecture $\kappa_{\epsilon}$ to also vary with $\omega(abc)$?
A commonly encountered form of the ABC-conjecture is the following:
For all $\epsilon > 0$, there is a constant $\kappa_{\epsilon} > 0$ (depending only on $\epsilon$) such that for all coprime ...
- 595
4
votes
3
answers
2k
views
How to compute $\prod_{n=1}^{\infty} (1-p^{-n})$
We know it converges for any prime $p$. I just want to know how to compute its exact value:
$$\prod_{n=1}^{\infty} (1-p^{-n})$$
- 43
4
votes
2
answers
417
views
Is the function $F(x) = \exp(x) + \exp(\exp(x))x$ a hypertranscendental function?
The function $F(x) = \exp(x) + \exp(\exp(x))x$ plays a role in the formulation of the Lagarias inequality:
$$\sigma(n) \le H_n + \exp(H_n) \log(H_n)$$
If we put $x = \log(H_n)$, then this inequality ...
- 3,064
4
votes
2
answers
2k
views
Arctangents and the golden ratio
Why is the golden ratio lurking in $(d/dx)\arctan\left( x + \frac{1}{x} \right)$
$$
= \frac{\left(\frac{1+\sqrt{5}}{2}\right)}{x^2 + \left(\frac{1+\sqrt{5}}{2}\right)^2} + \frac{\left(\frac{1-\sqrt{5}}...
4
votes
1
answer
353
views
Inequalities involving binary representation of integers
Let $N\geq 1$ be a positive integer and assume that $N=2^{n_1}+2^{n_2}+\cdots+2^{n_{p}}$, $n_{1}>n_{2}>\cdots>n_{p}\geq 0$, is the binary representation of $N$. I believe that the following ...
- 41
4
votes
1
answer
441
views
Two conjectural infinite series for $\pi$
I am looking for a proofs of the following two claims:
Claim 1.
$$\frac{2\pi}{\sqrt{3}}=\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{\Omega_1(n)}}{n}$$ where $\Omega_1(n)$ is the number of prime ...
- 2,661
4
votes
1
answer
315
views
What is the number of representations of a real number?
Let $f:\omega\to\mathbb N$ be a function such that $\sum_{n=0}^\infty\frac{f(n)}{2^n}<\infty$.
We identify each natural number $n\in\mathbb N$ with the set $\{0,\dots,n-1\}$.
Then the map $$\...
- 41.9k
4
votes
2
answers
1k
views
Product over the primes
I'm trying to estimate the product
$$\prod_{p\lt q\lt r\lt s}1-\frac{24}{(pqrs)^2}$$
where $p,q,r,s$ are primes.
This is for the purpose of calculating the density of Sloane's A070284 [1]. The idea ...
- 9,114
4
votes
1
answer
263
views
Extension by harmonics
Let $d \geq 2$ and $K$ some positive integer. Consider distinct points $\theta_1, \ldots, \theta_K\in \mathbb{T}^d$ and (not necessarily distinct) $z_1, \ldots, z_K \in \mathbb{C}$, does there exist ...
- 39k
4
votes
0
answers
189
views
Can alternative energy concepts improve bounds on large values of Dirichlet polynomials?
In the recent paper "New large value estimates for Dirichlet polynomials" by Guth and Maynard, the authors use the concept of additive energy to derive improved bounds for large values of ...
4
votes
0
answers
512
views
The Riemann zeta function and differential operators
I've revisited an old post of mine--Dirac's Delta Functions and Riemann's Jump Function J(x) for the Primes--dealing with Riemann's "jump" or "staircase" function (aka, Π(x)) that ...
- 10.5k
4
votes
0
answers
101
views
Injectivity of product functions on natural number sequences
Let $M = \{ a = (a_i)_{i} : a_i \in \mathbb{N}, a_1 \geq 2, a_i > a_j \forall i>j\}$ the set of all ascending natural number sequences, with $a_1$ at least 2.
