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3 questions
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Closed form expression for this zeta-like series involving GCD and LCM
I am looking for a closed form for this function $\Lambda:\mathbb{Q}^+\to\mathbb{R}^+$:
$$\Lambda(q) = \sum_{m,n\geq 1}\left(\frac{q\wedge\frac{m}{n}}{q\vee\frac{m}{n}}\right)^\alpha\left(\frac{m \...
- 634
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Partial sums involving Gregory coefficients that cannot be an integer
For integers $n\geq 1$ let $G_n$ be the Gregory coefficients or reciprocal logarithmic numbers, see the Wikipedia [Gregory coefficients.]
(https://en.wikipedia.org/wiki/Gregory_coefficients)
$${z\...
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Approximation of $\sum_{\substack{n\geq 1\\n\text{ is abundant}}}\frac{\sigma(n)}{n^3}$, where $\sigma(n)$ denotes the sum of divisors function
Let $\sigma(n)=\sum_{1\leq d\mid n}d$ the sum of divisors function then, from the theory of Dirichlet series, it is well-known the value of $$\sum_{n=1}^\infty\frac{\sigma(n)}{n^3},$$
in terms of ...