All Questions
Tagged with ca.classical-analysis-and-odes nt.number-theory
146 questions
-5
votes
1
answer
191
views
How do you prove the validity of this formula for $H(n)$? [closed]
I'm looking for a proof of the identity
$$\sum_{k=1}^{n}\frac{1}{k}=\frac{1}{2n}+\pi\int_{0}^{1} (1-u)\cot{\pi u}\left(1-\cos{2\pi n u}\right)\,du.$$
There is a generalization of this formula for $...
user130611
-2
votes
1
answer
208
views
Strong estimates for the zeta function on natural numbers
Let $$\zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s}$$
be the Riemann zeta function (here we just consider real $s$).
We do have a description given by
$$\zeta(s) = \frac{s}{s-1}-s\int_{1}^\infty \frac{...
- 749
4
votes
0
answers
101
views
Injectivity of product functions on natural number sequences
Let $M = \{ a = (a_i)_{i} : a_i \in \mathbb{N}, a_1 \geq 2, a_i > a_j \forall i>j\}$ the set of all ascending natural number sequences, with $a_1$ at least 2.
We now define for each $k \geq 2$ ...
- 749
34
votes
1
answer
3k
views
A remarkable almost-identity
OEIS sequence A210247 gives the signs of $\text{li}(-n,-1/3) = \sum_{k=1}^\infty (-1)^k k^n/3^k$, also the signs of the Maclaurin coefficients of $4/(3 + \exp(4x))$.
Mikhail Kurkov noticed that it ...
- 54.2k
6
votes
2
answers
389
views
asymptotic for li(x)-Ri(x)
Is it true that $$\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
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
13
votes
1
answer
604
views
A recursive formula
$$a_0 = 1, \ \ a_{n+1} = \ 1+\frac{n *a_n}{n+a_n} , \ \ n=0,1,2,3,4,...$$
I have built the above recursive formula. Some terms of sequence are:
1, 1, 3/2, 13/7, 73/34, 501/209, 4051/1546, 37633/...
- 1,305
30
votes
3
answers
3k
views
Lindelöf hypothesis claim
I was randomly browsing, when I found this puff piece claiming a proof of the Lindelöf hypothesis by Fokas. Note that the Wikipedia article says that he claimed, then withdrew his claim in 2017, but ...
- 96.4k
17
votes
3
answers
3k
views
Is there an "analytical" version of Tao's uncertainty principle?
Let $p$ be a prime. For $f: \mathbb{Z}/p \mathbb{Z} \rightarrow \mathbb{C}$ let its Fourier transform be:
$$\hat f(n) = \frac{1}{\sqrt{p}}\sum_{x \in \mathbb{Z}/p \mathbb{Z}} f(x)\, e\left(\frac{-xn}{...
- 1,235
5
votes
1
answer
728
views
Linear independence of exponentials
Let $X$ be the set of functions $e^{p(x)}$ of the real vector $x$, where $p$ is a multivariate polynomial with $p(0)=0$.
Is any finite subset of $X$ linearly independent? If yes, why? If no, is the ...
- 3,873
2
votes
2
answers
416
views
What is the solution, $f(n)$, of the following functional equation: $mf(m)+nf(n)=(m+n+xmn)f(m+n+xmn)$?
What is the solution, $f(n)$, of the following functional equation:
$$mf(m)+nf(n)=(m+n+xmn)f(m+n+xmn) ,$$
where $f$ takes on integer values, $m$ and $n$ are integers, and $x$ is an indeterminate? ...
- 153
33
votes
1
answer
3k
views
About the validity of a new conjecture about a diophantine equation
Let us consider the following conjecture:
Conjecture: There are no integer solutions of the equation $$x^{y-z}z^{x-y}=y^{x-z}$$ with $x,y,z$ distinct positive integers greater than or equal to $2$.
...
- 1,197
4
votes
1
answer
315
views
What is the number of representations of a real number?
Let $f:\omega\to\mathbb N$ be a function such that $\sum_{n=0}^\infty\frac{f(n)}{2^n}<\infty$.
We identify each natural number $n\in\mathbb N$ with the set $\{0,\dots,n-1\}$.
Then the map $$\...
- 41.8k
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
10
votes
1
answer
752
views
A conjecture about certain values of the Fabius function
The Fabius function is a smooth monotone function $F:[0,1]\to[0,1]$, satisfying functional equations
$$F(0)=0, \quad F(1-x)=1-F(x)\tag1$$
and
$$F'(x) = 2 \,F(2 x) \quad \text{for} \,\, 0<x<1/2.\...
