All questions with tag [math: analytic-number-theory]
0
Limit inferior of the quotient of two consecutive primes
I have recently read an article about the prime number theorem, in which Mathematicians Erdos and Selberg had claimed that proving $\lim \frac{p_n}{p_{n+1}}=1$, where $p_k$ is the $k$th prime, is a very helpful step towards proving the prime number theorem, although I don't know how, primarily because I have not gone through the proof of the the...
2022-07-25 20:45:34
0
What is the binomial sum $\sum_{n=1}^\infty \frac{1}{n^5\,\binom {2n}n}$ in terms of zeta functions?
We have the following evaluations:
$$\begin{aligned}
&\sum_{n=1}^\infty \frac{1}{n\,\binom {2n}n} = \frac{\pi}{3\sqrt{3}}\\
&\sum_{n=1}^\infty \frac{1}{n^2\,\binom {2n}n} = \frac{1}{3}\,\zeta(2)\\
&\sum_{n=1}^\infty \frac{1}{n^3\,\binom {2n}n} = -\frac{4}{3}\,\zeta(3)+\frac{\pi\sqrt{3}}{2\cdot 3^2}\,\left(\zeta(2, \tfra...
2022-07-25 20:43:11
1
Is there a simple way to prove Bertrand's postulate from the prime number theorem?
Is there a simple way to from the prime number theorem?
The prime number theorem immediately implies Bertrand's postulate for sufficiently large $n$, but it fails to establish a base case (the linked proof on Wikipedia explicitly gives the base case $n \ge 468$). In the other direction, Bertrand's postulate yields $\pi(n) \ge \log_2(n)$ which ...
2022-07-25 20:22:05
2
Question regarding Von-Mangoldt function.
Let $\psi(x) := \sum_{n\leq x} \Lambda(n)$ where $\Lambda(n)$ is the Von-Mangoldt function.
I want to show that if $$ \lim_{x \rightarrow \infty} \frac{\psi(x)}{x} =1 $$ then also $$\lim_{x\rightarrow \infty} \frac{\pi(x) \log x }{x}=1.$$
I tried to play a little bit with $\psi$, what I want to show is that:
$$\left| \frac{\pi(x) \log x}{x} -1 \...
2022-07-25 16:11:03
1
Always a prime between $x$ and $x+cf(x)$
What is the asymptotically slowest growing function $f(x)$, such that there exists constants $a$ and $b$, such that for all $x>a$, there is always a prime between $x$ and $x+bf(x)$?
$f(x)=x$ works, does $\sqrt{x}$ work, $\log(x)$, or $\log\log(x)$?
2022-07-25 16:09:47
0
Asymptotic bounds: $\ll$ vs. $\ll_{\epsilon}$?
I am feeling a bit slow today. In Analytic Number Theory it is usual to express asymptotic bounds by specifying the relation of the constant to a specific variable, i.e.
$\log n \ll_\epsilon n^\epsilon$
which means that $\log n \leqslant C_\epsilon n^\epsilon$ for sufficiently large $n$, where the constant $C_\epsilon$ depends only on the consta...
2022-07-25 10:51:22
1
Integers with odd number of prime factors
Let d (n) be the variety of integers much less after that n which has a weird variety of prime variables (2,3,5,7,8,11,12,13,17,18 ). Just how to confirm d (n)/ n have a restriction 1/2? Is there for all m an n such that $|n-2d(n)|>m$?
2022-07-24 05:53:57
1
A combinatorial number theory proof
How can I confirm the adhering to identity: $$\sum_{k=1}^{n}{\sigma_{\ 0} (k^2)} = \sum_{k=1}^{n}{\left\lfloor \frac{n}{k}\right\rfloor \ 2^{\omega(k)}}$$ where $\omega(k)$ is the variety of distinctive prime divisors of $k$.
2022-07-22 18:28:52
1
Exponentiation of a Dirichlet series
I'm trying to understand a proof in Chandrasekharan's Introduction to Analytic Number Theory. Specifically, the proof of the lemma on p.118 before Dirichlet's theorem on primes in arithmetic progressions.
Define
$$
Q(s) = \log P(s)
$$
for some particular branch of the logarithm for $\sigma > 1$. If
$$
Q(s) = \sum_{n=1}^{\infty} \frac{a_...
2022-07-22 18:28:52
3
What is so interesting about the zeroes of the Riemann $\zeta$ function?
The Riemann $\zeta$ function plays a substantial duty in number theory and also is specified by $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} \qquad \text{ for } \sigma > 1 \text{ and } s= \sigma + it$$ The Riemann hypothesis insists that all the non - unimportant absolutely nos of the $\zeta$ function push the line $\text{Re}(s) = \frac...
