All Questions
Tagged with ca.classical-analysis-and-odes nt.number-theory
146 questions
3
votes
0
answers
186
views
Is factorial the restriction of some elementary function?
Hölder's theorem says that Gamma function is very non-elementary, but it does not exclude the possibility that factorial is the restriction of some elementary function to natural numbers. The answer ...
- 623
3
votes
0
answers
216
views
The Fourier transform of a compactly supported smooth function on Lie groups over $\mathbb{Q}_S$, where $S$ contains finitely many primes and $\infty$
Let $G$ be a semisimple Lie group defined over global Field $\mathbb{Q}$. Let $S$ be a set of finitely many non-Archimedean places including Archimedean places. Let $P_{0}=M_{0}A_{0}N_{0}$ be the ...
- 31
3
votes
0
answers
237
views
Asymptotic expansions for the continued fraction $[1,x,x^2,x^3,\cdots]$
The $n$-th convergent is defined as
$$R_n(x) = \frac{P_n(x)}{Q_n(x)}=[1;x,x^2,\cdots,x^n]=1+\frac{1}{x+}\frac{1}{x^2+}\frac{1}{x^3+\cdots}\frac{1}{x^n}$$
where $P_n(x), Q(x)$ are polynomials ...
- 3,098
3
votes
0
answers
237
views
Reconstructing a sine wave using square waves and Möbius inversion: L² convergence?
Let $s$ be the (“square wave”) $1$-periodic real function such that $s(x) = 1$ if $0<x<\frac{1}{2}$ and $s(x) = -1$ if $\frac{1}{2}<x<1$ (and maybe $s(0)=s(\frac{1}{2})=0$ for the sake of ...
- 32.5k
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
$$K_r(n):=\prod_{\ell_1=1}^n\cdots\prod_{\ell_r=1}^n\left(
4\...
- 43.2k
3
votes
0
answers
316
views
Modified Jacobi’s theta function
Be $t\in\mathbb{R}_0^+$.
Jacobi’s theta function is $$\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
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
3
votes
0
answers
487
views
Euler divergent series $(-1)^nn!$ in $\mathbb{R}$ and $\mathbb{Q}_p$
Consider the series:
$$\sum\limits_{n=0}^{\infty}(-1)^nn!$$
We know that if $p$ is prime, this series converges in $\mathbb{Q}_p$. Let $s_p\in\mathbb{Q}_p$ be the sum of this series.
Also let
$$s_{\...
user21167
3
votes
0
answers
867
views
Proof that derivative of Hurwitz Zeta by the first argument is not expressable in terms of Hurwitz Zeta
The set of elementary functions is defined so that it to be closed against operation of differentiation. It is also evidently close against discrete differentiation.
In the discrete calculus there is ...
- 10.1k
2
votes
1
answer
543
views
On $\zeta(7)$ as the integration of the product of an indefinite integral due to Lobachevskii by a power of the inverse Gudermannian function
In this post I invoke certain function from a post of this site MathOverflow it is [1] (please see further references from the post, authors from the Springer link of the cited literature and answers ...
2
votes
1
answer
1k
views
Convergence of a general Bertrand series
Is the sum $$ S= \sum_{n=2}^\infty \frac{1}{ \log^1n \log^2n \log^3n \cdots\log^{TL(n)}n} $$
convergent?
Here
$\log^i n$ denotes the $i$'th iterate of $\log$ (in base 2) of $n$, so $\log^2n$ ...
- 1,306
2
votes
2
answers
218
views
Asymptotic estimate for $\sum_{\substack{ab\le x \\ a,b\in A}}f(a)f(b)$
Suppose that $A\subseteq \mathbb{N}$ and suppose that you have an estimate of the form
$$
\sum_{\substack{a\le x \\ a\in A}}f(a) \sim g(x).
$$
With this information is it possible to get an asymptotic ...
- 178
2
votes
2
answers
416
views
What is the solution, $f(n)$, of the following functional equation: $mf(m)+nf(n)=(m+n+xmn)f(m+n+xmn)$?
What is the solution, $f(n)$, of the following functional equation:
$$mf(m)+nf(n)=(m+n+xmn)f(m+n+xmn) ,$$
where $f$ takes on integer values, $m$ and $n$ are integers, and $x$ is an indeterminate? ...
- 153
2
votes
1
answer
297
views
Sets of integers "a little less dense" than the set of prime numbers
Given a set $A \subseteq \mathbb{N}$ of positive integers, put
$$
S_A: \mathbb{N} \rightarrow \mathbb{R}, \ \ \ \
N \mapsto \sum_{n \in A \cap \{1, \dots, N\}} \frac{1}{n}.
$$
There are obvious ...
- 19.6k
2
votes
1
answer
215
views
An integral transform computation
In Erdelyi, Tables of Integral Transforms, p. 344 Section 7.2.
they note that
\begin{align}
\frac{1}{2 \pi i} \int_{c-i\infty}^{c+i\infty} s^{\nu} e^{\alpha s^2} x^{-s} \, ds
= 2^{-\nu/2} \pi^{-...
