# All questions with tag [math: analytic-number-theory]

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## 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
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We have the following evaluations: \begin{aligned} &amp;\sum_{n=1}^\infty \frac{1}{n\,\binom {2n}n} = \frac{\pi}{3\sqrt{3}}\\ &amp;\sum_{n=1}^\infty \frac{1}{n^2\,\binom {2n}n} = \frac{1}{3}\,\zeta(2)\\ &amp;\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
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## 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 $x&gt;a$, 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
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## 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
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## 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 $|n-2d(n)|&gt;m$?
2022-07-24 05:53:57
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## 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
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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 $\sigma &gt; 1$. 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 &gt; 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)&amp;=\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}$$where \Phi is the \text{Euler's Totient}. 2022-07-20 17:48:51 0 ## Products of primes of the form an + b What is the asymptotic order of numbers divisible by no tops other than those of the kind an+b (a, b dealt with)? Undoubtedly (with the exception of the unimportant instances) they are of order purely in between that of he tops and also of all numbers. 2022-07-20 17:22:09 0 ## Equidistribution of roots of prime cyclotomic polynomials to prime moduli Here is a relevant - and longstanding, I'm told - conjecture. Let f \in \mathbb{Z}[x] be irreducible and of degree &gt; 1. Set E_p = \{x/p \: | \: 0 \leq x &lt; p, f(x) \equiv 0 \: (p) \} = normalised least positive residues of zeros of f in \mathbb{F_p} and E = \bigcup_{p} E_p \subset [0,1] Conjecture: E is equidistributed... 2022-07-20 01:23:37 1 ## Question about primes in square-free numbers For any kind of prime, what percent of the square - free numbers has that prime as a prime variable? 2022-07-17 16:58:13 0 ## Dirichlet series 'shifted' by a polynomial Let F(x) \in \mathbb{Z}[x] and$$ \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
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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) &amp;= 1+240\sum_{n=1}^{\infty} \frac{n^3q^n}{1-q^n}\\[1.5mm] E_6 (q) &amp;= 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