# Elementary functions with zeros only at the positive integers

Does there exist a (meromorphic) elementary function $f(z)$ that is zero at all the positive integers $z = 1, 2, 3, \ldots$ and only at those points?

Edit: an elementary function can be written as a finite composition of constants, rational functions, exponentials and logarithms.

Obviously a function with those zeros can be constructed using the gamma function or a Weierstrass product, but the question is whether there is an elementary function.

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sin(pi*x) maybe? – jef May 8 '10 at 18:45
oh sorry, misread the question. – jef May 8 '10 at 18:48

$1/\Gamma(1-z)$.
This is, however, the "most elementary" function with this property, for some reasonable notions of elementary (at least, assuming you want simple zeroes). For instance, any other function with the same zeroes has larger asymptotic growth; this is a version of the Hadamard Factorization Theorem. More precisely, if $f(z)$ has simple zeroes at the positive integers, then there is a polynomial $g(z)$ so that $f(z) = e^{g(z)}/\Gamma(1-z)$. So if you don't think the gamma function is elementary, the answer to your original question is "no". – Dylan Thurston May 9 '10 at 1:50