# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $\sum_{n=1}^{\infty} a_{n}= \sum_{n=1}^{\infty}b_{n}$ 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Berkovich analytifications and non-archimedean geometry of Transseries

In Transseries and Real Differential 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $v\in S$ $F(t)$ ...
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### 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 ...
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### 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 ...
$n\geq2$, prove that the product $$\prod_{1 \leq j<k\leq n \atop \gcd(j,n)=\gcd(k,n)=1}4 \sin^2\frac{(k-j)\pi}{n}=\dfrac{n^{\varphi(n)}}{\prod\limits_{p\mid n, p\; ... 1answer 64 views ### |A(n)|<B, \lim_{s \to 0^{+}}\sum_{n=1}^{\infty}a_{n}n^{-s}=a imply \lim_{x \to \infty}\sum_{n \leq x} a_{n}(1-\log{n}/\log{x})=a Suppose that |A(n)|<B and \lim_{s \to 0^{+}}\sum_{n=1}^{\infty}a_{n}n^{-s}=a where A(x)=\sum_{n \leq x}a_{n}. Then$$\lim_{x \to \infty}\sum_{n \leq x} a_{n}(1-\log{n}/\log{x})=a$$What I ... 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 ... 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, ... 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)} ... 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), ... 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 ... 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 ... 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 ... 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? 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 ... 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}$$an interesting phenomenon occurs ... 3answers 74 views ### If x = a + b, and only x is known, how to solve what is a-b? If x equals to a+b, how can I solve what is a-b, knowing only x? (approximation will do as well, if it cannot be solved exactly) 1answer 78 views ### Prove that \frac{m+n+2}{2(m+1)}{n\choose m}2^{(n-m)/2} is an integer Let m,n be positive integers, both odd or both even, with n\ge m. I think the following number$$\frac{m+n+2}{2(m+1)}{n\choose m}2^{(n-m)/2}$$is always an integer, but I have trouble proving it. 3answers 95 views ### The number  \frac{(m)^{(k)}(m)_k}{(1/2)^{(k)} k!} For a real number a and a positive integer k, denote by (a)^{(k)} the number a(a+1)\cdots (a+k-1) and (a)_k the number a(a-1)\cdots (a-k+1). Let m be a positive integer \ge k. Can ... 0answers 55 views ### An identity related to Chebyshev polynomial Let n=2m be a positive even integer. I can prove that$$1+\sum_{k=1}^m (-1)^k \frac{n^2(n^2-2^2)\cdots(n^2-(2k-2)^2)}{(2k)!}=(-1)^m$$using hypergeometric identity ... 1answer 66 views ### Ordinary generating function for Bernoulli polynomial I know the exponential generating function for the Bernoulli polynomial B_n(x):$$\frac{te^{tx}}{e^t-1}=\sum_{n=0}^\infty B_n(x)\frac{t^n}{n!}.$$But is there an ordinary generating function? i.e a ... 2answers 105 views ### A product of two sums of four squares I am dealing with a problem and I hope you can help me. I have already proved this: Let us suppose that integers m and n can be written as sum of squares of two integers. Prove that m*n can also ... 3answers 226 views ### Even integer approximations to multiples of pi I admit that I'm probably out of my depth with this question, but I can't help but feel curious. I wanted to show that, in the sequence \{\sin(n)\}, there is never a largest term (the sequence ... 2answers 256 views ### Analysis proof for repeating digits of rational numbers "Every rational number is either a terminating or repeating decimal". I knew there's a proof for this using number theory's theorems, but I wish to find a purely analysis proof, that is: the series ... 3answers 121 views ### Given n+1\mid2\sum_{k=1}^{n}{a_k}, find a_k. Let m be a positive integer. There are only 2 finite sequences of positive integers like a_1,a_2,...,a_m such that$$(\forall n \leq m)\left(n+1\mid2\sum_{k=1}^{n}{a_k}, \quad a_n\in [1,m],\quad ...
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 ...