Questions tagged [chebyshev-function]

Question 1

I have heard that the original proof of Chebyshev's inequality did not utilize Markov's inequality. I am interested: To know whether this is true/accurate? Any other alternative proof for Chebyshev's ...

Question 2

Let $T_t(x)$ denote the Chebyshev polynomial of the first kind $$ T_t(x) = \cos(t \arccos x). $$ How does one evaluate $T_t(x)$ at complex arguments of the form $$ x = \cos\theta - 2i q \sin\theta, $$ ...

Question 3

In a proof I'm going through, we arrive at the inequality $$ \psi(x) - \psi\left(\frac{x}{2}\right) \leq (\log 2)x + O(\log x), $$ where $$ \psi(x) = \sum_{p^a \leq x} \log p. $$ The proof then states ...

Question 4

I am trying to understand a line in a proof of Chebycheff's theorem: $Ax < \theta (x ) < Bx.$ Let $T(x) = \sum_{n \leq x} \psi (x/n)$, where $\psi (x) = \sum_{p^a \leq x} \log p$ is the ...

Question 5

I am trying to determine values of a lowpass filter by using Chebyshev polynomials. Here is a transfer function of the fifth order lowpass filter. $$H(s)=\dfrac{0.0626}{(s^2+0.1098s+0.936)(s^2+0.2873s+...

Question 6

I have this function $f(x) := \sin(2\pi x-4\cos(\pi x))$ and I am trying to find a Chebyshev polynomial approximation in the domain $[0,2]$. I currently defined my nodes to be \begin{equation} \text{...

Question 7

An upper bound for the primorial can be found based on the first chebyshev function. From $\vartheta(x) < 1.00028x$, it is clear that: $$\prod\limits_{p \le x}p \le e^{1.00028x} < (2.72)^x$$ I ...

Question 8

What is the growth rate of the 2nd Chebyshev function i.e. $Ψ(x)$ where $Ψ(x)$ $=$ $ln(lcm(1, 2, ... , x)$ $ln$ denotes the natural logarithm and $lcm(1, 2, ... , x)$ refers to the lowest common ...

Question 9

I'm trying to find the inverse function of $$U_{k-1}(\cos(\frac{\pi}{x}))=\sum_{n=0}^{\left\lfloor\frac{k-1}2\right\rfloor}\frac{(-1)^n \Gamma(k-n)}{n!\Gamma(k-2n)} \left(2\cos\left(\frac\pi x\right)\...

Question 10

Background: We define $$\theta(x) := \sum_{p\le x} \log p$$ where the sum is taken over primes $\le x$. Chebycheff’s Theorem: There exist positive constants $A$ and $B$ such that $$Ax < \theta(x) &...

Question 11

I have the following integration problem: $$ \int_0^1{ -m f(x) \left(\int_0^x{f(u)} du \right)}^{m-1} dx $$ I attempted to approximate $ \int_0^x{f(u)} du $ using Chebyshev interpolation, I took $n+1$...

Question 12

In Schoenfeld's paper "Sharper bounds for the Chebyshev functions $\theta(x)$ and $\psi(x)$. II," theorem 10, inequality (6.2) states "If the Riemann hypothesis holds, then $|\psi(x) - ...

Question 13

I am studying the prime number theorem and related stuff and was trying to solve this following problem: Suppose there exists a constant $c$ such that $\psi(x) = x + (c + o(1))\frac{x}{\log x}$ as $x \...

Question 14

Let, $$\Theta(n)=\sum_{p^{\alpha}\leq n} \ln p $$ be the second chebyshev theta function. Then is it true that, $$\ln x!=\sum_{k\geq 1}\Theta\left(\frac{x}{k}\right)$$ If yes how can I prove that? MY ...

Question 15

I found the following claim here: Chebyshev's theorem gives the lower bound $2^{(n/2)}$. Is this correct? If $n\#$ is the primorial of $n$, does it follow that: $$n\# \ge 2^{(n/2)}$$ As I understand ...

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Questions tagged [chebyshev-function]

Alternative proofs for chebyshev inequality

Evaluating Chebyshev polynomials at complex inputs of the form $\cos\theta−2iq\sin\theta$

Justifying the Error Term in an inequality for $\psi(x)$

Issue with Expressing $\sum_{n \leq x} \log n $ in Terms of the Chebycheff Function $\psi (x)$

Transfer function of a chebyshev lowpass filter

How to find Chebyshev Coefficient in the domain $[0,2]$

Estimating the upper bound for $\prod\limits_{p \le x}{p^{\frac{1}{p}}}$

Growth Rate of the 2nd Chebyshev function

Inverse function of $U_{k-1}(\cos(\frac{\pi}{x}))$?

Ramanujan's Proof of Chebycheff's Theorem

Numerical Integration: Why isn't Polynomial Approximation Working?

Why was the number $73.2$ used in "Sharper bounds for the Chebyshev functions $\theta(x)$ and $\psi(x)$. II" in theorem 10, inequality 6.2?

A result related to Chebyshev function $\psi(x)$.

Relation between factorial and chebyshev theta function

Does Chebyshev's theorem provide a lower bound of the primorial $n\#$ such that $n\# \ge 2^{n/2}$

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