We now define for each $k \geq 2$ ...
- 749
3
votes
4
answers
497
views
Asymptotic for Ramanujan's $\tau$-function
The Ramanujan's $\tau$-function is defined by
$$q\prod_{n=1}^\infty (1-q^n)^{24}=\sum_{n=1}^\infty \tau (n)q^n$$
where $|q|\lt 1$.
Is there a known asymptotic formula for $\tau (n)$ or $|\tau (n)|$, i....
- 317
3
votes
2
answers
597
views
lower bound for $\Re\zeta(1+it)$
Hi
is there any lower bound for $\Re\zeta(1+it)$.
I did try with computer until some ordinate and I saw $\Re\zeta(1+it)>0$.
If it is true, is there any reference to prove it.
thanks
- 163
3
votes
4
answers
3k
views
Non trivial zeros of the Zeta function
The Zeta-function can be written as the following infinite Hadamard product of its non-trivial zeroes:
$\zeta(s) = \pi^{\frac{s}{2}} \dfrac{\prod_\rho \left(1- \frac{s}{\rho} \right)}{2(s-1)\Gamma(1+...
- 4,169
3
votes
1
answer
332
views
Rate of convergence of ergodic averages related to irrational rotation
Let $\alpha$ be an irrational number, consider the basic dynamical system $T^{n}(0) = \{n \alpha\}$ where $\{.\}$ denotes the fractional part.
Let $a < b$ be two numbers in $[0, 1]$. Then by ...
3
votes
2
answers
1k
views
roots of analytic functions
Let $z$ be a complex variable and $f(z)$ be a formal power series with rational coefficients (an element in $\mathbb Q[[z]]$), with a finite radius of convergence, and assume $f(z)$ has a meromorphic ...
- 4,265
3
votes
3
answers
285
views
Limit connected with a periodic function
I am posting the following question from Math.Stackexchange:
Let $f$ be a $1$-periodic function, i.e., $f(x+1)=f(x)$, defined on the interval $(0, 1)$ by the formula
$$
f(x)=2x-1.
$$
For a real ...
- 217
3
votes
2
answers
621
views
Who needs a symmetric upper asymptotic density on the integers?
The upper asymptotic density on $\mathbf Z$, viz. the function
$$
{\sf d}^\ast: \mathcal P(\mathbf Z) \to [0,1]: X \mapsto \limsup_{n \to \infty} \frac{|X \cap [1,n]|}{n},
$$
has a ''symmetric ...
- 10.5k
3
votes
1
answer
247
views
Diophantine approximation by fractions whose numerator and denominator are both prime
Let $S \subset \mathbb{R}$ be the set of all real numbers $x$
for which there are infinitely many pairs of prime numbers $p$ and $q$ such that
$$
\left|x-\frac{p}{q}\right| < \frac{1}{q^2}.
$$
Do ...
- 19.6k
3
votes
2
answers
306
views
Asymptotics for the number of digits of the ratio of binomial coefficients
Let $a$ and $b$ be distinct positive real numbers. Let $(a_n)$ and $(b_n)$ be sequences of natural numbers such that $a_n\sim an$ and $b_n\sim bn$. All the limit relations here are for $n\to\infty$. ...
- 128k
3
votes
1
answer
652
views
Upper bound for the quality of an $abc$-triple
A triple of positive integers $(a,b,c)$ is an $abc$-triple if $a$ and $b$ are coprime and $c = a + b$. Define the quality or power of an $abc$-triple as $P(a,b,c) = \frac{\log c}{\log \text{rad}(abc)}$...
- 595
3
votes
2
answers
224
views
Rationality of the sum of the reciprocals of the values of a polynomial function at the positive integers
Let $f$ be a polynomial function of degree at least $2$ with integer coefficients,
and assume that $f(n)$ is nonzero for any positive integer $n$.
Question: Is it algorithmically decidable whether
$$
...
- 19.6k