All Questions
Tagged with nt.number-theory real-analysis
208 questions
1
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0
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135
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About the sum of prime reciprocals
Let $B$ be the Meissel-Mertens constant., $\theta$ the Chebyshev theta. Let $S(x)=\sum_{p\le x}1/p$, $p$ prime. Let $M(x)=S(x)-\log\log x-B$. Robin, and later Diamond and Pintz, showed that $M(x)$ ...
- 1,018
0
votes
0
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221
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Positivity of an integral
Let $f(x)=x-\theta(x)$, and for $x\ge 2$ let $J(x)=\int_x^{\infty}\frac{f(y)(1+\log y)}{y^2\log^2 y}dy$. Furthermore, $J(2)>0$. Suppose $J(x)>0$ when $f(x)=0$. Does it follow that $J(x)>0$ ...
- 1,018
6
votes
1
answer
223
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Asympotic density of a very simple sequence
Let $A=\{mn(m+n)\mid n,m\in \mathbb{N}_0\}$. Sorted, this is OEIS sequence A088915. What is its asymptotic behavior? It seems approximately $a(n)=O(n^{1.5})$, but not quite.
I'm actually even more ...
- 5,103
2
votes
0
answers
194
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Shattering with sinusoids
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) $z_1, \ldots, z_K \in \{-1,1\}$ such that $\sum\...
- 698
4
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0
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105
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On a much weaker version of the Normal conjecture
I would like to ask you about the following question. It is conjectured that every algebraic irrational number is normal (absolutely normal). I know the result by Bugeaud and Adamczewski about the non-...
- 515
2
votes
1
answer
166
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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
$$x_{n,0} = \frac{1}{n+1} ...
- 21
5
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1
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456
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Number defined by a recursive binary sequence
In a math column in Scientific American many years ago, I encountered a peculiar binary sequence I describe below. Unfortunately I can't find a reference on this, so I would be grateful for any ...
- 48.9k
8
votes
3
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747
views
How does the parity of $n$ affect the properties of $\mathbb{R}^n$? [closed]
Does the parity of the dimension of $\mathbb{R}^n$ affect its structure/properties? As in, does it make a difference if $n$ is even or odd?
5
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2
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874
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Searching for a proof for a series identity
The below identity I have found experimentally.
Question. Is this true? If so, may you provide a "slick" (or any) proof.
$$6\sum_{k=1}^{\infty}\frac{k^2q^k}{(1-q^k)^2}+12\left(\sum_{k=1}^{\infty}...
- 43.2k
-2
votes
1
answer
208
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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
$$\zeta(s) = \frac{s}{s-1}-s\int_{1}^\infty \frac{...
- 749
4
votes
0
answers
101
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Injectivity of product functions on natural number sequences
Let $M = \{ a = (a_i)_{i} : a_i \in \mathbb{N}, a_1 \geq 2, a_i > a_j \forall i>j\}$ the set of all ascending natural number sequences, with $a_1$ at least 2.
We now define for each $k \geq 2$ ...
- 749
6
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2
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509
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Summing Bernoulli numbers
Consider the Bernoulli numbers denoted by $B_n$, which are rational numbers.
It is known that the harmonic numbers $H_n=\sum_{k=1}^n\frac1k$ are not integers once $n>1$.
I am curious about the ...
- 43.2k
1
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1
answer
142
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About a Dirichlet series [closed]
I would like to know if the following assertion is true:
Let consider a real decreasing sequence $(t_n)$ of positive numbers with limit zero, if the series $\sum\limits_{n=1}^\infty(t_n)^a$ is ...
- 337
6
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2
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389
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asymptotic for li(x)-Ri(x)
Is it true that $$\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
41
votes
6
answers
9k
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"Long-standing conjectures in analysis ... often turn out to be false"
The title is a quote from a Jim Holt article entitled, "The Riemann zeta conjecture and the laughter of the primes" (p. 47).1
His example of a "long-standing conjecture" is the Riemann hypothesis,...
1
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0
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156
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Fejer-Jackson-like inequality with divisor sum
A question was recently asked about a generalization of the Fejer-Jackson inequality $$\sum_{k=1}^n \frac{\sin kx}{k}\gt 0 \quad \forall\: n\in\mathbb{Z}^+\: \text{and}\: 0\lt x\lt\pi$$
to ...
- 105
3
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0
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169
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Why is the smallest (fractional) absolute central moment of a Gaussian distribution almost at $\sqrt{3}/2$?
Let $X$ be a standard normal random variable. What $\alpha$ minimizes $E|X|^{\alpha}$?
Numerically, $\alpha$ turns out to be equal to $\sqrt{3}/2-\varepsilon$ where $\varepsilon$ is of the order $10^...
7
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1
answer
683
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The Gauss Circle Problem asymptotic in dimension
The circle problem in $k$ dimensions: "For $n>0$, how many points $z\in \ \mathbb{Z}^k$ have $\|z\|^2\leq n$?"
