1
vote
0answers
91 views

Curious set of $n\sin^2(n)$

I consider this set $S_{0}=\{ n \in \mathbb{N}: n\sin^2(n) < 1 \}$. And I have some questions. See the elements of $S_{0}$= $\{ 1,3,6,19,22,25,44,47,66,69,88,110,132,154,157,176,179,$ (common ...
0
votes
0answers
12 views

Polynomial Roots of Bivariates

I've got a few polynomials that I am trying to get some results for (shown below). They come from the characteristic equation of a matrix. I have two variables in the polynomials, $\eta$ (which is ...
2
votes
3answers
115 views

Questions for first year students at the University. [closed]

I will help teach in a introductory class in mathematics for engineers in applied math at the University. Anyone have any good and cool favorite questions or know where I can find some? Anything is ...
3
votes
2answers
45 views

How to find all the positive integer number $n$ such that $\sum_{i=1}^{n}a_{i}=0$ and $|a_{i}|=1$ has a solution in $\mathbb{C}$ with $a_i+a_j\neq 0$

Find all the $n$ for which there exist complex numbers $a_{1},a_{2},\cdots,a_{n}$ such that: (1):$$|a_{i}|=1,i=1,2,\cdots,n$$ (2):$$a_{1}+a_{2}+\cdots+a_{n}=0$$ (3): for any $i\neq ...
1
vote
1answer
52 views

Is this a non empty perfect set with no rational numbers?

Let $A$ be the Cantor set and $B = \pi\, (-A \cup A)$. Is B a perfect set with no rational numbers?
1
vote
1answer
37 views

Find the rule of a sequence

I have a sequence $\{x(n), n=0,1,2,\ldots\}$ as follows: $x(0) = 1$ $x(1) = 1- e^{-a}$ $x(2) = \dfrac 12(1 - 4e^{-a} + 3e^{-2a})$ $x(3) = \dfrac{1}{6}(1-12e^{-a}+27e^{-2a}-16e^{-3a}) $ $x(4) = ...
0
votes
1answer
46 views

How would one find a) All the primitive characters modulo 8, b) All the non-primitive characters modulo 8?

Preferably explained in novice terms! I can start it off by having the multiplicative group modulo 8 with elements $[1], [3], [5], [7]$ and not sure where to go now. I see there is a similar question ...
2
votes
1answer
63 views

How prove the constant term of $\left(1+x+\frac{1}{x}\right)^p\equiv1\pmod {p^2}$

if $p>3$ is odd prime number,show that: the constant term of $$\left(1+x+\dfrac{1}{x}\right)^p\equiv1\pmod {p^2}$$ My try: since ...
0
votes
0answers
58 views

A partial sum involving Euler's function

This is Exercise 2.1.17 of the book "H. Montgomery and R. Vaughan. Multiplicative Number Theory— I. Classical Theory". For $x\ge 2$, $\sum_{n\le x}\frac{\mu(n)^2}{\varphi(n)}=\log ...
0
votes
1answer
25 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 ...
0
votes
0answers
39 views

Number of lattice points in an annulus

Consider the lattice spanned by two nonzero complex numbers $\xi_{1}$ and $\xi_{2}$ such that their ratio is not real. Let $w = m\xi_{1} + n\xi_{2}$. Let $A(n)$ be the number of lattice points such ...
4
votes
3answers
43 views

What is the convex-hull of the set $\{ (n,\varphi(n)) : n\in \mathbb N \} \subset \mathbb R^2$

I know that set $$ E=\{ (n,\varphi(n)) : n\in \mathbb N \} \subset \mathbb R^2 $$ has infinitely many points on the line $y=x-1$, which suggests this line to be included in the upper part of the ...
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 ...
1
vote
0answers
21 views

Berkovich analytifications and non-archimedean geometry of Transseries

In Transseries and Real Differential Algebra by Joris van der Hoeven it is said that Transseries admit a rich non-Archimedean geometry (somewhere on page 13), but since the book isn't about that, ...
0
votes
2answers
109 views

Does sum of all natural numbers contradict another rule?

I must say that I am not a mathematician, just a enthusiast who likes to read all the "weird" results in mathematics. I read that sum of all natural number equals to $-1/12$ and I am also aware that ...
3
votes
0answers
116 views

Ratio of maximal to minimal jump in the set of angle multiples (corrected)

(This is the corrected version of the question I asked here: Ratio of maximal to minimal jump in the set of angle multiples.) Let $S^1$ be the unit circle in the complex plain. Let $d:S^1\times ...
1
vote
1answer
23 views

Ratio of maximal to minimal jump in the set of angle multiples

Let $S^1$ be the unit circle in the complex plain. Let $d:S^1\times S^1\to\mathbb{R}$ be the distance function given by the arc length. Let $\theta\in S^1$ be an element of infinite order, that is ...
0
votes
1answer
89 views

a Problem about Sequence [duplicate]

Let $a_1$ be an integer. Then we assume $$ a_{n+1} = \begin{cases} 3a_n+1,&\text{$a_n$ is odd}\\ \frac{a_n}{2},&\text{$a_n$ is even} \end{cases} $$ Now we prove that for any ...
1
vote
1answer
32 views

