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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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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 ...
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31 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-...
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49 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 ...
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78 views

Chebyshev function: variational formulation

The Wikipedia article on the Chebyshev function $\psi(x)$ states that, evaluated at $x=e^t$, it minimizes the functional $$J[f] = \int_0^\infty \dfrac{f(s)\zeta'(s+c)}{\zeta(s+c)(s+c)}ds - \int_0^\...
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The problem of the Chebyshev approximation in complex sense

Let there be given the set of points H of the real line,and the real functions $f(x)$ and $F(x; \lambda_1 ,\lambda_2 .... ,\lambda_n)$ of the real variable $x$, which are bounded on this set. We have ...
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38 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 $\...
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1answer
128 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: $\...
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23 views

$n^{1/2}-f(n)=O(n^\epsilon)$ for all $\epsilon>0$?

Define $f(n)$ as follows: $f(1)=0$ $f(n)=\frac{(n^{1/2}+1)\Lambda(n)+f(n-1)\psi(n-1)}{\psi(n)}$ for $n>1$ where $\Lambda(n)$ is the Von Mangoldt function and $\psi(n)$ is the second Chebyshev ...
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40 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 $...
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1answer
82 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}}...
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134 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 ...
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1answer
82 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 ...
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22 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....
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1answer
56 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 $...
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1answer
119 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\...
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49 views

What are the Dirichlet transforms of $\Lambda(n+1)$ and $\frac{\Lambda(n+1)}{\log(n+1)}$?

This question assumes the following definitions. (1) $\quad\psi(x)=\sum\limits_{n\le x}\Lambda(n)\qquad\text{(second Chebyshev function)}$ (2) $\quad\Pi(x)=\sum\limits_{n\le x}\frac{\Lambda(n)}{\log(...
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Using the 2nd Chebyschev function for describing sums of primes

The question is as follows: Show that for any $n\ge1$, we have $$\psi(n)=\sum_{p\ge n}\left \lfloor\frac{\log n}{\log p}\right \rfloor \log p$$ where $\psi(x)=\sum_{p^m\le x}\log p$, where the sum ...
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63 views

How is it proved that $\sum_{p\le x}\frac1p=\int_2^x\frac{d(\vartheta(t))}{t\log t}$?

Let $\vartheta(x)=\sum_{p\le x}\log p$ be the first Chebyshev function. I'm self-studying analytical number theory and I'm looking for a proof that $$\sum_{p\le x}\frac1p=\int_2^x\frac{d(\vartheta(t))}...
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39 views

How to relate the two versions of Bombieri-Vinogradov theorem to one another?

When searching for the Bombieri-Vinogradov theorem on the internet, it appears under different forms, but with some minor differences there are essentially these two: $$ \sum_{q\leq Q}\max_{(a,q)=1}\...
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1answer
62 views

An inequality on a function of Chebyshev

Let's denote $\theta(x)$ as the sum of natural logarithms of primes up to $x$ and denote $\pi(x)$ as the number of primes up to number $x$.Does the inequality $$\pi(2x)-\pi(x) \geq \frac{\theta(2x)-\...
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29 views

Using balanced multi-sets to estimate Chesbyschev's function $\psi(x)$

I've found this gem when I was looking for estimates on $\pi(x)$. In the link, the author proves that if you have a balanced multi-set (sum of the inverse of the terms is zero) and a few other ...
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38 views

Minimization of Maximum Eigenvalue Error for Richardson Extrapolation

I am asked to solve the following problem: Consider k-steps of Richardson's method with different parameters $\omega_1, . . ., \omega_k $. The error equation is then: $$ e_k = (I - \omega_kA)...(I - \...
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142 views

How to improve Chebyshev bound on the prime counting inequality?

So, I've understood the proof of A*x/logx < pi(x) < B*x/logx for (A,B) = (0.5,2), but how can I make this difference smaller? Does any one know the methods ...
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1answer
163 views

Gauss - Chebyshev Quadrature confusion. Please help.

So I'm reading my lecturer's notes on Gauss-Chebyshev Quadrature (lecturer uses the word Formulation instead of Quadrature) and there is a point where he lost me completely. Here are his notes and ...
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2answers
318 views

How would I use Chebyshev's inequality for this problem?

