# Tagged Questions

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### Interchanging limits with the prime counting function

How does one justify that $$\lim_{s \to 1} \lim_{x \to \infty} \frac{\pi(x)}{x^s} = \lim_{x \to \infty} \lim_{s \to 1} \frac{\pi(x)}{x^s}, \quad s > 1,$$ without using the fact that the primes have ...
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### Extrema of the Ratio of Consecutive Primes

Let $p_i$ denote the $i$th prime number. We know that $\frac{p_{n+1}}{p_n}\rightarrow 1$ as $n\rightarrow\infty$. Therefore, if we pick some real number $c>1$, there should be some positive integer ...
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### Prime Zeta Function

Does $$\sum_{p \text{ prime}} \frac{1}{p^s} \sim \log \zeta(s) \quad \text{as} \quad s \to 1^+$$ imply $$\sum_{p \leq n} \frac{1}{p} \sim \log H_n \quad \text{as} \quad n \to \infty,$$ where $H_n$ is ...
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### Asymptotic behavior of $\pi (x)-\frac{x}{\log x}$

What is the asymptotic behavior of the function given below. $$f(x)=\pi (x)-\frac{x}{\log x}$$ $$f(x)=O(g(x))$$ What can be $g(x)$? Also what is the asymptotic behavior of the $h(x)=f(x)-g(x)$. My ...
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### On the sum of prime powers

Has anybody investigated the asymptotic growth rate of functions in the form of $$f(z,n)=\sum\limits_{p\le n}p^z$$ For $Re(z)\ge -1$. Of course $f(0,n)=\pi (n)$ has an ocean of research surrounding ...
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### A question about prime gaps

Recently, I have been reading the Wikipedia article about prime gaps (http://en.wikipedia.org/wiki/Prime_gap) and I came across the following: Hoheisel was the first to show that there exists a ...
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### prime zeta function when $0<s<1$ [duplicate]

I would like to know if there is a good estimate for the sum which concerns all primes not exceeding $x$: $$\sum\limits_{p\leq x}\frac{1}{p^s}$$$$0<s<1$$. Only this. Thanks in advance!
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### Asymptotic formula for almost primes

I have developed a formula for almost primes which is far more accurate asymptotically than Landau's well known $$\pi_k(n) \sim \left( \frac{n}{\log n} \right) \frac{(\log\log n)^{k-1}}{(k - 1)!}$$ ...
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### $\pi(x)$ asymptotic as integral $1/\log t$

From the prime number theorem we know that $\pi(x)\sim x/\log x$, i.e. $\dfrac{\pi(x)\log x}{x}\rightarrow 1$ as $x\rightarrow \infty$. How can we use that to show that ...
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### Asymptotics of LCM

Let $\operatorname{LCM}(x_1,x_2,\ldots,x_n)$ be the least common multiple of the integers $x_i$. How can one find the asymptotics of $\operatorname{LCM}(f(1),f(2),\dots,f(n))$ as $n$ approaches ...
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### Sums of Dirichlet-Characters over prime numbers (part 2)

This is kind of related to my previous question that was poorly stated because of misreading my own notes that I have taken on the papers I am currently reading, so no surprise that it eventually ...
### Sequence of numbers with prime factorization $pq^2$
I've been considering the sequence of natural numbers with prime factorization $pq^2$, $p\neq q$; it begins 12, 18, 20, 28, 44, 45, ... and is A054753 in OEIS. I have two questions: What is the ...
More formally, find an asymptotic for $N\to\infty$ of $$\frac{\sum_{1\le k\le N} M(k)}{N}$$ where $$M(p_1^{d_1}p_2^{d_2}\cdots p_k^{d_k}) = d_1+d_2+\cdots+d_k$$ For example, \$M(24) = M(2^3\cdot3) = ...