# Tagged Questions

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### On the vanishing of the generalized von Mangoldt function $\Lambda_k(n)$ when $n$ has more than $k$ prime factors

It is a well-known fact that the generalized von Mangoldt function, defined by $$\displaystyle \Lambda_k(n) = \sum_{d | n} \mu(d) \left(\log \frac{n}{d}\right)^k$$ vanishes whenever $n$ has more ...
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### How to do such a partitioning?

Assume: $$P \subseteq \{1,2,\dots,N\},\quad |P| = K, \qquad x \in \mathbb{R}_+^K , \qquad w = e^{-j\frac{2\pi}N}$$ and, $$f(l) = \sum_{i=1}^K \sum_{j=1}^K x_i x_j w^{(p_i-p_j)l}$$ I am going to ...
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### shortest relation between poly-Bernoulli numbers and Euler numbers

Poly-Bernoulli numbers which introduced by M.Kaneko are $B_k^{(n)}$ which satisfies in generating function ${Li_{k}(1-e^{-x}) \over 1-e^{-x}}=\sum_{n=0}^{\infty}B_{n}^{(k)}{x^{n}\over n!}$ where Li ...
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### Walsh Fourier Transform of the Möbius function

This question is related to this previous question where I asked about ordinary Fourier coefficients. Special case: is MÃ¶bius nearly Orthogonal to Morse ! Harold Calvin Marston Morse (24 March ...
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### Exact formulas for the partition function?

I am curious, what kind of exact formulas exist for the partition function $p(n)$? I seem to remember an exact formula along the lines $p(n) = \sum_k f(n, k)$, where $f(n, k)$ was some extremely ...
A friend of mine and I were wondering what relationship exists between the partition numbers $p_{n}$ and the Bell numbers $B_{n}$ (and also possibly the Bell polynomials ...