For large $n$, the answer is $\approx n^{k/2}\cdot \operatorname{Vol}(B^k(0,1))+\...
9
votes
1
answer
458
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Summing moments and Riemann zeta values
Let $d\mu_n(x)=\cos^{2n}x\,dx$ and consider the averages of moments
$$\alpha_n=\frac{\int_0^{\pi/2}x^4d\mu_n(x)}{\int_0^{\pi/2}d\mu_n(x)}.$$
Then, I have encountered a curious evaluation
$$\sum_{n=1}^{...
- 43.2k
2
votes
0
answers
76
views
Bounding algebraic numbers away from the Gaussian integers
Let $\alpha$ be an algebraic number with degree $\leq d$ and (absolute multiplicative) height $\leq H$. Then we can say a couple of things about such $\alpha$:
(1) We know the set of all $\alpha$ ...
- 31
3
votes
1
answer
631
views
Is the sequence $(\log(n!)\mod1)_{n\in\mathbb N}$ dense in the interval $[0,1]$?
This question was raised in the comment by Todd Trimble at how to proof there is a natural number n, the first four digits of n! Is 2018?. I thought the question may be posted separately, as even ...
- 128k
1
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0
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137
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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
2
votes
2
answers
172
views
An analogue of the equidistribution theorem?
Suppose that $(n_k)_{k\in \mathbb{N}}$ is a given increasing sequence of positive integers.
Does there exist an (irrational) number $a$ such that
$\{an_k\}:=(a n_k)\text{mod }1 \rightarrow 1/2$ as $...
- 21
2
votes
0
answers
114
views
Is there an explicit version of Morse Lemma used in stationary phase method?
In the proof of the stationary phase method (at least the one I have seen) Morse lemma shows up, which states: Let $g:\mathbb R^n\to \mathbb R$ be a function of class $C^\infty$ for which $0$ is a ...
- 3,625
1
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1
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250
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Question about the stationary phase method and the smooth function used
A statement of the stationary phase method I know is the following.
Suppose $\phi(x_0) = \phi'(x_0) = 0$ and $\phi''(x_0) \not = 0$. If $\psi$ is a smooth function supported in a sufficiently small ...
- 3,625
0
votes
1
answer
195
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Approximating the sum $\sum_{n \leq X} a_n$ with a smooth sum $\sum_{n \geq 1} a_n w(X)$
I have a sequence $a_n$ such that $0 \leq a_n \leq \log n$, and I am considering
$\sum_{n \leq X} a_n$. However, I prefer using smooth weights so I would like to approximate it with $\sum_{n \geq 1} ...
- 3,625
4
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0
answers
187
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Asymptotic formula, polynomial, irrational number and uniformly distribution
Problem 1
Given a irrational number $\alpha$ and two polynomials with positive integer coefficients $P(n),Q(n)$, is it possible to get the asymptotic estimate and reasonable error term for:
$$\...
- 697
33
votes
1
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3k
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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
5
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0
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280
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Proving that a certain function (related to a volume of a region) has a bounded derivative
Let $F$ be a homogeneous form in $n$ variables with integer coefficients.
Let $D$ be a closed box in $\mathbb{R}^n$ (product of closed and bounded intervals). Assume that the partial $\partial F/\...
- 3,625
1
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0
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134
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Convolution integral of series involving the non-trivial zeros of $\zeta(s)$
Let us consider the convolution $$f\left(x\right):=\int_{2}^{x-2}\sum_{\rho_{1}}\frac{u^{\rho_{1}}}{\rho_{1}}\sum_{\rho_{2}}\frac{\left(x-u\right)^{\rho_{2}}}{\rho_{2}}du,\,x>4$$ where $\rho_{i},\,...
- 219
122
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5
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27k
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Is the series $\sum_n|\sin n|^n/n$ convergent?
Problem. Is the series $$\sum_{n=1}^\infty\frac{|\sin(n)|^n}n$$convergent?
(The problem was posed on 22.06.2017 by Ph D students of H.Steinhaus Center of Wroclaw Polytechnica. The promised prize for ...
- 4,535
1
vote
2
answers
257
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Pairwise disjoint subsets of $\mathbb{N}$ with positive upper density
For $A\subseteq\mathbb{N}$ we define the upper density to be $$\text{ud}(A) = \lim\sup_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}$$
Is there an infinite set ${\cal S}$ of pairwise disjoint subsets ...
- 48.9k
4
votes
1
answer
315
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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 $$\...
- 41.8k
3
votes
2
answers
303
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Basic question related to Stieltjes integral
I am reading this paper. I am stuck on something, which I think is something basic but I haven't been able to figure it out yet, and I was hoping someone could explain it to me.
Let
$$
\sigma(u) = \...
