Mathematics Stack Exchange
Mathematics Stack Exchange

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.

This tag is intended for questions about Chebyshev functions $\vartheta(x)$ and $\psi(x)$, which are often used in number theory. These functions are defined as $$\vartheta(x)=\sum_{p\le x} \log p$$ and $$\psi(x) = \sum_{p^k\le x}\log p=\sum_{n \leq x} \Lambda(n) = \sum_{p\le x}\lfloor\log_p x\rfloor\log p.$$

For questions about Chebyshev polynomials, use the tag.

105 questions
11
votes
1 answer

Looking for help understanding the Möbius Inversion Formula

I was recently told that the Möbius Inversion Formula can be applied to the Chebyshev Function. Let $\vartheta(x)$,$\psi(x)$ be the first and second Chebyshev functions so that: $$\vartheta(x) = \sum_{p\le{x}}\log p$$ $$\psi(x) =…
Larry Freeman
  • 10,189
8
votes
2 answers

When is Chebyshev's $\vartheta(x)>x$?

Various bounds and computations for Chebyshev's functions $$ \vartheta(x) = \sum_{p\le x} \log p, \quad \psi(x) = \sum_{p^a\le x} \log p $$ can be found in e.g. Rosser and Schoenfeld, Approximate Formulas for Some Functions of Prime Numbers Dusart,…
Zander
  • 11,118
  • 1
  • 27
  • 55
7
votes
1 answer

Chebyshev function: variational formulation

The Wikipedia article on the Chebyshev function $\psi(x)$ states that $\psi(e^t)$ minimizes the functional $$J[f] = \int_0^\infty \dfrac{f(s)\zeta'(s+c)}{\zeta(s+c)(s+c)}ds - \int_0^\infty \int_0^\infty e^{-st}f(s)f(t)ds dt,$$ so that $f(t) =…
bsbb4
  • 3,751
5
votes
1 answer

Trying to understand Theorem 2.27 in a recent paper on the Chebyshev function

In February 2013, Sadegh Nazardonyavi and Semyon Yakubovich posted on arxiv: Sharper estimates for Chebyshev's functions $\vartheta$ and $\psi$. I have a question about Theorem 2.27 on page 22. My question regards the argument for…
Larry Freeman
  • 10,189
5
votes
1 answer

Understanding a famous proof by Jitsuro Nagura: Need help understanding one step in the main theorem

I am going through the proof by Jitsuro Nagura which shows that there is always a prime between $x$ and $\frac{6x}{5}$ where $x \ge 25$. Nagura uses the following definitions: $$\vartheta(x) = \sum_{p \le x} \log p$$ $$\psi(x) =…
Larry Freeman
  • 10,189
5
votes
3 answers

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 $n_k$ such that $$ \lim_{k\to\infty} \frac{P_k}{n_k} =…
user52227
  • 1,714
4
votes
0 answers

Trying to show that $\ln(x!) - \ln(\lfloor\frac{x}{2}\rfloor!) - \ln(\lfloor\frac{x}{3}\rfloor!) - \ln(\lfloor\frac{x}{6}\rfloor!) \ge \psi(x)$

I've been told that the approach below will not work. I would be interested if someone could help me to understand what will go wrong. Let: $$\psi(x) = \sum\limits_{p^k \le x} \ln p$$ So that (see here): $$\ln(x!) = \sum_{k=1}\psi(\frac{x}{k})$$ So…
Larry Freeman
  • 10,189
4
votes
1 answer

Can the Möbius inversion formula be applied to the second Chebyshev function?

Is this a valid application of the Möbius Inversion Formula: Define: $$\psi\left(x\right) = \sum\limits_{p^k \le x} \log p$$ So that: $$\log x! = \sum\limits_{k=1}^{\infty}\psi\left(\frac{x}{k}\right)$$ Then, applying the Möbius Inversion Formula…
Larry Freeman
  • 10,189
4
votes
0 answers

Generalizing Jitsuro Nagura's result: my resulting upper bound for the second chebyshev function is too low. What am I doing wrong?

I've been reading through Jitsuro Nagura's classic proof that there is a prime between $x$ and $\frac{6x}{5}$ and it seems to me that it should be possible to improve on his upper bound for the second Chebyshev function. In Nagura's paper, he…
Larry Freeman
  • 10,189
4
votes
1 answer

Looking for help in understanding Jitsuro Nagura's analysis of the upper bound for $\psi(x)$

I'm working on understanding Nagura's analysis of the upper bound for $\psi(x)$ which is done in Lemma 2. I am unclear on one step of his reasoning. With Lemma 1, he establishes for $x \ge 2000$: $$\log\Gamma(\lfloor{x}\rfloor+1) -…
Larry Freeman
  • 10,189
4
votes
2 answers

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-values at each 'step': $${\psi}^{}_{0} (x)=x-\frac{\zeta…
Richard Burke
  • 925
4
votes
3 answers

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. In the distributional framework for prime counting functions the…
Steven Clark
  • 8,744
4
votes
1 answer

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 understand. Thanks
SAl
  • 41
3
votes
0 answers

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" because it seems less stringent than the prime number theorem:…
Patrick Danzi
  • 1,589
1
2 3 4 5 6 7