Powers of primitive roots

by v_Enhance, Aug 4, 2012, 7:34 PM

Let $A$ be an (possibly infinite) multiset of positive integers greater than one such that the multiplicity of each element is finite, and for each $d>1$, there are finitely many elements of $A$ divisible by $d$. Define $f: \mathbb N \to \mathbb C$ \[ f(n) = \sum_{\substack{a \in A \\ 1 \le k \le a}} \left[ e^{\frac{2\pi ik}{a}} \right]^n. \] Is it possible that $f$ can be expressed as a nonconstant polynomial with complex coefficients?
This post has been edited 1 time. Last edited by v_Enhance, Aug 4, 2012, 7:36 PM

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Yes. Let $A$ simply have every integer $n$ $\phi(n)$ times, then $f(n) = n$ for all $n \in \mathbb{N}$.
EDIT : oops nevermind.
This post has been edited 3 times. Last edited by dinoboy, Aug 4, 2012, 8:17 PM

by dinoboy, Aug 4, 2012, 8:12 PM

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Is that equation just a giant smokescreen for "the sum of divisors of $n$ in $A$"? If so, by the weird divisible-by-$d$ condition there are finitely many even elements in $A$, so finitely many elements of the form $2^k$, and if we plug any power of two above the largest one into $f$ then we get the same value, so $f$ attains that value for infinitely many arguments and has to be constant.

Something seems wrong though...

by math_explorer, Aug 5, 2012, 12:40 PM

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darn, yes it was. For some reason I actually thought this would be easier to work with; I guess this is clearly not the case. (The motivation being that I hardly believed a sum of exponentials could ever be a polynomial.)

hmm the powers of two thing seems to work though.

also the random $d$ condition was just to make sure that $f$ was always finite.

by v_Enhance, Aug 5, 2012, 1:03 PM

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$f$ is always finite iff, for every $d$, it is divisible by finitely many elements of $A$, not the other way around. And that would be implied by the finite multiplicity of each element, because integers have finitely many divisors.

...?

by math_explorer, Aug 7, 2012, 2:22 AM

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Or maybe I just fail :P

by v_Enhance, Aug 7, 2012, 3:04 AM

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