The Lumber Room

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A “Möbius” inversion formula

[Not the Möbius inversion formula, but something similar.]
As usual, define the Möbius function μ on the natural numbers as
$\mu(n) = \begin{cases}0 \text{ if n is not square free,}\\ -1\text{ if n is squarefree with an odd number of prime factors,}\\ 1\text{ if n is squarefree with an even number of prime factors.}\\\end{cases}$
Let $f$ be any function defined on the natural numbers, and let $F$ be the function defined as $\displaystyle F(n) = \sum_{j=1}^{n}{f(\left\lfloor{n/j}\right\rfloor)}$.
Then it is true that $\displaystyle f(n) = \sum_{k=1}^{n}{\mu(k)F(\left\lfloor{n/k}\right\rfloor)}$.

Note that f need not be multiplicative; it can be any arbitrarily defined function. I have no idea why it is true. Help?

Written by S

Fri, 2008-11-28 at 18:55:44

Posted in Uncategorized

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Asimov on ‘The Last Question’

I like tracking down quotes, but find it terribly hard to track down quotes by Asimov about his writing: there are so many anthologies, and so many comments he has made about a single story in different places. Here, for example, are two comments I remember having read about what is definitely one of his two most famous stories, The Last Question:

• The first was easier to track down; it’s on Wikipedia with a date. In The Best of Isaac Asimov, published in 1973, he says:

‘The Last Question’ is my personal favorite, the one story I made sure would not be omitted from this collection.
Why is it my favorite? For one thing I got the idea all at once and didn’t have to fiddle with it; and I wrote it in white-heat and scarcely had to change a word. This sort of thing endears any story to any writer.
Then, too, it has had the strangest effect on my readers. Frequently someone writes to ask me if I can write them the name of a story, which they think I may have written, and tell them where to find it. They don’t remember the title but when they describe the story it is invariably ‘The Last Question’. This has reached the point where I recently received a long-distance phone call from a desperate man who began, ‘Dr. Asimov, there’s a story I think you wrote, whose title I can’t remember–‘ at which point I interrupted to tell him it was ‘The Last Question’ and when I described the plot it proved to be indeed the story he was after. I left him convinced I could read minds at a distance of a thousand miles.
No other story I have written has anything like this effect on my readers—producing at once an unshakeable memory of the plot and an unshakeable forgettery of the title and even author. I think it may be that the story fills them so frighteningly full, that they can retain none of the side-issues.

Written by S

Fri, 2008-11-28 at 06:47:42

Posted in Uncategorized