0
votes
1answer
92 views

Computing infinite product over primes

How can I compute $$ \prod_p \left(1+\frac{k}{p}\right)\exp(-k/p) $$ where $0<k<e$ and the product is over all primes $p$? Background L. G. Sathe proved [1] that there are $$ ...
8
votes
2answers
308 views

A series with only rational terms for $\ln \ln 2$

We all know that $$ \ln 2 = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}. $$ Do you know a series with only rational terms for $$\ln \ln 2 = ?$$ Let's exclude base expansions with non ...
2
votes
0answers
34 views

Approximating real vectors in the unit hypercube by rational vectors in the unit hypercube

I am currently looking for an error bound for an problem where I have to approximate a real vector in the unit hypercube using a rational vector in the unit hypercube. Specifically: given a ...
0
votes
0answers
50 views

a special function for count

Let be $f:(\mathbb{N} \setminus \left\{0,1 \right\})^2 \rightarrow \mathbb{N}$ function that $f(a,k)=\text{total numbers of }n \in \mathbb{N} \text{ that } \frac{a^n}{n^k} \le 1$ . My question is: ...
2
votes
0answers
70 views

Points in a general Cantor set

We often look at the Cantor set with the construction that keeps removing the middle thirds of the remaining line segments at each iteration. Corresponding to this construction, we can determine ...
26
votes
4answers
500 views

Geometric intuition for $\pi /4 = 1 - 1/3 + 1/5 - \cdots$?

Following reading this great post : What are some interesting cases of $\pi$ appearing in situations that are not / do not seem geometric?, Vadim's answer reminded me of something an analysis ...
1
vote
1answer
110 views

Can any real number be expressed as an integral combination of $e$ and $\pi$?

Prove or disprove: For any real number $x$, there exist integers $a$ and $b$ such that $ae + b\pi=x$. It certainly seems improbable, but how does one prove it?
1
vote
1answer
39 views

Does sequences related to function for $lcm(1,2,3 \cdots n)$ exists?

This just came out of curiosity let $$L(n)=lcm(1,2,3 \cdots n)$$ and I know that we can write this with the help of some product involving primes and all . But what I am interested is in Does ...
2
votes
3answers
68 views

What is meant by a function that varies in only one direction?

My professor mentioned in class today about a "function that varies in only one direction". To clarify, he said he meant a function which is a function of a linear combination of the variables. That ...
3
votes
1answer
77 views

Asymptotic of a sum evaluation as $ x \to \infty $

Let be the sum $$ \sum_{n\le x}[x/n]=g(x) $$ where $ [x] $ means floor function. My best try for asymptotic is $ g(x) \sim x\log (x)+\gamma x +1$ where I have used the asymptotic $ [x] \sim x $ ...
7
votes
1answer
180 views

The closed form of $\sum_{n=1}^{\infty} \left(\frac{1}{\lfloor\sqrt{3n}\rfloor^2}-\frac{1}{3n}\right)$

I need some ideas to exploit for finding the closed form of $$\sum_{n=1}^{\infty} \left(\frac{1}{\lfloor\sqrt{3n}\rfloor^2}-\frac{1}{3n}\right)$$
1
vote
0answers
47 views

Rational approximation bound for real numbers in (0,1)

I am working on a problem that related to rational approximation of real numbers. I am looking for a bound of the form: Given a positive real number, $\alpha \in (0,1)$, there exist positive ...
3
votes
0answers
152 views

Conjecture on OEIS A167055

OEIS A167055 Numbers n such that $12n + 5$ is prime. $0, 1, 2, 3, 4, 7, 8, 9, 11, 12, 14, 16, 19, 21,...$ are items of OEIS A167055. I conjecture that the set of the sum of every two items of this ...
9
votes
3answers
210 views

Does $\sum_{p\in\mathbb P}\frac {( - 1)^{[\sqrt p\,]}}{p}$ converges?