- 21
2
votes
1
answer
232
views
Proof of a binomial identity
Computations with Maple suggest the following binomial identity
\begin{equation*}
\forall{p,j}: \sum_{k=j+1}^{p+1} (-1)^j \dfrac{1}{k}\binom{k-1}{j} =
\sum_{k=j+1}^{p+1} (-1)^{k-1} \dfrac{1}{...
- 1,020
2
votes
1
answer
300
views
the sum of fractional parts times the ordinary powers
Is there any way to compute/express $\sum\limits^m_{i=0}\{\frac{q*i}{m}\}(\frac{i}{m})^n$ ? Here $q,m,n$ are natural numbers, one can assume $gcd(q,m)=1$. Furthermore, $n$ can be treated as a ...
- 2,232
2
votes
1
answer
433
views
Proof for an explicit formula for the even Euler numbers
The EULER numbers $E_n$, $n \in \mathbb{N}$, are defined via the TAYLOR
expansion of the hyperbolic secant:
\begin{equation}
\text{sech}(x) = \sum_{n=0}^{\infty} \frac{E_n}{n!} x^n =
\sum_{n=0}...
- 1,020
2
votes
1
answer
166
views
Approximate sequence of numbers
Let $n \in \mathbb N$ and $k_n \in \left\{0,..,n \right\}$ then we define the numbers
$$x_{n,k_n} = \frac{k_n+n^2}{n^3+n^2}.$$
It is easy to see that these numbers satisfy
$$x_{n,0} = \frac{1}{n+1} ...
- 21
2
votes
0
answers
121
views
Solving a system of differential-like equations for reverse Euler-Maclaurin summation
Aim
A particular instance of a rational zeries that has as of yet not been evaluated is:
\begin{align}
Z:= \sum_{n=1}^{\infty} \frac{\zeta(2n)}{(2n)!}. \label{EM1} \tag{EM1}
\end{align}
This sum ...
- 4,829
2
votes
0
answers
158
views
What rational zeta series with non-integer arguments appear in mathematics?
Background
Rational zeta series are series of the form $$\sum_{n=2}^{\infty} q_{n} \zeta(n + p, m), \label{1} \tag{1} $$ where $\zeta(x,m)$ is the Hurwitz zeta function and $q_{n}, \ p \in \mathbb{Q} \...
- 4,829
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
2
votes
0
answers
79
views
For $\Phi$ a majorant of $1_{[-1/2,1/2]}$, how small can the total variation of $\widehat\Phi$ be?
Let $\Phi:\mathbb{R}\to \mathbb{R}$ be a real-valued, symmetric, non-negative function such that $\Phi(t)\geq 1$ for $|t|\leq 1/2$. Assume furthermore that $\Phi$ and $\widehat\Phi$ are both in $L^1\...
- 20.2k
2
votes
0
answers
66
views
A sub-logarithmic complexity in Analysis and N.Th
The question will be about complexity $\ \mathcal C(p)\ $ being positive and the same for all primes $\ p.$
Function $\ \mathcal Q\ $ is defined in the set of finite sequences of positive rational ...
- 4,786
2
votes
0
answers
113
views
Inequality about exponential integrals
I am reading about Dirichlet polynomials in the book Analytic Number Theory by Iwaniec-Kowalski.
During the proof of Theorem 9.1 for any positive real numbers $T, N$ they define a piecewise linear and ...
- 3,062
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
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
$$
\mathfrak{b}^\star: \mathcal{P}(\mathbf{N}^+) \to \mathbf{R}: X\mapsto \inf_{X\subseteq A:A\in \mathscr{A}}\...
- 1,511
2
votes
0
answers
99
views
Does there exist $k\ge2$ s.t. $X \subseteq \mathbf N^+$ has positive upper Banach density if the counting function of $X$ is $\gg n/\log^{[k]}(n)$?
Does there exist an integer $k \ge 1$ such that ${\sf bd}^\ast(X) > 0$ whenever $X \subseteq \mathbf N^+$ and $\pi_X(n) \gg \frac{n}{\log^{[k]}(n)}$ as $n \to \infty$? Here, ${\sf bd}^\ast$ is the ...
- 10.5k
2
votes
0
answers
248
views
Linear forms with best approximation vectors lying in a subspace
Setup: For $u \in \mathbb{R}^n$, let $\rho(u)$ be the Euclidean length, $\sqrt{u_1^2 + \ldots + u_n^2}$. For $x \in \mathbb{R}$ let $\|x\| = \min_{k \in \mathbb{Z}} |x - k|$, and for $x \in \mathbb{R}^...
- 616
2
votes
0
answers
341
views
How well can you approximate a function by a band-restricted function?
Say I have a compactly supported $C^1$ function $f:\mathbb{R} \to \mathbb{R}$.
Let $R>0$. Let $\nu$ be some reasonable measure on $\mathbb{R}$ -- take, for instance, (a) $d\nu(t)=dt$ or (b) $d\nu(...
