Elementary functions with zeros only at the positive integers - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T09:43:23Z http://mathoverflow.net/feeds/question/23946 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23946/elementary-functions-with-zeros-only-at-the-positive-integers Elementary functions with zeros only at the positive integers Fredrik Johansson 2010-05-08T18:42:08Z 2010-05-08T21:36:48Z <p>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?</p> <p>Edit: an <a href="http://en.wikipedia.org/wiki/Elementary_function" rel="nofollow">elementary function</a> can be written as a finite composition of constants, rational functions, exponentials and logarithms.</p> <p>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.</p> http://mathoverflow.net/questions/23946/elementary-functions-with-zeros-only-at-the-positive-integers/23947#23947 Answer by rpotrie for Elementary functions with zeros only at the positive integers rpotrie 2010-05-08T18:45:22Z 2010-05-08T18:45:22Z <p>I don´t completely understand what you mean by elementary, but you can look at <a href="http://en.wikipedia.org/wiki/Weierstrass_factorization_theorem" rel="nofollow">http://en.wikipedia.org/wiki/Weierstrass_factorization_theorem</a>. Sorry if the question had not to do with this. </p> http://mathoverflow.net/questions/23946/elementary-functions-with-zeros-only-at-the-positive-integers/23948#23948 Answer by Dylan Thurston for Elementary functions with zeros only at the positive integers Dylan Thurston 2010-05-08T18:45:39Z 2010-05-08T18:45:39Z <p>\$1/\Gamma(1-z)\$.</p>