Does $$\sum_{p\in\mathbb P}\frac {( - 1)^{[\sqrt p\,]}}{p}$$ converges ? I know that the following $\sum_{p\in\mathbb P} \frac{1}{p}$ diverges, we can find proofs on Wikepedia Divergence of the ...
7
votes
3answers
198 views

If $\sum\frac1{a_n}$ is convergent, then irrational?

$\{a_n\}$ is a strictly increasing sequence of positive integers such that $$\lim_{n\to\infty}\frac{a_{n+1}}{ a_n}=1$$ If $\sum\limits_{n=1}^\infty\frac1{a_n}$ is convergent, can one conclude ...
3
votes
1answer
67 views

The function $f(t)=2+\sin(t)+\sin(t\sqrt2)$

The function $f$ defined on $\mathbb{R}$ by $$f(t)=2+\sin(t)+\sin(t\sqrt2)$$ can never reach $0$. Can we find some sequence $(t_n)_{n\geq0}$ such that $$\lim_{n \to \infty}f(t_n)=0 \ \ \ ?$$ Or in ...
2
votes
2answers
78 views

Non uniform continuity of a function and almost periodicity

We say that a continuous function $f:\mathbb{R}\to\mathbb{C}$ is almost periodic in the sense of Bohr if: For every sequence $(t'_n)_{n\geq0}$, there's a sub-sequence $(t_n)_{n\geq0}$ such that ...
2
votes
1answer
32 views

About the implicit funtion in a holomorphic situation.

Let $f(x,y)$ be a polonomial with integral coefficients which has a zero $(a,b)\in \mathbb{R}^2$ such that the partial derivative respect to $y$ at this point is nonzero. Then by the implicit function ...
16
votes
1answer
347 views

Integral = $\pi/2$ !!

I am trying to calculate the integral $$ I_n=\int \limits_0^\infty \prod_{k=1}^n \frac{\sin \frac{x}{2k-1}}{\frac{x}{2k-1}}dx. $$ (I have literature on this, if people want). Note, we can write the ...
1
vote
2answers
94 views

Looking for references

I am looking for reference on the following problem. Let $S= \{ ax_1+bx_2 \mid x_1 \in X_1 , x_2 \in X_2, \}$ where $X_1,X_2\subseteq \mathbb{R}^n$ and $a,b \in \mathbb{R}$. Note that $a$ and $b$ ...
5
votes
1answer
200 views

Exponentials of rational numbers

Does there exist an $$0<x<1$$ such that $$\forall q \in \mathbb{Q^+}$$ $$q^x \in \mathbb{Q^+}$$
0
votes
1answer
26 views

$\sum_{n=1}^{\infty}|a_{n} x^{n}| < \infty \implies \sum_{n=1}^{\infty} |a_{n}| (|x|^{n-1}+ |x^{n-2}y|+…+|y^{n-1}|) <\infty $?

Let $\{a_{n}\}_{n\in \mathbb N} \subset \mathbb C$ so that the series, $\sum_{n=1}^{\infty} a_{n} x^{n},$ converges absolutely for all $x\in \mathbb R$. Now, we let, $x, y \in \mathbb R$ with $x\neq ...
1
vote
3answers
47 views

$\sum_{n=1}^{\infty} a_{n}= \sum_{n=1}^{\infty}b_{n}$ and $\sum_{n=1}^{\infty} |a_{n}| < \infty \implies \sum_{n=1}^{\infty} |b_{n}| < \infty$?

Suppose that $\{a_{n}\}_{n\in \mathbb N}, \{b_{n}\}_{n\in \mathbb N} \subset \mathbb C$ such that both the series, $\sum_{n=1}^{\infty} a_{n}$ and $\sum_{n=1}^{\infty} b_{n}$ converges, and its sum ...
1
vote
0answers
38 views

Zero to power Zero (Zero ^ Zero) indeterminable or not? [duplicate]

I want to know Zero power to Zero equal to 1 or Indeterminable. I think it cannot be exist. Please explain with proper mathematical definitions.
2
votes
1answer
113 views

Existence of a simultaneous rational approximation of real numbers in (0,1)

I have a simple question the rational approximation of real vectors. Dirichlet's simultaneous approximation theorem states: Given any $d$ real numbers $\alpha_1,\ldots,\alpha_d$ and for every ...
5
votes
2answers
60 views

Non analytic numbers

We know that some real numbers (actually, most of them) are not algebraic and the proof of this fact is beautiful: algebraic numbers, like polynomials with integer coefficients, are countable, ...
0
votes
1answer
46 views

How to show that a real number has a finite decimal representation?