2022-07-22 18:07:12
3
Asymptotic formula for the logarithm of the hyperfactorial
Background: I was trying to derive an asymptotic formula for the following:
$$\sum_{m\leqslant n}\sum_{k\leqslant m}(m\ \mathrm{mod}\ k),$$
which I think I succeeded in doing (I will skip some steps below to come to my question sooner). We have
\begin{align}
\sum_{m\leqslant n}\sum_{k\leqslant m}(m\ \mathrm{mod}\ k)&=\sum_{m\leqslant n}\...
2022-07-22 15:27:32
2
Calculating the Zeroes of the Riemann-Zeta function
Wikipedia states that
The Riemann zeta function $\zeta(s)$ is defined for all complex numbers $s \neq 1$. It has zeros at the negative even integers (i.e. at $s = −2, −4, −6, ...)$. These are called the trivial zeros. The Riemann hypothesis is concerned with the non-trivial zeros, and states that: The real part of any non-trivial zero of the Ri...
2022-07-22 15:10:05
0
Divisor summatory function for squares
The Divisor summatory function is a function that is an amount over the divisor function. $$D(x)=\sum_{n\le x} d(n) = 2 \sum_{k=1}^u \lfloor\frac{x}{k}\rfloor - u^2, \;\;\text{with}\; u = \lfloor \sqrt{x}\rfloor$$ http://en.wikipedia.org/wiki/Divisor_summatory_function#Dirichlet.27s_divisor_problemI am seeking a formula or a reliable algorithm ...
2022-07-21 09:10:25
1
To show $\lim_{\xi \to 1} (\xi -1) L(\xi,\chi_{0}) = \frac{\Phi(N)}{N}$
How to show if $\chi_{0}$ is the unimportant $\text{Dirichlet Character}$ after that $$\lim_{\xi \to 1} (\xi -1) L(\xi,\chi_{0}) = \frac{\Phi(N)}{N}$$ where $\Phi$ is the $\text{Euler's Totient}$.
2022-07-20 17:48:51
0
Products of primes of the form $an + b$
What is the asymptotic order of numbers divisible by no tops other than those of the kind $an+b$ ($a$, $b$ dealt with)? Undoubtedly (with the exception of the unimportant instances) they are of order purely in between that of he tops and also of all numbers.
2022-07-20 17:22:09
0
Equidistribution of roots of prime cyclotomic polynomials to prime moduli
Here is a relevant - and longstanding, I'm told - conjecture. Let $f \in \mathbb{Z}[x]$ be irreducible and of degree > 1. Set
$E_p = \{x/p \: | \: 0 \leq x < p, f(x) \equiv 0 \: (p) \}$ = normalised least positive residues of zeros of $f$ in $\mathbb{F_p}$
and
$E = \bigcup_{p} E_p \subset [0,1]$
Conjecture: $E$ is equidistributed...
2022-07-20 01:23:37
1
Question about primes in square-free numbers
For any kind of prime, what percent of the square - free numbers has that prime as a prime variable?
2022-07-17 16:58:13
0
Dirichlet series 'shifted' by a polynomial
Let $F(x) \in \mathbb{Z}[x]$ and
$$
\xi(s) = \sum^\infty_{n=1}g(n)n^{-s}
$$
be the Dirichlet series associated an arithmetic function $g(n)$. Define a new Dirichlet series
$$
\xi_F(s) = \sum^\infty_{n=1}g(n)F(n)^{-s}.
$$
I call $\xi_F(s)$ the Dirichlet series obtained from $\xi(s)$ by shifting by $F(x)$. (Maybe this is known by another na...
2022-07-16 17:06:28
1
Is $\eta^{24}(\tau)\,j(\tau) = {E_4}^3(q)$?
Given the j-function $j(\tau)$,
$j(\tau) = 1728J(\tau)$,
where $J(\tau)$ is Klein’s absolute invariant, the Dedekind eta function $\eta(\tau)$, and the following Eisenstein series,
$\begin{align}
E_4 (q) &= 1+240\sum_{n=1}^{\infty} \frac{n^3q^n}{1-q^n}\\[1.5mm]
E_6 (q) &= 1-504\sum_{n=1}^{\infty} \frac{n^5q^n}{1-q^n}\\[1.5mm]
...
2022-07-14 05:22:24
0
How to show that the Laurent series of the Riemann Zeta function has $\gamma$ as its constant term?
I suggest the Laurent collection at $s=1$. I intend to do it by confirming $\displaystyle \int_0^\infty \frac{2t}{(t^2+1)(e^{\pi t}+1)} dt = \ln 2 - \gamma$, based upon the indispensable formula given up Wikipedia. Yet I can not address this indispensable other than by utilizing Mathematica. Attempted intricate analytic means yet no good luck....
2022-07-14 05:22:11