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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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Understanding the proof of Theorem 2.4 in Montgomery & Vaughan's Multiplicative Number Theory

Theorem 2.4 in the book of Multiplicative Number Theory by Montgomery and Vaughan proves that for $x ≥ 2, ψ(x) ≍ x$. In the last step of the proof it is concluded $ψ(x) ≥ (\log 2)x + O(\log x)$ and $ψ(...
Ali's user avatar
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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 ...
Larry Freeman's user avatar
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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 ...
Sumana Bagchi's user avatar
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53 views

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 ...
Ok-Virus2237's user avatar
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167 views

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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1 answer
130 views

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) &...
stoic-santiago's user avatar
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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 $n+1$...
Zayn's user avatar
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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) - ...
mathlander's user avatar
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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 \...
Casey's user avatar
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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 ...
RAHUL 's user avatar
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2 votes
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123 views

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 ...
Larry Freeman's user avatar
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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 ...
user3108815's user avatar
2 votes
2 answers
124 views

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 ...
gary mp's user avatar
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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. ...
D.R.'s user avatar
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1 answer
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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.
Sarah's user avatar
  • 63
3 votes
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252 views

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&...
Patrick Danzi's user avatar
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1 answer
272 views

What is the error term in Chebyshev's $\theta$ function?

What are possible error terms in Chebyshev's $\theta$ function? Chebyshev's $\theta$ function is given by $$\theta(x)=\sum_{p\leq x}\log(p),$$ where $\log(p)$ is the natural log of primes $p$. ...
lilliege's user avatar
1 vote
0 answers
235 views

How do you turn an upper bound of the second Chebyshev function into a corresponding lower bound?

In Table 5.1 on page 103 of the book The Riemann Hypothesis for Function Fields: Frobenius Flow and Shift Operators by Machiel van Frankenhuijsen the author states: $$\textit{Riemann hypothesis} \iff ...
vuur's user avatar
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Orthogonality of Chebyshev Second Kind

I was reading on Chebyshev functions, and I found lots of resources on proving the orthogonality of Chebyshev polynomials of the first kind: $\int_{-1}^1 T_m(x) T_n(x) \frac{dx}{\sqrt{1-x^2}} = \...
James David's user avatar
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1 answer
186 views

Properties of Chebyshev’s $\psi$ Function

Given the definition $\psi(x)=\sum_{n\le x}\Lambda(n)$ How can one arrive to the conclusion below: $$\psi(x)=\sum_{p\le x}\left\lfloor{\frac{\log x}{\log p}}\right\rfloor\log p $$
Walid Abdelal's user avatar
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2 answers
200 views

Chebyshev’s $\theta_1(x) = \sum_{n \leqslant x } (x-n) \Lambda_1 (n)$ [closed]

Let $\theta_1 = \int_{1}^{x}\theta (t)dt$, for $x \gt 1$ where $\theta(x)$ is the Chebyshev’s function. Letting $\Lambda_1(n) = \log n,\;$ if $n$ is prime, then $\Lambda_1(n)=0.$ Otherwise prove that: ...
Walid Abdelal's user avatar
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0 answers
417 views

Is there a random variable $X$ for which $P [µx - 2σ ≤ X ≤ µx + 2σx] = 0.6$?

Is there a random variable X for which $$P [µx - 2σ ≤ X ≤ µx + 2σx] = 0.6$$ I have tried this: $$P|X- µx ≤ 2σx|=P((X- µx)² ≤ 4(σx)²) ≥ 1- 1/r²= 1- 1/4= 3/4 ≠ 0.6.$$ So there is no random variable ...
Eduardo Dimas's user avatar
2 votes
0 answers
79 views

Correct reasoning for a finite number of twin primes next to the lcm of first N numbers?

Does the following argument make sense? Let ${\rm lcm}(1,...,N)$ be the least common multiple of the first $N$ natural numbers. In: Highest Twin primes such that the number in between twins is the $\...
rtomas's user avatar
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232 views

Establish an upper bound for $\zeta'(s)\over\zeta(s)$?

Using Perron's formula, I am able to show that $$ \psi(x)={1\over2\pi i}\int_{a-i\infty}^{a+i\infty}\left[-{\zeta'(s)\over\zeta(s)}\right]{x^s\over s}\mathrm ds $$ where $\psi(x)$ is the Chebyshev's ...
TravorLZH's user avatar
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1 answer
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How can we prove $\sum_{p\leq x}\ln(p) < 2x$?

