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.

Filter by
Sorted by
Tagged with
1
vote
0answers
61 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 ...
0
votes
1answer
68 views

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 ...
2
votes
0answers
34 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 ${...
2
votes
1answer
44 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)=\...
2
votes
1answer
32 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 $p\geq 3$ ...
2
votes
0answers
43 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 ...
0
votes
1answer
40 views

Can anyone helps 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 $...
2
votes
2answers
229 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 ...
1
vote
1answer
74 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 ...
1
vote
1answer
65 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 ...
1
vote
1answer
58 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 ...
0
votes
0answers
337 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: ...
0
votes
1answer
69 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. (...
1
vote
1answer
72 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 ...
0
votes
2answers
107 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 ...
3
votes
0answers
141 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-...
0
votes
2answers
78 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 ...
7
votes
1answer
183 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^\...
0
votes
0answers
50 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 $\...
2
votes
1answer
159 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: $\...
1
vote
0answers
56 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 $...
1
vote
1answer
99 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}}...
2
votes
0answers
138 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 ...
0
votes
1answer
179 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 ...
0
votes
0answers
28 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....
0
votes
1answer
66 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 $...
2
votes
1answer
268 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\...
2
votes
0answers
65 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(...
2
votes
0answers
64 views

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 ...
0
votes
0answers
70 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))}...
1
vote
0answers
52 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}\...
0
votes
1answer
82 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)-\...
2
votes
0answers
32 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 ...
1
vote
0answers
48 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 - \...
2
votes
1answer
233 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 ...
0
votes
1answer
256 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 ...
0
votes
2answers
709 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 ...
2
votes
0answers
81 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)=...
3
votes
1answer
58 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, ...
0
votes
0answers
91 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 $...
-1
votes
2answers
181 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 ...
0
votes
2answers
181 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 $\mathbb{E}[g(x)]$ ...
3
votes
1answer
124 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 ...
0
votes
1answer
86 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}\ ...
3
votes
0answers
189 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 ...
0
votes
1answer
128 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 ...
0
votes
1answer
146 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 ...
1
vote
1answer
539 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 $50\%$ ...
2
votes
0answers
135 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 $\psi(x)$ ...
0
votes
1answer
122 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 $$(p,q)=\int_{-1}^{1} p(t)q(t)dt$$ $W = Span\{1,t,t^2\}$ ...