Stronger condition then ultrametric condition on metric space

A metric space $(X,d)$ is called an ultrametric space if it is a metric space and fulfills the stronger triangle inequality (see Wikipedia) $$ d(x,y) \le \max\{ d(x,z), d(z, y) \}. $$ Examples are ...
2
votes
1answer
50 views

Lattice points in spheres

Let $\mathbb{R}^n$ have the standard Euclidean metric and call a point $P = (x_1, \ldots,x_n)\in\mathbb{R}^n$ a lattice point if for all $i$, $x_i\in\mathbb{Z}$. Allowing small number theoretic ...
1
vote
1answer
57 views

Find the sum of the series

For any integer $n$ define $k(n) = \frac{n^7}{7} + \frac{n^3}{3} + \frac{11n}{21} + 1$ and $$f(n) = 0 \text{if $k(n)$ is an integer ; $\frac{1}{n^2}$ if $k(n)$ is not an integer } $$ Find $\sum_{n = ...
3
votes
1answer
101 views

Lower bound on a number theoretic function

Let $n$ be a positive odd integer, let $$n_j = \Bigl\{\frac{n}{2^{j+1}}\Bigr\}\,,$$ where $\{x\}$ denotes the fractional part of $x$, and finally let $k = \lceil \log_2 n\rceil$. Consider the ...
6
votes
2answers
182 views

How find this$\frac{1}{{{p}_{1}}}+\frac{1}{{{p}_{2}}}+…+\frac{1}{{{p}_{n}}}<10$

Let ${{p}_{1}},{{p}_{2}},...,{{p}_{n}}$ be the prime numbers less than ${{2}^{100}}$. Prove that $$\frac{1}{{{p}_{1}}}+\frac{1}{{{p}_{2}}}+...+\frac{1}{{{p}_{n}}}<10$$ This problem is from this ...
2
votes
1answer
93 views

Absolute convergence of Euler products and infinite series

We know that given a multiplicative function $f$ for which the series $\sum_{n=1}^\infty f(n)$ converges absolutely then so does the Euler product $\prod_{p}\sum_{k=0}^\infty f(p^k)$, but does the ...
3
votes
0answers
38 views

Borel lemma on rationality

I am looking for an enlightening proof of the following fact: Let $F(t)\in \mathbb{Z}[[t]]$ and suppose $S$ is a finite set of places on $\mathbb Q$ containing $\infty$. If for every $v\in S$ $F(t)$ ...
5
votes
1answer
155 views

Prime harmonic series

We have following identity: ($p$ is a prime number) $$\left(1+\frac{1}{p}\right)\sum_{k=0}^n\frac{1}{p^{2k}}=\sum_{k=0}^{2n+1}\frac{1}{p^k}$$ Now, How to derive the following inequality from the above ...
6
votes
1answer
210 views

Newman's “Natural proof”(Analytic) of Prime Number Theorem (1980)

I am trying to understand this short proof by newmann. I faced some problems while grasping this very proof. Please help me out. 1 . I am not clear, why in step (1)'s proof he says that from unique ...
0
votes
1answer
105 views

prove the product $\sin^2\frac{(k-j)\pi}n$

$ n\geq2 $, prove that the product $$\prod_{1 \leq j<k\leq n \atop \gcd(j,n)=\gcd(k,n)=1}4 \sin^2\frac{(k-j)\pi}{n}=\dfrac{n^{\varphi(n)}}{\prod\limits_{p\mid n, p\; ...
1
vote
1answer
64 views

$|A(n)|<B$, $\lim_{s \to 0^{+}}\sum_{n=1}^{\infty}a_{n}n^{-s}=a$ imply $\lim_{x \to \infty}\sum_{n \leq x} a_{n}(1-\log{n}/\log{x})=a$

Suppose that $|A(n)|<B$ and $\lim_{s \to 0^{+}}\sum_{n=1}^{\infty}a_{n}n^{-s}=a$ where $A(x)=\sum_{n \leq x}a_{n}$. Then $$\lim_{x \to \infty}\sum_{n \leq x} a_{n}(1-\log{n}/\log{x})=a$$ What I ...
2
votes
1answer
153 views

Harmonic analysis in number theory

When I was reading Folland's A course in abstract harmonic analysis, I was told these materials have wonderful applications to number theory. However, I do not see really a lot of examples there. Can ...
1
vote
1answer
152 views

Unexpected Probability Theory Uses

I am a french student in mathematical engineering. I had to go trough an intensive 3 year "preparation" to pass a "concours" to go to High School. In mathematics, I have been taught a lot of algebra, ...
1
vote
0answers
50 views

$\theta(x) = O(x)$ in the prime number theorem

In the Newman short proof of the prime number theorem (http://www.maths.dur.ac.uk/~dma0hg/prime_number_theorem_zagier.pdf) Zagier states that the fact that $2^{2n} >= e^{\theta(2n) - \theta(n)}$ ...
4
votes
0answers
149 views

Understanding Newman's proof of the prime number theorem

I am trying to understand D.J. Newman's proof of the prime number theorem, as presented by D. Zagier. I am not too familiar with analysis, and so there are some things I don't understand. In (III), ...
0
votes
1answer
94 views

Which numbers of [0,1) have a unique base g expansion?