- 6,819
18
votes
3
answers
1k
views
A curious series related to the asymptotic behavior of the tetration
The tetration is denoted $^n a$, where $a$ is called the base and $n$ is called the height, and is defined for $n\in\mathbb N\cup\{-1,\,0\}$ by the recurrence
$$
{^{-1} a} = 0, \quad {^{n+1} a} = a^{\...
- 6,819
3
votes
0
answers
122
views
How large do $r$-dimensional "Kasteleyn-Temperley-Fisher" numbers grow?
I brought up a couple of combinatorial and number-theoretic items with this MO question. Now, I shall inquire on growth estimates. Recall
$$K_r(n):=\prod_{\ell_1=1}^n\cdots\prod_{\ell_r=1}^n\left(
4\...
- 43.2k
1
vote
1
answer
990
views
Is the existence of $\lim_{n\to\infty}\cos(n!\pi x)$ for given arbitrary irrational $x$ an open problem?
Motivated by a recent MSE question about the sequence of function $\cos(n!\pi x)$, I have read related several related questions:
On the behaviour of $\sin(n!\pi x)$ when $x$ is irrational.
Is there ...
user14319
1
vote
1
answer
224
views
Approximating the central binomial coefficient
Let $\ c\in \mathbb R.\ $ Let
$$ D_n(c)\ := \frac {4^n}{\binom {2\cdot n}n\cdot\sqrt{4\cdot n + c}} $$
Then, more or less by the Wallis product theorem, we have this well-known convergence:
$$ \...
- 7,500
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
$$
...
- 19.6k
2
votes
1
answer
300
views
the sum of fractional parts times the ordinary powers
Is there any way to compute/express $\sum\limits^m_{i=0}\{\frac{q*i}{m}\}(\frac{i}{m})^n$ ? Here $q,m,n$ are natural numbers, one can assume $gcd(q,m)=1$. Furthermore, $n$ can be treated as a ...
- 2,232
2
votes
0
answers
256
views
various Hardy-Littlewood Tauberian theorems $ \sum \frac{a_n nx^n}{1-x^n} \to \frac{A}{1-x} $ implies $\sum_{k=0}^\infty a_k = A$
I am seeing the Hardy-Littlewood Tauberian Theorem phrased in several different ways. Are they all equivalent?
A If $\sum a_n x^n \sim \frac{1}{1-x}$ then $\sum_{k=0}^n a_k = n$
B If $\sum a_n e^{...
- 22.8k
1
vote
0
answers
133
views
Ramanujan sum type
I try to show
$$\sum _{k=1}^{\infty } \frac{e^{-2 k} k}{e^{-2 k}+1}=\frac{\pi ^2}{48}-\frac{\pi ^2-6 \left(\psi _{e^{\frac{\pi ^2}{2}}}^{(1)}(1)+\psi _{e^{\frac{\pi ^2}{2}}}^{(1)}\left(\frac{-2 i+\pi }...
- 149
3
votes
0
answers
316
views
Modified Jacobi’s theta function
Be $t\in\mathbb{R}_0^+$.
Jacobi’s theta function is $$\Theta(t):=\sum\limits_{k=-\infty}^{+\infty} e^{-\pi k^2 t}$$ with $$\Theta(\frac{1}{t})=\sqrt{t}\Theta(t)$$
Therefore $$\sum\limits_{k=1}^\infty ...
- 293
2
votes
0
answers
123
views
A limsup representation for the upper Buck density
The upper Buck density, introduced by R.C.Buck in 1946, is defined to be the function
$$
\mathfrak{b}^\star: \mathcal{P}(\mathbf{N}^+) \to \mathbf{R}: X\mapsto \inf_{X\subseteq A:A\in \mathscr{A}}\...
- 1,511
10
votes
2
answers
1k
views
Algebraic independence of exponentials
First of all, a happy new year. Be it better than 2015,
healthy, wealthy, fruitful and cross-fertilizing
for you, familly and friends.
In order to cope with families of solutions of evolution ...
- 3,738
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
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 ...
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
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
2
votes
0
answers
99
views
Does there exist $k\ge2$ s.t. $X \subseteq \mathbf N^+$ has positive upper Banach density if the counting function of $X$ is $\gg n/\log^{[k]}(n)$?
Does there exist an integer $k \ge 1$ such that ${\sf bd}^\ast(X) > 0$ whenever $X \subseteq \mathbf N^+$ and $\pi_X(n) \gg \frac{n}{\log^{[k]}(n)}$ as $n \to \infty$? Here, ${\sf bd}^\ast$ is the ...