A probability distribution has a mean of 50 and a standard deviation of $2$. Use Chebyshev's inequality to find the minimum probability that an outcome is between $42$ and $58$. ~~ How would I use ...
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73 views

Is $\sum_{\rho}\frac{x^{\rho+1}}{\rho\left(\rho+1\right)}$ differentiable term by term?

Let $\rho$ be the non trivial zeros of the Riemann zeta function and $x>1$. I would like to prove (or disprove) that $$\frac{d}{dx}\left(\sum_{\rho}\frac{x^{\rho+1}}{\rho\left(\rho+1\right)}\right)=...
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1answer
53 views

Lower bound for the limit of a ratio related to the distribution of primes.

Let $\theta(x)=\sum_{p\leq x}\ln{p}$ denote the first Chebyshev function and let $f(x):=\sum_{p\leq x}p$. I'm hoping to show $$\liminf_{x\to\infty}\frac{\theta(x)^2}{f(x)\ln{f(x)}}\geq1.$$ That is, ...
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77 views

Question regarding Chebyshev function bound with product

So I stumbled upon this while trying to prove something else and cannot find a simple (read:elementary) proof for this (I'm old fashioned and still use $\log$ for natural log): For large enough even $...
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2answers
114 views

Big O and Asymptotic Notation [closed]

So I am currently working in the Apostol Book "Introduction to Analytic Number Theory", and am attempting to prove one of the major results regarding the prime number theorem. Specifically I am trying ...
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1answer
112 views

Proof of increasing positive valued function

Let $g$ be a monotonically increasing positive valued function defined on $\mathbb{R}$. Show that $\mathbb{P}(X\geq a) \leq \frac{\mathbb{E}[g(X)]}{g(a)}$. -I've tried expanding $\mathbb{E}[g(x)]$ ...
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1answer
121 views

The quality of specific triples based on the first $2n$ prime numbers

I'm interested in the abc conjecture. Sometimes I do tasks as examples, with the purpose to understand more about this conjecture. I am thinking about the definition of quality that provide us this ...
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1answer
71 views

Questions on Convergence of $\int_1^\infty\left(\psi(x)-x\right)\ x^{-s-1}\ dx$ and $\int_1^\infty\left(\psi'(x)-1\right)\ x^{-s}\ dx$

Relationships (1) and (2) below are valid for $\Re(s)>1$. (1) $\quad\int_1^\infty\psi(x)\ x^{-s-1}\ dx=-\frac{\zeta'(s)}{s\ \zeta(s)}\,,\quad \Re(s)>1$ (2) $\quad\int_1^\infty\psi'(x)\ x^{-s}\ ...
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165 views

Question on Integral Transform Related to Riemann Zeta Function $\zeta(s)$

The question below assumes the following definitions. $\quad\zeta(s)$ - Riemann zeta function $\quad\psi(x)$ - second Chebyshev function $\quad J(x)$ - Riemann prime-power counting function The ...
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1answer
125 views

Questions on Implications of Riemann Hypothesis with Respect to $\psi\left(e^u\right)$

I've been told the Riemann hypothesis implies (1) below. (1)$\quad\psi\left(e^{\ u}\right)-e^{\ u}=o\left(e^{\ u\ (1/2+\epsilon)}\right)$ I've also been told (1) above implies (2) below converges ...
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1answer
121 views

Questions on Fourier Transform of $\frac{\psi[e^u]-e^u}{e^{u(1/2+\epsilon)}}$

In a response to one of my earlier questions which I believe was related to Evolution of Zeta Zeros from Fourier Transform of $e^{-t/2}\left(\psi'[e^t]-1\right)$, it was suggested I instead focus on ...
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1answer
455 views

Chebyshev's Theorem and probability?