I've been reading this paper for Merten's second theorem. Every thing is just fine but I think that theorem 11 (3.5.2) can have an easier proof. Is there any better approach or an explanation that why ...
Parsa Noori's user avatar
2 votes
0 answers
117 views

Proving the first Chebyshev function $\vartheta(x)=O(x)$ using prime factorisation of ${2n \choose n}$

An exercise in chapter 1 of A course in Analytic Number Theory by Marius Overholt prompts me to find an upper bound of the binomial coefficient ${2n\choose n}$, to deduce the prime factorisation of ${...
Ma Joad's user avatar
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1 vote
1 answer
190 views

Approximating $\vartheta(x)=\sum_{p\le x} \log(p)?$

Consider the first Chebyshev function $\vartheta(x)=\sum_{p\le x} \log(p)$ where the sum runs over the primes less than or equal to $x$. I wanted to approximate $\vartheta(x).$ My attempt was $f(x)=\...
zeta space's user avatar
2 votes
1 answer
54 views

$\sum_{p,m\geq 3}(-1)^{m(p-1)/2}e^{-p^my}\log p = O(y^{-1/3})$

Show that for sufficiently small $y$ we have $\sum_{p,m\geq 3}(-1)^{m(p-1)/2}e^{-p^my}\log p = O(y^{-1/3})$ where $m\geq 3$ represents all positive integers from $3$ onwards, while $p\geq 3$ ...
DesmondMiles's user avatar
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2 votes
0 answers
183 views

How to show convergence of the series sum over zeroes of the zeta function

In the explicit formula for the Chebyshev function, one of the terms that appears is $$ \sum_{\rho} \frac{x^\rho}{\rho} $$ where the sum is taken over the nontrivial zeroes of the zeta function. I'm a ...
Dark Malthorp's user avatar
0 votes
1 answer
104 views

Can anyone help me understanding this integral solution?

I was reading a research article in which authors used Chebyshev-Gauss Quadrature on an integral which is given as in the attached image. Can anyone help me out with any citation to understand how $...
Ali Raza's user avatar
2 votes
2 answers
291 views

How to deduce a result assuming prime number theorem

I am unable to derive this result assuming prime number theorem. Can someone please tell how to do it. Edit -> Here is a proof from stackexchange >but I couldn't think how last line is true. Can ...
user avatar
1 vote
1 answer
97 views

Doubt in proof of Chebysheff theorem

While self studying analytic number theory from Introduction to sieve methods and it's applications by M Ram Murthy and Alina Carmen,I have a doubt in theorem 1.4.1 proof by Chebysheff. My doubt ...
user avatar
1 vote
1 answer
129 views

Doubt in proof of Bertrand Postulate

I studied proof of Bertrand Postulate from M Ram Murthy Problems in analytic number theory and completely understood it . In M Ram Murthy Book , Statement of Bertrand Postulate is (1) - For n ...
user avatar
1 vote
1 answer
99 views

A doubt in Ramanujan's proof of Chebycheff's Theorem in number theory

I am self studying analytic number theory from An introduction to Sieve Methods and its applications by Alina Carmen and M Ram Murthy . I have a doubt on page 7 in Theorem of chebyscheff whose ...
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0 answers
508 views

Chebyshev's Inequality and CLT

I am very confused as to how to approach questions 2 and 3 and the example problems in my notes don't have this specific problem structure. Any help would be great, here is the question below: ...
layover's user avatar
0 votes
1 answer
81 views

Tchebychev's theorem, Statistics, Probability

A typical large dog runs with a mean speed of $58.667$ feet/sec and a standard deviation of $2 $ feet per second. My reaction time is $0.7$ seconds, with a standard deviation of $0.1$ seconds. (...
mohit kumar's user avatar
1 vote
1 answer
102 views

Chebyshev Filter Low Pass Conceptual Question

If I have a 2D signal (say a 2D image) thats defined on $[-1,1]^2$. I sample the 2D signal on discrete Chebyshev Points (Chebyshev-Legendre Points), say there is 60 grid points per side. The ...
Book Book Book's user avatar
0 votes
2 answers
196 views

Chebyshev function - show that $\psi(x)>(x-2)\log2-\log(x+1)$

The question I'm trying to do is this: Assume $x>2$ and $n=\lfloor x/2\rfloor$. Show that $\psi(x)>(x-2)\log2-\log(x+1)$, given the inequality $2n\log2-\log(2n+1)<\psi(2n)$. All I've ...
Xtrfyable's user avatar
4 votes
1 answer
493 views

von Mangoldt's formula for Chebyshev $\psi$ function

Chebyshev's $\psi$ function is defined for primes $p$ as $$\psi(x)=\sum _{p^k\leq x} \log (p)$$ von Mangoldt found an explicit formula for this, with the exception that the function takes half-...
Richard Burke-Ward's user avatar
0 votes
2 answers
175 views