Good evening, i know that is question is rather standard, but unfornunately I have not much knowledge of number theory. Take $2 \leq g\in \mathbb{N}$. I know that every $x \in [0,1)$ can be ...
1
vote
0answers
70 views

Extending a rational entry matrix to an orthogonal matrix.

Suppose $M \leq N$ are positive integers and let $r_1,\ldots,r_M$ represent the rows of an $M\times N$ matrix $A$ in which: (i) the rows of $A$ are orthogonal and having the same norm, (ii) the ...
1
vote
0answers
92 views

Modified Arithmetic-Geometric Mean

Let $\{x_n\}$ and $\{y_n\}$ be defined iteratively, $x_0:=\beta >1, \ y_0:= 1$ and $x_{n+1}= \frac{x_n+y_n}{2}$, $y_{n+1} = (x_n.y_n)^{\frac{1}{2}}$; i.e. they are respectively the arithmetic and ...
1
vote
0answers
129 views

Is zero a cluster point of $n\sin n$?

I have seen somewhere that if $0\le \alpha<1$, then zero is a cluster point of the sequence $n^\alpha \sin n, n=1,2,\cdots$. My question is what if $\alpha=1$? Or $\alpha>1$?
75
votes
9answers
4k views

How far can one get in analysis without leaving $\mathbb{Q}$?

Suppose you're trying to teach analysis to a stubborn algebraist who refuses to acknowledge the existence of any characteristic $0$ field other than $\mathbb{Q}$. How ugly are things going to get for ...
5
votes
1answer
126 views

Interesting phenomenon with the $\zeta(3)$ series

I noticed that if one takes certain partial sums of the series for $\zeta(3)$: $$\zeta(3) = \sum_{n=1}^{\infty} \frac{1}{n^3} \approx \sum_{n=1}^{N} \frac{1}{n^3}$$ an interesting phenomenon occurs ...
0
votes
3answers
74 views

If $x = a + b$, and only $x$ is known, how to solve what is $a-b$?

If $x$ equals to $a+b$, how can I solve what is $a-b$, knowing only $x$? (approximation will do as well, if it cannot be solved exactly)
1
vote
1answer
78 views

Prove that $\frac{m+n+2}{2(m+1)}{n\choose m}2^{(n-m)/2}$ is an integer

Let $m,n$ be positive integers, both odd or both even, with $n\ge m$. I think the following number $$\frac{m+n+2}{2(m+1)}{n\choose m}2^{(n-m)/2}$$ is always an integer, but I have trouble proving it.
3
votes
3answers
95 views

The number $ \frac{(m)^{(k)}(m)_k}{(1/2)^{(k)} k!}$

For a real number $a$ and a positive integer $k$, denote by $(a)^{(k)}$ the number $a(a+1)\cdots (a+k-1)$ and $(a)_k$ the number $a(a-1)\cdots (a-k+1)$. Let $m$ be a positive integer $\ge k$. Can ...
1
vote
0answers
55 views

An identity related to Chebyshev polynomial

Let $n=2m$ be a positive even integer. I can prove that $$1+\sum_{k=1}^m (-1)^k \frac{n^2(n^2-2^2)\cdots(n^2-(2k-2)^2)}{(2k)!}=(-1)^m$$ using hypergeometric identity ...
0
votes
1answer
66 views

Ordinary generating function for Bernoulli polynomial

I know the exponential generating function for the Bernoulli polynomial $B_n(x)$:$$\frac{te^{tx}}{e^t-1}=\sum_{n=0}^\infty B_n(x)\frac{t^n}{n!}.$$ But is there an ordinary generating function? i.e a ...
1
vote
2answers
105 views

A product of two sums of four squares

I am dealing with a problem and I hope you can help me. I have already proved this: Let us suppose that integers $m$ and $n$ can be written as sum of squares of two integers. Prove that m*n can also ...
7
votes
3answers
226 views

Even integer approximations to multiples of pi

I admit that I'm probably out of my depth with this question, but I can't help but feel curious. I wanted to show that, in the sequence $\{\sin(n)\}$, there is never a largest term (the sequence ...
3
votes
2answers
256 views

Analysis proof for repeating digits of rational numbers

"Every rational number is either a terminating or repeating decimal". I knew there's a proof for this using number theory's theorems, but I wish to find a purely analysis proof, that is: the series ...
0
votes
3answers
121 views

Given $n+1\mid2\sum_{k=1}^{n}{a_k}$, find $a_k$.

Let $m$ be a positive integer. There are only 2 finite sequences of positive integers like $a_1,a_2,...,a_m$ such that $$(\forall n \leq m)\left(n+1\mid2\sum_{k=1}^{n}{a_k}, \quad a_n\in [1,m],\quad ...
2
votes
0answers
204 views

Uses of Taylor series expansion [duplicate]

In calculus we use taylor series expansion at large number of places.I recently one of the application in number theory(To find solution of polynomial in finite field of order $p^{n}$, where p is ...