I have a question from the last year's Statistics exam and I could not answer it. I hope you can help me, thanks from now :) "A company owns $100$ televisions. Each television has an $50\%$ ...
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117 views

Questions on Evaluation of Integral for Recovering Second Chebyshev Function $\psi[y]$

The context of this question is the relationship between evaluation of the partial integral for $\psi(x)$ defined in (1) below and partial evaluation of von Mandgoldt's explicit formula for $\psi(x)$ ...
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1answer
98 views

A question about extending polynomial span to monomial basis

I have a final next week and our instructor gave us some examples with solutions but I could not understand some operations. Inner product is $$(p,q)=\int_{-1}^{1} p(t)q(t)dt$$ $W = Span\{1,t,t^2\}$ ...
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1answer
911 views

Finding the lower bound using Chebyshev's inequality

My Question is: Given that Chebyshev's inequality is $$P(|X-µ|≥ kσ)≤\frac{1}{k^2}$$ Find a lower bound on the probability that X is within two standard deviations from its mean. I've found an upper ...
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1answer
114 views

Why is the sum of chebyshev gaussian quadraure weights not $2$?

When I sum the weights of the chebyshev gaussian quadrature over the interval $[-1,1]$. I get about $1.57$. I don't understand why it is not equal to the domain size of the integral (i.e. $2$). My ...
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1answer
122 views

Primes in Arithmetic Progression and Chebyshev Bias

I was reading a paper on the Chebyshev bias and they defined $$\psi(x,q,a) = \sum_{\substack{n\leq x \\ n \equiv a \mod{q}}} \Lambda(n)$$ where $\Lambda$ is the von Mangoldt function. They also ...
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3answers
810 views

What is Relationship Between Distributional and Fourier Series Frameworks for Prime Counting Functions?

I've defined three general methods for derivation of formulas for prime counting functions where each prime counting function is represented by an infinite series of Fourier series, and illustrated ...
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1answer
73 views

Calculate an upper bound for $\left|e^{\psi(N)}-\sum_{p_k\leq e^{\psi(N)}}(p_{k+1}-p_k)^2\right|$

Let $p_k$ the kth prime number, and $\Lambda(n)$ the von Mangoldt function. Thus $\psi(n)=\sum_{k\leq n}\Lambda(k)$ the second Chebyshev function. Also one can defines $g_n$ to be the gap between ...
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0answers
34 views

Convergence of $\sum_{n=1}^\infty\frac{\psi(n)}{e^n}\sin ns$ on an horizontal closed strip

Let $\psi(x)=\sum_{k\leq x}\Lambda(k)$ the Second Chebyshev function, and $\epsilon>0$. I would like to ask Question. Can you prove or disprove that the series $$\sum_{n=1}^\infty\frac{\psi(n)}{...
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1answer
64 views

On computations related with $\lim_{x\to\infty} e^{-x}\sum_{\rho}\frac{(e^x)^\rho}{\rho}=0$

When I've reproduced the shape of the function $\sigma(x)$ of Apostol's section 4.10, a view of the page 98 is avaible as a Google Book (Apostol, Introduction to Analytic Number Theory, Springer 1976),...
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1answer
977 views

Prove the “Chebyshev's theorem”

I know the Chebyshev's theorem for primes that is : There is a $p$ between $n, 2n$ if $n>1$ Can you prove it easily? Actually I'm just 13 years old and I couldn't find an answer that I can ...
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0answers
122 views

Number-theoretic asymptotic looks false but is true?

Question Let $p_r$ be the $r'th$ prime. Is it true that, $$\sum_{r=1}^\infty s^r \ln(p_r) \sim \frac{s}{(1-s)} $$ I know this looks bizarre but kindly consider the argument below. I'm also ...
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1answer
380 views

What would the Riemann Hypothesis mean for the Prime Number Theorem?

The Prime Number Theorem states $\pi(n)\sim \dfrac{n}{\ln n}$. Would there be an equally simple expression if Riemann's Hypothesis were proved true? From Chebyshev Function, would $\pi(n)\sim \dfrac{...
4
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3answers
339 views

On the asymptotic growth of the products of prime numbers

Something must be known about the asymptotic growth of the products of prime numbers. Let $p_n$ be the sequence of prime numbers and define $$P_k=\prod_{n=1}^k p_n$$ I'm looking for a sequence $n_k$ ...
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0answers
430 views

Cubic Approximation to $e^x$ using Chebyshev Polynomial

Was trying to solve this: $C_r=\frac{2}{\pi}\int_{-1}^1\frac{e^xT_r(x)}{\sqrt{1-x^2}}dx$ where $r=0,1,2,3$ $T_r(x) =cosr[{cos}^{-1}x]$ While solving, I equated $x=cos\theta$ Therefore ...