- 3,625
0
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1
answer
110
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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
11
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1
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466
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A property of real numbers concerning integer parts of multiples
For a given positive real number $\alpha$, define the set $T_\alpha$ by
$T_\alpha = \{ [n\alpha] \mid n = 1,2,\dots \}$. What is a necessary and sufficient condition (in terms of $\alpha$ and $\beta$...
- 111
13
votes
3
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1k
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iterated harmonic numbers vs Riemann zeta
Define the $m$-th iterated harmonic sums in the manner: $\bar{H}_0(n):=1$ and for
$m\geq1$ by
$$\bar{H}_m(n):=\sum_{k=1}^n\frac{\bar{H}_{m-1}(k)}k.$$
For example, $\bar{H}_1(n)=\sum_{k=1}^n\frac1k$ ...
- 43.2k
18
votes
5
answers
3k
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Bernoulli sum meets golden number
Let $B_n$ denote the Bernoulli numbers and let $\phi=\frac{1+\sqrt{5}}2$ be the golden ratio.
I encountered the following infinite sum and would like to ask:
Question. Is this true? If so, any ...
- 43.2k
2
votes
2
answers
261
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Prove a family of series having integer coefficients
I encountered a certain family of infinite series in some work, which is given by
$$F_r(x)=\frac1{2^r}\sum_{k=0}^r\binom{r}k\frac1{1+x(2k-r)^2}.$$
I've convincing date to believe the following is true,...
- 43.2k
1
vote
2
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111
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A two-parameter inequality on product of linear terms
I would like to ask about a certain inequality that I need and which came out of some work in here.
Question. For integers $n\geq1$ and $k\geq3$, is this true? If so, any proof?
$$6\prod_{j=1}^k(...
- 43.2k
89
votes
7
answers
13k
views
If I exchange infinitely many digits of $\pi$ and $e$, are the two resulting numbers transcendental?
If I swap the digits of $\pi$ and $e$ in infinitely many places, I get two new numbers. Are these two numbers transcendental?
user10290
3
votes
0
answers
232
views
When polynomial f(t+1/t) can be factored as g(t)·g(1/t)?
In venue of my old question When polynomial f(x^2) can be factored as g(x)·g(-x)? and this recent answer to a different question, I wonder:
How to characterize polynomials $f(x)$ with rational ...
- 34.3k
15
votes
4
answers
3k
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No Tonelli or Fubini
Whenever we can interchange summation (perhaps due to Tonelli-Fubini), good things happen. Otherwise, one has to struggle evaluating double sums in just one way, because the alternative results in a ...
9
votes
2
answers
244
views
If normal with respect to prime base then normal for all bases
I tried to find it on internet but couldn't so m asking this here. I want to ask if a number is normal with respect to all prime number base then do we know that it is normal with respect to any base. ...
- 335
27
votes
3
answers
2k
views
Kasteleyn's formula for domino tilings generalized?
It seems a marvel when a bunch of irrational numbers "conspire" to become rational, even better an integer. An elementary example is $\prod_{j=1}^n4\cos^2\left(\pi j/(2n+1)\right)=1$.
Kasteleyn's ...
- 43.2k
5
votes
0
answers
170
views
operation on Ord., Exp., Dri. generating functions
The ordinary, exponential and Dirichlet generating functions for a sequence $\{a_n\}_{n\geq0}$ are given (at least on the formal side), respectively, by
$$F(x)=\sum_{n\geq0}a_nx^n, \qquad E(x)=\sum_{n\...
- 43.2k
5
votes
2
answers
341
views
a modification on an infinite Bernoulli convolution
The distribution $\nu_{\lambda}$ of the random series $\sum\pm\lambda^n$ is the infinite convolution product of $\frac12(\delta_{-\lambda^n}+\delta_{\lambda^n})$. This problem has been studied ...
- 43.2k
3
votes
3
answers
233
views
sequencial shift on families =flipped powers. How?
Consider the following family of functions
$$f_n(w):=\sum_{k=0}^{\infty}\frac{(-1)^{k-1}}{k!}(k+n)^{k-1}w^k.$$
QUESTION 1. Does the following hold?
$$f_n(w)=-\frac1{n(f_{-1}(w))^n}.$$
Deeper ...
- 43.2k
1
vote
0
answers
99
views
simultaneous smallness
QUESTION. Given reals $0 < \epsilon, \delta < 1$, is it always possible to find $m, n \in \mathbb{N}$ such that
$$\begin{cases} \qquad \,\,\,\, \,(1-\delta^m)^n < \epsilon \\
1-(1-(\frac{\...
- 43.2k
106
votes
5
answers
10k
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integral of a "sin-omial" coefficients=binomial
I find the following averaged-integral amusing and intriguing, to say the least. Is there any proof?
For any pair of integers $n\geq k\geq0$, we have
$$\frac1{\pi}\int_0^{\pi}\frac{\sin^n(x)}{\...
- 43.2k