How to show that a real number has a finite decimal representation (one that ends with an infinite sequence of zeros) if and only if it can be represented as a rational number m/n where n has no prime ...
2
votes
0answers
51 views

Expected error due to the tablemakers' dilemma

[note: to me, this does not seem like a question for m.se, but on mathoverflow it has been retroactively closed, with very little indication of why or what might be corrected... and waiting for ...
0
votes
1answer
36 views

Convergence of a modified sum of prime reciprocals for all $s \in \mathbb{C}$?

It is known that $\displaystyle \sum^\infty_{p \in \mathbb{P}} \frac{1}{p^s}$, with $\mathbb{P}$ the set of primes, only converges for $\Re(s) > 1$. The following sum of primes seems to converge ...
0
votes
0answers
75 views

Real numbers correspond bijectively to decimal expansions

Prove that positive real numbers correspond bijectively to decimal expansions no terminating in an infinite string of 9's as follows. The decimal expansion of $x \in R$ is $N.x_1x_2...$, where $N$ is ...
4
votes
3answers
90 views

limit of rational sequence

Let $a$ and $b$ be two real number. Assume that there exits two real sequences $a_n, b_n$ such that $$ \lim a_n=1, \lim b_n=1$$ and all $$\frac{a- a_n}{b-b_n} $$are rational numbers. Is it true ...
0
votes
0answers
58 views

linear diophantine equation and limit

Let $a_1, a_2, \dots, a_l$ are natural numbers and $\text{gcd}(a_1, a_2, \dots, a_l)=1$. Let $A_n$ is a number of non-negative integer solutions of $a_1x_1+a_2x_2+\dots+a_lx_l=n$. Prove that $\lim ...
4
votes
1answer
171 views

More identities of the Ramanujan Double factorial type.

Ramanujan discovered the following identity $$x=\sum_{n=0}^\infty (-1)^n ...
4
votes
1answer
127 views

Estimating integrals involving $\pi(x)$

While solving an exercise in analytic number theory, I ran into difficulty of estimating an integral of the form $\displaystyle\int_{1}^{x} \frac{\pi(t)}{t} dt$ where $\pi(x)$ is the prime counting ...
1
vote
1answer
40 views

Measure of a set??

I looking to find a measure of the following set: $A=\{ (a_1,a_2,...a_k) \in \mathbb{R}^k : |1+a_1z_1+a_2z_2+...+a_kz_k|<\delta \}$ and where $z_i \in \mathbb{Z}$. I believe the measure of this set ...
3
votes
5answers
210 views

Are transcendental numbers dense in $\mathbb R$? What is algebraic number? Is $cos(\pi/13)$ algebraic?

Are transcendental numbers dense in $\mathbb R$? What is algebraic number? Is $\cos(\pi/13)$ algebraic?
1
vote
0answers
58 views

Cardinality of sum-set

Let $X\subset R$ and $Y \subset R$ where X and Y have finite cardinalities. Let also, $a,b \in R$. How to show that $|aX+bY|=|X||Y|$ almost everywhere (measure of $(a,b) \subset R^2$ such that ...
0
votes
1answer
66 views

General polynomial form of a factorial?

Is anyone aware if there is a general polynomial form of a factorial? For instance; $2!=2^2-2^1$ $3!=3^3-3(3^2)+2(3^1)$ $4!=4^4-6(4^3)+11(4^2)-6(4^1)$ $5!=5^5-10(5^4)+35(5^3)-50(5^2)+24(5^1)$ ...
5
votes
1answer
120 views

express rational number as sum of squares of unit fraction

Let $q$ be a rational number with $0\lt q\leq\dfrac{\pi^2}6-1$. Then show that there exists a set $S\subset \{2,3,4,\dotsc\}$, such that $$q=\sum_{n\in S}\frac1{n^2}$$ I have no clue about it. Could ...
2
votes
2answers
200 views

Can We Represent Every Real Number Using Only Finite Memory?

