# Tagged Questions

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### Finding formulas for sums

I know that $\sum_{d \mid n} \mu(d) = 0$ whenever $n >1$, and I know that $\sum_{d \mid n} \phi(d) = n$. How can I use this in order to give a formula for $\sum_{d \mid n} \mu(d)\phi(d)$?
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### Find the limit of $\sum \frac{1}{log^n(n)}$

Working on convergence and divergence of infinite series, I recently focused my attention on the summation $$\displaystyle\sum\limits_{n=2}^{\infty} \frac{1}{log^n(n)}$$ While proving the convergence ...
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### Is there a closed form expression for the sum of all the proper divisors of an integer?

I have already found a summation formula here: http://math.stackexchange.com/a/22723, and also a very interesting recursive formula here: http://math.stackexchange.com/a/22744. Any ideas on how 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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### Evaluating the sum $\lim_{n\to \infty}\sqrt[2]{2+\sqrt[3]{2+\sqrt[4]{2+\cdots+\sqrt[n]{2}}}}$

The following nested radical $$\lim_{n\to \infty}\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}$$ is known to converge to 2. We can consider a similar nested radical where the degree of the radicals increases: ...
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### Gosper summable

I'd like to know why the following is NOT gosper summable: $$\sum_{k\in \Bbb{Z}} \frac{p(k)}{\prod_{j=0}^{m-1}(k+a+j)}$$ where $m>0, m\in\Bbb{Z}$ and $p(k)$ is a polynomial of degree $k=m-1$.
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### Sum a series of series where each value increments by one

Can anyone suggest an elegant way to sum a series of numbers like this: (1, 2, 3, 4) (2, 3, 4, 5) (3, 4, 5, 6) (4, 5, 6, 7) That is for $n$ sets ...
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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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### Sum of digits of number from 1 to n

Is there any general formula for calculating the sum of digits of number from 1 to n? n < 10^9
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### How find this sum $S=\sum_{i=1}^{m}(-1)^{a_{i}}\cdot 2^{m-i}$ and $2^i\equiv a_{i}\pmod n$

Question: let $n$ is give odd positive integer numbers,and $a_{i}\neq 1,0\le a_{i}\le n-1$, and $$2^i\equiv a_{i}\pmod n,i=1,2,\cdots,m-1$$ where $m(m\le n)$ such $2^m\equiv 1\pmod n$ ...
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### Inverse logarithmic integral

If the expansion of the logarithmic interval is$$\text{li}(n) = \log \log n + \gamma + \sum_{k=1}^\infty \dfrac{(\log n)^k}{k! k}$$ what is the inverse of the function?
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### Simplifying a sum- combinatorics

Hello fellow mathematicians, I have been working avidly as a high school project to prove Legendre's conjecture. The question below and the other questions I have posted are directly linked to a ...
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### Summation in 104 Number Theory problems

There's a paragraph of 104 Number Theory problems, on page $9$ that says: From the formula $\prod_{i=1}^\infty\frac{p_i}{p_i-1} = \infty ,$ using the inequality $1+t \le e^t$, $t \in \mathbb{R}$ we ...
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### Integrality Conjectures

Here are some interesting conjectures I would like to prove. For all positive integers $a=bc,m,n$ the following expressions are integers: $$c\sum_{k=1}^{am}k\left\{\frac{kbn}{am}\right\}$$ ...
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### Dedekind Sum Integrality Result

Can we prove the following is always an integer? $$6b\sum_{k=1}^bk\left\{\frac{ka}{b}\right\}$$ where $\{x\}=x-\lfloor x\rfloor$ denotes the fractional part operator. UPDATE: Through the ...
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### Evaluation of finite sum

How can one prove the following equality (for fixed positive integer $a$): $$12\sum_{k=1}^{an^2-1}k\left\{\frac{k(an-1)}{an^2}\right\}=3a^2n^4-a^2n^2-2$$ where $\{x\}=x-\lfloor x\rfloor$ denotes the ...
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### Summation of Dedekind sums to zero

I'm trying to show that: $$\sum_{a=0}^b\text{GCD}(a,b)s(a,b)=0$$ More generally, can we also show: $$\sum_{a=0}^b\text{GCD}(a,b)^ls(a,b)=0$$ where $s$ is the Dedekind sum. Any ideas?