- 20.2k
1
vote
1
answer
224
views
Approximating the central binomial coefficient
Let $\ c\in \mathbb R.\ $ Let
$$ 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
1
vote
1
answer
101
views
Finding $\lim_{n \to \infty} \sum_{k=2}^{n-2} \zeta(k) \zeta(n-k) x^{k-1} = x^{-1} - \psi_{0}(-x) - \gamma$ from the generating function of $\zeta(•)$
In equation (130) of this page, the identity $$\lim_{n \to \infty} \sum_{k=2}^{n-2} \zeta(k) \zeta(n-k) x^{k-1} = x^{-1} - \psi_{0}(-x) - \gamma \label{1} \tag{1} $$ is stated. Here, $\zeta(\cdot)$ is ...
- 4,829
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
178
views
Relation between $\sum_{k\ge 0}\binom {n+k}{k}a_k $ and $\sum_{k\ge 0}\binom {n+k}{k}\frac{a_k}{k}$
Let $\{a_k\}(k\ge 0)$ be a sequence of nonzero real numbers which changes signs infinitely often. Suppose $|a_k|\to 0 $ and $|a_k|$ decreases fast. Let $n$ be a positive integer. What's the relation ...
- 365
1
vote
1
answer
204
views
Local nonarchimedean Sobolev inequality
Let $B$ be the unit ball in $\mathbb{R}^n$. A local version of the Sobolev inequality on $\mathbb{R}^n$ says that for any $p\in[1,\infty]$ there exist constants $C >0$ and $k \in \mathbb{N}$ such ...
1
vote
2
answers
641
views
Sharp upper bounds for sums of the form $\sum_{p \mid k} \frac{1}{p+1}$
Are there known sharp upper bounds (in terms of $k$ or $\omega(k)$, the number of distinct prime divisors of $k$) for sums of the form $\sum_{p \mid k} \frac{1}{p+1}$ for $k > 1$ subject to the ...
- 595
1
vote
1
answer
123
views
How to show something is "true in mean square"?
I am looking at the conjecture, that for every $\varepsilon,B >0$, then
$$\Big| \sum_{\substack{n \in \mathbb{N}\\n \leq x}} n^{-it} - \int_1^x u^{-it} du \, \Big| \leq Cx^{1/2}|t|^{\varepsilon}$$ ...
- 11
1
vote
1
answer
447
views
Closed form series for reciprocal cubic function
consider a cubic of the form f(x)=$x^3-2x+z$
Is it possible to derive a power series of coefficients for the function $x^y/f(x)$, for some $y=0,1,2...$ that does not require the use of Faà di Bruno'...
- 367
1
vote
0
answers
138
views
Integral of $|1/\zeta(\sigma+i T)|$ (or $|(1/\zeta(\sigma+i T))^{(k)}|$) on a horizontal half-line in the left upper quadrant
Let $T_0\geq 20$. Let $L$ be the half-line from $-\infty + i T$ to (say) $-1/2 + i T$. Since $|\zeta(s)|$ is roughly proportional to $(T/2 \pi e)^\sigma$ for $s=\sigma+ i T$ on $L$, it is clear that ...
- 20.2k
1
vote
0
answers
133
views
Ramanujan sum type
I try to show
$$\sum _{k=1}^{\infty } \frac{e^{-2 k} k}{e^{-2 k}+1}=\frac{\pi ^2}{48}-\frac{\pi ^2-6 \left(\psi _{e^{\frac{\pi ^2}{2}}}^{(1)}(1)+\psi _{e^{\frac{\pi ^2}{2}}}^{(1)}\left(\frac{-2 i+\pi }...
- 149
1
vote
0
answers
222
views
p-adic asymptotic analysis
There is a huge field of asymptotic expansions and such over the real and complex fields (see Bender and Orszag, or Copson, or Whittaker and Watson). How different is the theory over p-adic fields? ...
- 96.4k
0
votes
1
answer
999
views
Generalizations of a product formula for the gamma function
Hello and Happy holidays.
I am interested in generalizations of the following product formula for the gamma function
$\Gamma(z)= \int_{0}^{\infty} t^{z-1}e^{-t}dt$ when $n \geq 2$:
\begin{align}
\...
- 265
0
votes
1
answer
216
views
A Newton identity and the primes--the Faber partition polynomials and modular arithmetic
[Edit, July 6, 2022: Removed erroneous characterization of Faber polynomials as an Appell sequence.]
Dress and Siebeneicher in their tale of the Burnside family express an opinion (1.2) that, if I ...
- 10.5k
-1
votes
1
answer
96
views
Limiting points of elementary set
I consider the following set
$$A:=\left\{ \frac{3mn}{2(m^2+mn+n^2)}; m,n \in \mathbb Z; \text{ and }m,n \text{ are not both zero}\right\}$$
Is it possible to identify the closure of $A$ in the reals?
- 124
-2
votes
1
answer
209
views
Strong estimates for the zeta function on natural numbers
Let $$\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
-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