<s>[INST] Analytic Function with positive integers as zeros? Do you know any nontrivial analytic function f(z) with zeros only at positive integer values of the argument z = 1, 2, 3, 4, ... ? 
If yes, please give some example.
PS: I already thought of $f(x)=\frac{1}{\Gamma(-x+1)}$. Any other nice options?
EDIT: To avoid trivial solutions due to restriction of definition range, please consider the required function to be defined in the whole complex plane.
 [/INST] $$ f(x) = 3^x - 2^x $$
has 0 as a zero.
$$ f(x) = 4^x - 2 \cdot 3^x + 2^x$$
has 0 and 1 as a zero.
$$ f(x) = 5^x - 3\cdot4^x + 3\cdot3^x - 2^x $$
has 0, 1, and 2 as a zero.
$$ f(x) =  6^x - 4\cdot5^x + 6\cdot4^x - 4\cdot3^x + 2^x $$
has 0, 1, 2, and 3 as a zero.
See the pattern. Continuing
$$ f(x) = 7^x - 5\cdot6^x+10\cdot5^x-10\cdot4^x+5\cdot3^x-2^x$$
has 0, 1, 2, 3, and 4 as a zero.
$$ f(x) = 8^x-6\cdot7^x+15\cdot6^x-20\cdot5^x+15\cdot4^x-6\cdot3^x+2^x $$
has 0, 1, 2, 3, 4, and 5 as a zero.
The pattern is the base decreases by one. Signs alternate. The coefficients are the numbers of the Pascal's triangle.
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