0
votes
1answer
13 views

Maximal value of $\vert r^2-n\vert$ with a special condition

let $M,n\in \mathbb{N}$ and $R=\lbrace r\in \mathbb{N} \mid \vert r- \sqrt{n}\vert <M<2\sqrt{n}\rbrace $. I have to show that the maximal value of $\vert r^2-n\vert $ for $r\in R$ is at most ...
1
vote
1answer
63 views

Asymptotic estimate of sum of reciprocals of $\log(p_k)$

Fix $n$. The sum $S(n)=\sum_{p_k\leq n} \frac{1}{\log p_k}$ taken over primes less than or equal to n, clearly diverges as $n$ goes to infinity, being lower bounded by the sum of the reciprocals of ...
0
votes
2answers
57 views

Let$\ x$ be a real number between$\ 0$ and$\ 1$. Is it possible to write$\ e^{x}$ as a function of$\ \Gamma \left(x+1\right)$?

In particular, I'm looking for a relation between$\ e^x$ and$\ \frac{1}{ \Gamma \left(x+1\right) }$, which would be of help for a proof.
3
votes
1answer
29 views

Can we have $\sum_{n\leq [x]}e^{-\sqrt{\frac{\log x}{r}}}\ll \frac{x}{e^{c \sqrt{\log x}}}$ for some constant $c>0$, where $x>1.$

Let positive interger $n$ is square-free, that is $n=p_1p_2\cdots p_r$ some $r$. Can we have $$\sum_{n\leq [x]}e^{-\sqrt{\frac{\log x}{r}}}\ll \frac{x}{e^{c \sqrt{\log x}}}$$ for some constant ...
0
votes
0answers
29 views

Proof Using Mathematical Induction that a set S equals Natural Set

I am trying to prove that if $S$ is subset of $\mathbb{N}$, such that a) $2^k \in S$ for all $k \in \mathbb{N}$. b) If $k \in S$ and $k \ge 2$, then $k-1 \in S$. Prove that $S = N$. I am trying ...
0
votes
1answer
30 views

Prove for all $x \in \mathbb{R}$, there is some $y \in [0,1)$ such that $x \equiv y \mod \mathbb{Z}$

So my logic is as such choose any $x$ say $99.05$. Then I can find $y \in [0,1)$ such that $99.05-y \in \mathbb{Z}$ doesn't $y$ have to be $0.05$? Congruences are a little more difficult when you let ...
2
votes
1answer
97 views

Resources to investigate rational numbers

I have been told that resources like Mathematica's Number Recognition (which I've never tried myself) and the Inverse Symbolic Calculator (ISC) can be used to find possible closed forms for real ...
2
votes
0answers
41 views

Bernoulli number analog identity

I am doing work on Bernoulli numbers. Let $$b(x)=\sum_{k=0}^\infty{\frac{B_k}{k!}x^k}=\frac{x}{e^x-1}$$ From this, by taking derivatives of both sides we get the identity $$b(x)^2=(1-x)b(x)=xb'(x)$$ ...
0
votes
3answers
85 views

How is the set of all closed intervals countable?

I am trying to figure out the answer to the problem: Show that the set of all closed intervals $[a,b]$ with $a,b \in \mathbb{Q}$ is countable. Now I know that the interval $[0,1)$ for example is ...
0
votes
0answers
51 views

is $x_{n}\ll \overline{x}_{n}^{2}$?

Let $(x_{n})_{n\ge 1}$ be an increasing sequence of positive integers and $\displaystyle{\overline{x}_{n}:=\dfrac{1}{n}\sum_{i=1}^{n}x_{i}}$. Suppose furthermore that $\forall\varepsilon\gt 0, \ \ ...
8
votes
2answers
137 views

Can $\sum_1^n 1/k$ be arranged so that it is an integer for infinitely many $n?$

It's well known that when $n>1:$ $$\sum_{k=1}^n \frac{1}{k}\not\in \mathbb{N}$$ But if we are allowed to rearrange the series, we can for instance can get: ...
0
votes
0answers
18 views

Identifying or bounding the zeros of the composition of two generating functions

Given two generating functions $$ G(a_n;x)=\sum_{n=0}^\infty a_nx^n \quad\text{ and }\quad H(b_n;x)=\sum_{n=0}^\infty b_nx^n, $$ what techniques are available for locating, or finding bounds on, the ...
0
votes
1answer
95 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 $$ ...
9
votes
2answers
336 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
41 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
80 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
518 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
113 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
40 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
70 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
84 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
188 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
48 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
157 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
213 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
205 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
87 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 ...
20
votes
1answer
374 views

How to find $\int_0^\infty \prod_{k=1}^n \frac{\sin \frac{x}{2k-1}}{\frac{x}{2k-1}}\mathrm dx$

I am trying to calculate the integral $$ I_n=\int \limits_0^\infty \prod_{k=1}^n \frac{\sin \frac{x}{2k-1}}{\frac{x}{2k-1}}\mathrm dx. $$ (I have literature on this, if people want). Note, we can ...
1
vote
2answers
95 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
39 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
122 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
49 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
38 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
100 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
94 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
60 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
194 views

More identities of the Ramanujan Double factorial type.

Ramanujan discovered the following identity $$x=\sum_{n=0}^\infty (-1)^n ...
4
votes
1answer
133 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
236 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
61 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
72 views

General polynomial form of a factorial?

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