Chebyshev's Inequality to solve amount of stock needed

A mail order company offers their first $1000$ customers a ladies' or mens' watch. Suppose that both sexes are equally attracted by the offer. How many ladies' and mens' watches are needed in order to ...
SABOY's user avatar
  • 1,838
7 votes
1 answer
239 views

Chebyshev function: variational formulation

The Wikipedia article on the Chebyshev function $\psi(x)$ states that $\psi(e^t)$ minimizes the functional $$J[f] = \int_0^\infty \dfrac{f(s)\zeta'(s+c)}{\zeta(s+c)(s+c)}ds - \int_0^\infty \int_0^\...
bsbb4's user avatar
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0 answers
53 views

Is it true that if $x \ge 2000$, then the least common multiple of $\{1,2,3,\dots,x\}$ is greater than $2.499^x$

As I was reading through Jitsuro Nagura's proof, I am seeing that he showed for $x \ge 2000$: $$\psi(x) \ge 0.916x$$ where $\psi(x) = \sum\limits_{m=1}^{\infty}\vartheta(\sqrt[m]{x})$ and where $\...
Larry Freeman's user avatar
2 votes
1 answer
203 views

Is this a valid conclusion from Chebyshev's first and second function

I am reading through a paper on Arxiv.org. I am confused by a step in Theorem 2.3.1 on page 39. Kyle Balliet, the author of the paper, starts with standard definitions of the Chebyshev functions: $\...
Larry Freeman's user avatar
2 votes
0 answers
84 views

How do I write $\sum\limits_{p \leq q \text{ prime}}\log (p-x)$ using Chebyshev's function(s)?

Question: If $x\in \mathbb{N}$ and $p$ and $q$ are prime numbers how do I write $\sum\limits_{p \leq q \text{ prime}}\log (p-x)$ using Chebyshev's function(s) ? I have some partial results: The case $...
Anthony's user avatar
  • 3,758
1 vote
1 answer
127 views

How can I write $\sum\limits_{p \leq q \text{ prime}}\log (p-1)$ using Chebyshev's function(s)?

Question: If $p$ and $q$ are prime numbers how do I write $\sum\limits_{p \leq q \text{ prime}}\log (p-1)$ using Chebyshev's function(s) ? I would like to think $\sum\limits_{p \leq q \text{ prime}}...
Anthony's user avatar
  • 3,758
2 votes
0 answers
152 views

Is the sequence: $S_n=\Im(n\cdot\exp(\frac{2\pi\cdot i}{\log_{n}(p_n\#)})):n\in\Bbb N$ monotonic for $P_n>41893$?

How many times does the sequence: $S_n=\Im\left(n\cdot\exp\left(\frac{2\pi\cdot i}{\log_{n}(p_n\#)}\right)\right):n\in\Bbb N$ oscillate? I.e. is it monotonic increasing after $S_{4379}$ which ...
it's a hire car baby's user avatar
0 votes
1 answer
336 views

Linear vs Orthogonal polynomial for fitting

I am neither a mathematician nor do I possess a large knowledge in maths so this could also be completely false. But I read that if you use an orthogonal function like the Chebychev-polynomial to ...
Justanotherchemist's user avatar
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0 answers
41 views

Finding new expectation using Chebyshev.

I've tried to use similar questions for this, but I ended up with just a horrible answer. I have a question about a woman serving soup, she originally had 30 litres for 80 pupils, with expectation 0....
user584391's user avatar
0 votes
1 answer
75 views

Question about the hint and answer.(inequality,and the relation between hint and question itself)

Question: Given two sequences of random variables {$X;n=1,2,...$} and {$Y;n=1,2,...$} and a random variable $X$,suppose that with probability one $|X_n-X| \le Y_n$ all $n$ and that $E[Y_n] \to 0$ as $...
Shine Sun's user avatar
  • 565
3 votes
1 answer
500 views

What's the sum of the inverses of the Primorial numbers?

What's the sum of the inverses of the primorial numbers? Let the $n^{th}$ primorial number be the product of the first $n$ primes $\displaystyle n\#= \prod_{p\leq p_n}p$ So $N\#=2,2\cdot3,2\cdot3\...
it's a hire car baby's user avatar