Suppose the mean noon-time temperature for September days in San Diego is 24∘ and the standard deviation is 4.6. (Temperature in this problem is measured in degrees celsius) Using Chebyshev’s theorem, what is the minimal probability (in percents) that the noon-time temperature of a september day...

Prove that the Gaussian $Q$ function is bounded on the top by $1/2x^2$, i.e. $Q(x)\le 1/2x^2$. for $x\ge 0$ using the Chebyshev inequality and the Nakagami $m$ distribution with $m=0.5$ that reduces it to half normal distribution.

Does that sound about right? Assume that $X$ is uniform on the interval $[0,1]$. Plot $P\big(\big|X\big|\geq \varepsilon\big)$ and $\dfrac{E\big(X^2\big)}{\varepsilon^2}$ as functions of $\varepsilon$. How does this relate to Chebushev's inequality? Attempt: $ f(x) = \left\{ \begin{array}{l...

Give an example to a function $ f: (a,b) \to \mathbb R $ where $ f_x $ is not bounded, but every point $x \in (a, b) $ has a neighborhood where the function is bounded. (If I translated neighborhood wrong, please correct me) I gave: $ f_x = \frac 1{x - b} $ Because the only point having no ne...

Since nobody objected to the idea that this tag - if we keep it - should only be for questions about Chebyshev functions in number theory, I went ahead and created tag-wiki and tag-exceprt. I have also started retagging questions which do not belong here, see this list. They are mostly about Che...

I am trying to prove the following If $\pi(x) = \operatorname{Li}(x) + O(x^{\frac{1}{2}}\log(x))$ then $\psi(x) = x + O(x^{\frac{1}{2}} \log^2(x))$ I have tried using $\psi(x) = \theta(x) + O(x^{\frac{1}{2}} \log^2(x))$ and then bounding $\theta(x)$ but could not reach anywhere. I have tried ...

Let $\pi, \vartheta$ be respectively the prime counting function and the first chebyshev function. As you know, $ \pi(x) \sim x/\log x$, and $\vartheta(x) \sim x$, so that, at first order, seems $\pi(x) \log x \sim \vartheta(x)$. Is it easy to state that $\pi(x) \log x > \vartheta(x)$ (just use ...

How is Chebyshev's first function $$\vartheta(N)=\sum_{p\leq N}\log p$$ useful in counting primes? Can it alone be used to analytically derive the prime number theorem?

How do I show that if $\psi(x)=x+O(x^{1/2}\log^2(x))$ then $\pi(x)=\int_2^x \frac{dt}{logt} + O(x^{1/2}\log x)$ Where $\psi(x)$ is Chebyshev's second function and $\pi(x)$ is the prime counting function

Chebyshev's Theta Function is defined as $\vartheta(x)=\sum_{p\le x} \log p$ I am trying to find a table of this function for large-ish values of $x$, preferably in a form I could download to a spreadsheet. By "large-ish" I mean for $x$ over 16000. I have found a table in a 1962 paper by J.Ross...

What are the best known (unconditional) bounds on the following: $$\mid\psi(x) - x\mid$$ (With a known constant factor)

In February 2013, Sadegh Nazardonyavi and Semyon Yakubovich posted on arxiv: Sharper estimates for Chebyshev's functions $\vartheta$ and $\psi$. I have a question about Theorem 2.27 on page 22. My question regards the argument for this: $$\vartheta(x) < 1.000027651, \;\;(x > 0)$$ I can follow...

I've been told that the approach below will not work. I would be interested if someone could help me to understand what will go wrong. Let: $$\psi(x) = \sum\limits_{p^k \le x} \ln p$$ So that (see here): $$\ln(x!) = \sum_{k=1}\psi(\frac{x}{k})$$ So that: $$\ln(x!) - \ln\left(\left\lfloor\f...

Is this a valid application of the Möbius Inversion Formula: Define: $$\psi\left(x\right) = \sum\limits_{p^k \le x} \log p$$ So that: $$\log x! = \sum\limits_{k=1}^{\infty}\psi\left(\frac{x}{k}\right)$$ Then, applying the Möbius Inversion Formula gives: $$\psi\left(x\right) = \sum_{k=1}^{\in...

I've been reading through Jitsuro Nagura's classic proof that there is a prime between $x$ and $\frac{6x}{5}$ and it seems to me that it should be possible to improve on his upper bound for the second Chebyshev function. In Nagura's paper, he establishes the following inequality: $$\psi\left(x\...

I'm working on understanding Nagura's analysis of the upper bound for $\psi(x)$ which is done in Lemma 2. I am unclear on one step of his reasoning. With Lemma 1, he establishes for $x \ge 2000$: $$\log\Gamma(\lfloor{x}\rfloor+1) - \log\Gamma(\lfloor{\frac{x}{2}}\rfloor+1) - \log\Gamma(\lfloor...

I was recently told that the Möbius Inversion Formula can be applied to the Chebyshev Function. Let $\vartheta(x)$,$\psi(x)$ be the first and second Chebyshev functions so that: $$\vartheta(x) = \sum_{p\le{x}}\log p$$ $$\psi(x) = \sum_{n=1}^{\infty}\vartheta(\sqrt[n]{x})$$ Then applying the M...

I am going through the proof by Jitsuro Nagura which shows that there is always a prime between $x$ and $\frac{6x}{5}$ where $x \ge 25$. Nagura uses the following definitions: $$\vartheta(x) = \sum_{p \le x} \log p$$ $$\psi(x) = \sum_{m=1}^{\infty}\vartheta(\sqrt[m]{x})$$ Then as part of the ...

Let $X$ be a Banach space and $T\in\mathscr{L}(X,X)$ with $\|T\|_*<1$. Use the Contraction Mapping Principle to show (where $I$ is the identity map on $X$) that $I-T\in\mathscr{L}(X,X)$ is injective and surjective. Attempt: Since $\mathscr{L}(X,X)$ is a normed linear space and $I,T\in\mathscr{L}...

I'm doing an introductory linear algebra course and I'm stuck on this question. Show that with respect to any inner product, u+v is orthogonal to u-v if and only if ||u|| = ||v||. I'm trying to prove the forward implication and I don't know where to go from < u+v,u-v >=0 I tried working with t...

Functions $g$ and $f$ are linear and injective. How do I go about finding the kernel of $g \circ f$? I'm asking because I want to prove that $\ker(f) = \ker(g \circ f)$.

I am attempting to use LU Decomposition to calculate the determinant of a matrix. Given: $$ A = \begin{pmatrix} 1 & 2 \\ 5 & 6 \end{pmatrix} $$ When using this calculator: Here the values give me: $$ L = \begin{pmatrix} 1 & 0 \\ 0.2 & 1.0 \end{pmatrix} $$ $$ U = \begin{pmatrix} 5.00 & 6.00 \...

If you got a Matrix $A$. Is there a estimation how big the largest element in the inverse of the matrix is? If it helps the matrix is unimodular.

I am trying to determine a formula to calculate how efficient honeybee comb is at space usage. I want to determine the number of full and partial cells that exist within a defined volume of honey comb. The rectangular box area is 46.5cm long, 15cm wide, 20cm high Each cells is 4.9mm across fro...