This question arises from a comment I recently read in another question. My question is whether we can represent every real number using only finite memory. I will clarify what I mean by represent ...
2
votes
1answer
121 views

Different proofs of $\lim_{x\to \infty}\left(1+ \frac{1}{n}\right)^n =e$

I recently was teaching my friend about the number $e$. I introduced him the number by using the compound interest thing . Then I wrote down the general result -$$\lim_{x\to \infty}\left(1+ ...
26
votes
1answer
518 views

Show $(1+\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\frac{1}{9}+\frac{1}{11}-\cdots)^2 = 1+\frac{1}{9}+\frac{1}{25}+\frac{1}{49} + \cdots$

Last month I was calculating $\displaystyle \int_0^\infty \frac{1}{1+x^4}\, dx$ when I stumbled on the surprising identity: $$\sum_{n=0}^\infty (-1)^n\left(\frac{1}{4n+1} +\frac{1}{4n+3}\right) = ...
2
votes
1answer
77 views

Irrational Natural Density

Are there any sets of natural numbers with irrational natural density? I.e., does there exist a set $A\subset \mathbb{N}$ such that $$ \lim_{n->\infty} \frac{|A \cap [1,n]|}{n} \not\in ...
6
votes
1answer
53 views

Can You Construct a Syndetic Set with an Undefined Density?

Let $A \subset \mathbb{N}$. Enumerate $A = \{A_1, A_2,...\}$ such that $A_1 \le A_2 \le ...$. We say that $A$ is syndetic if there exists some $M \geq 0$ such that $A_{i+1} - A_i \le M$ for all $i ...
3
votes
1answer
107 views

Diophantine Equation Question

Find all ordered pairs $(x,y)$ of rational numbers $x, y$ such that the equations $2x^5 = x^2y^4 + 9y^5$ and $6y^3 = 3x^3 + xy^3$ hold simultaneously. My try: Multiplying the second equation by ...
1
vote
1answer
57 views

Prove that for $n\ge1$, $\xi-\frac{h_n}{k_n}=(-1)^nk_n^{-2}\left(\xi_{n+1}+\langle 0,a_n,a_{n-1},…,a_2,a_1\rangle\right)^{-1}$

Prove that for $n\ge1$, $$\xi-\frac{h_n}{k_n}=(-1)^nk_n^{-2}\left(\xi_{n+1}+\langle 0,a_n,a_{n-1},...,a_2,a_1\rangle\right)^{-1}$$ In addition, show that ...
3
votes
3answers
122 views

What does the integer span of one irrational, and one (possibly irrational) real number look like in $\mathbb{R}$?

My title was rejected a few times, here is what it was initially: If you take two real numbers- one irrational and one possibly irrational - how close does their $\mathbb{Z}$ span come to any ...
2
votes
1answer
76 views

Cosine and sine dense in unit circle

We may assume the following theorem: Theorem: A real number $\lambda$ is irrational iff the set $\{m+\lambda n\mid m,n\in\mathbb{Z}\}$ is a dense subset of $\mathbb{R}$. Assume $\lambda$ is ...
2
votes
1answer
52 views

Density with irrational number and trig function

We may assume the following theorem: Theorem: A real number $\lambda$ is irrational iff the set $\{m+\lambda n\mid m,n\in\mathbb{Z}\}$ is a dense subset of $\mathbb{R}$. Consider the points ...
6
votes
2answers
137 views

If $x\notin\mathbb Q$, then $\left|x-\frac{p}{q}\right|<\frac{1}{q^2}$ for infinitely many $\frac{p}{q}$?

This appears on problem 1 of chapter 1 in Stein & Shakarchi's Real Analysis: Given an irrational $x$, one can show (using the pigeon-hole principle, for example) that there are infinitely many ...