- 10.5k
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
1
vote
0
answers
222
views
p-adic asymptotic analysis
There is a huge field of asymptotic expansions and such over the real and complex fields (see Bender and Orszag, or Copson, or Whittaker and Watson). How different is the theory over p-adic fields? ...
- 96.4k
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
0
answers
133
views
Reference for a lemma on the asymptotic upper density of special sets with large gaps and intervals
Update. Based on Anthony Quas' comment below, the proof can be made sensibly shorter and the lemma can be slightly generalized by weakening the old assumption (iii).
In a joint paper that I am ...
- 10.5k
12
votes
2
answers
552
views
On the independence of lower and upper asymptotic and Banach densities
Given a set $X \subseteq \mathbf N^+$, denote by $\mathsf{d}_\ast(X)$ and $\mathsf{d}^\ast(X)$, respectively, the lower and upper asymptotic (or natural) density of $X$, viz. $$\mathsf{d}_\ast(X) := \...
- 10.5k
6
votes
2
answers
1k
views
A (likely) positivity property of the Lerch zeta-function
The problem is to show that $\Re L(b/2,1/2,p+1)>0$ for all real $b\ne0$ and all real $p>-1$, where
$$L(\lambda,c,s):=\sum_{k=0}^\infty\frac{\exp(2\pi i\lambda k)}{(k+c)^s}$$
is the Lerch zeta-...
- 128k
2
votes
0
answers
248
views
Linear forms with best approximation vectors lying in a subspace
Setup: For $u \in \mathbb{R}^n$, let $\rho(u)$ be the Euclidean length, $\sqrt{u_1^2 + \ldots + u_n^2}$. For $x \in \mathbb{R}$ let $\|x\| = \min_{k \in \mathbb{Z}} |x - k|$, and for $x \in \mathbb{R}^...
- 616
10
votes
2
answers
1k
views
Prove that the Dirichlet eta function is monotonic
Let us consider $\eta(p):= \sum\limits_{n=1}^\infty \frac{(-1)^{n+1}}{n^p}$ for $p>0$. Has anyone come along with an elementary proof that $\eta(x)$ is monotonically increasing on this set? By ...
13
votes
1
answer
782
views
Cesaro(?)/Euler(?) - summation of the $s(p)=\sum_{k=0}^\infty (-1)^{H(k)} (1+k)^p$ for $p=1,2,3,...$ (where $H(k)$ is the Hamming-weight)
In another thread (in MO) there was a question about a series where the signs at the terms alternate with the "Hamming-weight", that means according to the number of bits in the binary representation ...
- 5,342
6
votes
1
answer
426
views
Asymptotic expansion of the Mordell integral
my question concerns the Mordell integral
$$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},$$
which frequently occurs in ...
- 189
33
votes
4
answers
3k
views
A translation of the Cantor set contained in the irrationals
Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here.
What would ...
- 32.5k
10
votes
0
answers
2k
views
Questions on de Branges' work on the Riemann hypothesis
According to Wikipedia, Louis de Branges de Bourcia has obtained some notable
results, such as a proof of the Bieberbach conjecture in 1985, which is now
known as de Branges' theorem. Initially, his ...
- 163
7
votes
1
answer
507
views
Is the mapping $f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}$ surjective?
Is the mapping
$$
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
13
votes
1
answer
1k
views
Is $\varliminf_{n \rightarrow +\infty} |n \sin n| = 0$ correct, where $n$ is an integer?
Is it true that $\varliminf_{n \rightarrow +\infty} |n \sin n| = 0$, where $n$
runs over the integers?
The existence of the limes inferior follows from Dirichlet's approximation theorem,
but the ...
- 399
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
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
2
votes
0
answers
341
views
How well can you approximate a function by a band-restricted function?
Say I have a compactly supported $C^1$ function $f:\mathbb{R} \to \mathbb{R}$.
Let $R>0$. Let $\nu$ be some reasonable measure on $\mathbb{R}$ -- take, for instance, (a) $d\nu(t)=dt$ or (b) $d\nu(...
- 20.2k
20
votes
1
answer
1k
views
Provable zero-free region for any entire function that analytically is similar to zeta(s)
Is there an entire function $f:\mathbb C\rightarrow\mathbb C$ such that for some $\delta>0$:
$f(z)$ is bounded when $\Re z>1+\delta$
$f(z)$ is unbounded when $\Re z=1$
$f(z)$ grows polynomially ...
- 1,243
42
votes
7
answers
5k
views
How should an analytic number theorist look at Bessel functions?
(And a related question: Where should an analytic number theorist learn about Bessel functions?)
Bessel functions occur quite frequently in analytic number theory. One example, Corollary 4.7 of ...
- 7,337