# Tagged Questions

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### Asymptotic behavior of sums of consecutive powers (bivariate)

Are there some (bivariate) closed form formulas for the asymptotic behaviour of the sum: $\sum_{k=1}^{n} k^d$ where $n$ and $d$ are large integers? I am especially interested in a lower bound of ...
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### Question about finite sums and integer recursions.

Let $n$ be a positive integer and let $g(n)$ be a given strictly increasing integer function such that $0<g(n)<n$ for all $n$. Also the sequence $|g(n) - n|$ is unbounded as $n$ grows. Let ...
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### asymptotic approximation for number of partitions of integer that do contain 1 nor 2

Hardy and Ramanujan provided a famous asymptotic approximation to $P(n)$ the number of partitions of an integer $n$ when $n$ gets large. I wonder if there is an asymptotic approximation to ...
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### Upper bound for the sum $\sum_{k=1}^N \frac{1}{\varphi(k)}$

Is there an upper bound for the sum $$\sum_{k=1}^N \frac{1}{\varphi^{\alpha}(k)}$$ where $\varphi(n)$ is the Euler totient function and $\alpha\geq 1$ a real constant? In particular, I'm interested ...
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### Asymptotic of a sum evaluation as $x \to \infty$

Let be the sum $$\sum_{n\le x}[x/n]=g(x)$$ where $[x]$ means floor function. My best try for asymptotic is $g(x) \sim x\log (x)+\gamma x +1$ where I have used the asymptotic $[x] \sim x$ ...
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### Asymptotics of the differences between successive zeta zeros

Does anyone know what the asymptotic of the differences between successive zeta zeros is? Update It appears that $\zeta(n)$ is not a bad asymptotic, when the data range is stretched: ...
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### What is the order of the product of $\frac{p-1}{p}$ under the square root of a prime?

Is there any known asymptotical order for $$\prod_{p_k\ \text{prime}}^{\sqrt{p_n}} \frac{p_k-1}{p_k}$$
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### On the sum of relatively prime number $<N$

Let $A(N)$ be a function which is the sum of all numbers relatively prime and $<N$ and $B(N)$ the sum of remaining $N−\phi(N)$ numbers. Then I have the following questions- Q-1 For what values of ...
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### Asymptotics for Mertens function

It seems that the cumulative mean of the Mertens function is very similar in behaviour to $x$ raised to the power of the first zeta zero. I tentatively notate it as: ...
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### Which series of numbers effectively translates the factorial to the exponential function?

We have the relation of the Bernoulli numbers $$B_{2n} = (-1)^{n+1}\frac {2(2n)!} {(2\pi)^{2n}} \left(1+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}+\cdots\;\right).$$ For $n>1$, the right hand sum ...
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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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### Estimating integrals involving $\pi(x)$

While solving an exercise in analytic number theory, I ran into difficulty of estimating an integral of the form $\displaystyle\int_{1}^{x} \frac{\pi(t)}{t} dt$ where $\pi(x)$ is the prime counting ...
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### asymptotic of a product

So the question that I'm working on is the following. Show that $\Pi_{p\leq z}(1-\dfrac{1}{p})=\dfrac{C(1+\mathcal{o}(1))}{\log z}$. First off I take logs and just work with the sum and thisis what ...