# Tagged Questions

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### Asymptotic expansion of $\zeta(s)$

It is well known that $$\zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}, \quad \Re[s] > 1, \tag{1}$$ but, if $p \leq N$ denotes the primes less than or ...
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### Bounds on a rapidly increasing sequence

I read about a sequence similar to this one here on Stack Exchange a while back, somebody used it as an example for something that I can't recall! However, when I read about it it made me come up with ...
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### On proving the convergence of $1/n^2\sum_{1\le k\le n}\varphi(k)$

Let $$\Phi_n=\frac{1}{n^2}\sum_{k=1}^n\varphi(k).$$ How one can show that $\Phi_n$ is convergent sequence? (Here, $\varphi$ denotes the Euler's totient function.) And please, without any monster ...
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### Is there a “natural” subsequence of positive integers $k_1 < k_2 < \ldots$ such that $\sum_{i=1}^n \frac{1}{k_i} = \Theta (\log \log \log n)$?

The harmonic series partial sums grow like $\log n$, and the sum of inverses of the first $n$ primes grows like $\log \log n$. Is there an example of a "nautral" subset of the positive integers (say ...
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### The asymptotic expansion for the weighted sum of divisors $\sum_{n\leq x} \frac{d(n)}{n}$

I am trying to solve a problem about the divisor function. Let us call $d(n)$ the classical divisor function, i.e. $d(n)=\sum_{d|n}$ is the number of divisors of the integer $n$. It is well known that ...
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### Power towers: to infinity and all the way back

In the following, let $n$ be a positive integer, all other variables be real (furthermore, $a>1$), all functions be real-valued, and logarithms of negative arguments be undefined. Let ...
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### Growth rate of Taylor convergents near pole

For any fixed $z_0\in\mathbb{C}\setminus \{0\}$ and $\beta\in\mathbb{R}^{+}$, prove that $$\left.T_n\left(\log^{\beta}z;z_0\right)\right|_{z=0}\sim\log^{\beta} n$$ Note: I observed that this holds ...
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### Prove: For all $n\geq 1$, $a_{n+1}-a_n<8^ka_n^{\left(1-\frac{1}{k}\right)^3}$.

Set $S=\left\{\left.x^k+y^k+z^k\right|x,y,z\in Z^+\cup \{0\}\right\}$, k is a positive integer, sort elements of $S$ increasingly, that $a_1<a_2<a_3<\text{...}<a_n<\text{...}$. Prove: ...
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### infinite sum with each summand converging to zero almost surely

Suppose $X_n$ is a random variable such that $X_n=O(b_n)$ almost surely, with $b_n\to 0$ as $n\to \infty$. Let $C$ be a real constant and $S_{j,k}(X_n)=\sum_{i=j}^kC^iX_n^i$ for ...
### $\sum _{p\leq n}\frac{1}{p}=C+\ln \ln n+O\left(\frac{1}{\ln n}\right)$
$\sum _{p\leq n}\frac{\ln p}{p}=\ln n+O(1),n\geq 2,$ where $p$ is a prime number, prove: $$\sum _{p\leq n}\frac{1}{p}=C+\ln \ln n+O\left(\frac{1}{\ln n}\right)~~~(1)$$ one examination ...
I have a rather general question: If there are two integer sequences such that $$\lim_{n\to\infty}A(n)/n=\lim_{n\to\infty}B(n)/n=c$$ is there anything else that can be said about them necessarily? ...