Questions tagged [chebyshev-function]
For questions about Chebyshev functions $\vartheta(x)$ and $\psi(x)$, which are often used in number theory. For questions about Chebyshev polynomials, use the (chebyshev-polynomials) tag.
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Estimating the upper bound for $\prod\limits_{p \le x}{p^{\frac{1}{p}}}$
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 ...
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Implementation reciprocal of a floating point number using Chebyshev approximation in CKKS
I am trying to obtain the reciprocal of a floating point value $x$ using the Chebyshev approximation, where $x$ is mostly in the order of $10^3$ to $10^5$. Subsequently, I am trying to implement that ...
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Growth Rate of the 2nd Chebyshev function
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 ...
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Inverse function of $U_{k-1}(\cos(\frac{\pi}{x}))$?
I'm trying to find the inverse function of $$U_{k-1}(\cos(\frac{\pi}{x}))=\sum_{n=0}^{\left\lfloor\frac{k-1}2\right\rfloor}\frac{(-1)^n \Gamma(k-n)}{n!\Gamma(k-2n)} \left(2\cos\left(\frac\pi x\right)\...
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Ramanujan's Proof of Chebycheff's Theorem
Background: We define $$\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) &...
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Numerical Integration: Why isn't Polynomial Approximation Working?
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 $n+1$...
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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?
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) - ...
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A result related to Chebyshev function $\psi(x)$.
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 \...
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Relation between factorial and chebyshev theta function
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 ...
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Does Chebyshev's theorem provide a lower bound of the primorial $n\#$ such that $n\# \ge 2^{n/2}$
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 ...
- 10.5k
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Expressing any even natural number as a sum of primorials with coefficients
I'm having a hard time trying to solve the following problem:
Given any random even natural number, $x$, prove that it can or cannot be written as the product of some integer, $b$, times the primorial ...
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Asymptotics of sum of Chebychev function
Show that $\sum_{n\leq x}\frac{θ(n)}{n^2}=\ln x+O(1)$ where $θ$ is the Chebychev function. (We are searching for a solution without the prime number theorem, just Chebychev bounds or something like ...
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References for Littlewood's "infinitely many crossovers" theorem from 1914
I was looking into Littlewood's 1914 result that pi(x) and Li(x) cross infinitely many times, and I came across this Wikipedia page: https://en.wikipedia.org/wiki/Skewes%27s_number#Riemann's_formula. ...
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Chebyshev's inequality, how is it applied in this problem?
Data set:
Problem:
Work attempted:
There is no way that .21 % is at least 3.48, especially when 29/30 are between the bounds of (3.48, 3.96). It's not clear to me what I'm doing wrong.
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Is there an elementary proof for the weak prime number theorem?
Let $ \pi(n) $ be the prime counting function, by "weak prime number theorem" I mean:
$$\lim_{n \to \infty}\frac{\sum_{k=1}^n \frac{\pi(k)}{k}}{\pi(n)}=1 \tag{1}$$
I call it "weak&...
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