0
votes
1answer
24 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
43 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
25 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
42 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
49 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
16 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
96 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
112 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
86 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
31 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
43 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
54 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
98 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
176 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
81 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
36 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 $v\in S$ $F(t)$ ...
4
votes
1answer
143 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
187 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
101 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
137 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
134 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
47 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
141 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
91 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
64 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
76 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
121 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 $\alpha>1$?
68
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
121 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
73 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
74 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
52 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
62 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
102 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
202 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
233 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
188 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 ...
9
votes
3answers
702 views

Curious facts about ordinal numbers

I have some notes about curious facts about ordinal numbers, for example that their addition is not commutative, multiplication is not distributive from the right hand side and that the exponent rule ...
0
votes
1answer
57 views

Asymptotic density and sum of the reciprocals

Let A and B be two infinite proper-subsets of the set of positive integers. Let A(n) denote the number of those elements of the set A , which does not exceed n ; we use similar definition for B(n) . ...
5
votes
3answers
224 views

From $\sum_p \frac{\log p}{p^s} = \frac{1}{s-1} + O(1)$ conclude that $\sum_p \frac{1}{p^s} = \log \frac{1}{s-1} + O(1)$

I'm reading a book on analytic number theory. It asks me to prove: $$ \sum_p \frac{\log p}{p^s} = \frac{1}{s-1} + O(1) \tag{A}$$ and conclude, via integration, that: $$ \sum_p \frac{1}{p^s} ...
8
votes
1answer
179 views

limit connected with a periodic function

Let $f$ be a $1$-periodic function, i.e., $f(x+1)=f(x)$, defined on the interval $(0, 1)$ by the formula $$ f(x)=2x-1. $$ For a real number $x$ consider the series $$ \sum_{n=1}^\infty\frac{f(nx)}{n}. ...
2
votes
2answers
72 views

A problem of constructing set to get finite summation

Given the function $f$, for any set $U$ with countable elements , define \begin{align} f(U)=\sum_{x \in U} \frac{1}{x} \end{align} Construct a set $A$, whose elements are positive integers, and ...
23
votes
1answer
2k views

Why is it hard to prove whether $\pi+e$ is an irrational number?

From this list I came to know that it is hard to conclude $\pi+e$ is an irrational? Can somebody discuss with reference "Why this is hard ?" Is it still an open problem ? If yes it will be helpful ...
2
votes
1answer
83 views

Numbers such that their sum of $k^{th}$ powers is always zero.

Let $x_1, \ldots,x_n$ be complex numbers such that for any $k$, $$ \sum_{i=0}^n x_i^k = 0.$$ I'd like to show that this implies $x_1 = x_2 = \cdots = x_n = 0.$ I was suggested to use this strategy. ...
1
vote
1answer
58 views

Minima of Binary Forms of Degree n

Does anyone know any upper bounds or known results on LOWER BOUNDS for binary forms i.e. if you have F(X,Y)=$X^n+YX^{n-1}+Y^2X^{n-2}+...+XY^{n-1}+Y^{n}$, I need to find a lower bound for F interms ...
2
votes
1answer
111 views

$\left|{\sin(\pi \alpha N)}/{\sin(\pi \alpha)}\right| \leq {1}/{2 \| \alpha \|}$

How does one prove the following inequality? $$\left|\frac{\sin(\pi \alpha N)}{\sin(\pi \alpha)}\right| \leq \frac{1}{2 \| \alpha \|}$$ Here $\| \alpha \|$ denotes the distance to the nearest ...