Now asked on MO here
This paper discusses how to prove that $\sum\limits_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6} $. The first proof on this paper is Euler's original proof:
$$\frac{\sin(\sqrt x)}{\sqrt x} = 1- \frac{x}{3!}+ \frac{x^2}{5!}- \frac{x^3}{7!}+\dots$$The roots of $\frac{\sin(\sqrt x)}{\sqrt x}$ n are the numbers $\pi^2, 4\pi^2, 9\pi^2, 16\pi^2, \dots$ Now Euler knew that adding up the reciprocals of all the roots of a polynomial results in the negativeof the ratio of the linear coefficient to the constant coefficient. In symbols, if$$(x − r_1)(x − r_2)···(x − r_n) = x^{n} + a_{n−1}x^{n−1} + \dots + a_1x + a_0$$ then $$\sum_{k=1}^n \frac{1}{r_k}= \frac{-a_1}{a_0}$$Assuming that the same law must hold for a power series expansion, he applied it to $\frac{\sin(\sqrt x)}{\sqrt x}$ ,concluding that$$\frac{1}{6}=\sum\limits_{n=1}^\infty \frac{1}{(\pi n)^2}$$Why is this not considered a valid proof today? The problem is that power series are not polynomials,and do not share all the properties of polynomials.
The next page the author wrote this :
$$\frac{1}{1 − x}=1+ x + x^2 + x^3 + \dots$$holds for all $x$ of absolute value less than $1$. Now consider the function $g(x)=2 − 1/(1 − x)$. Clearly, $g$has a single root, $1/2$. The power series expansion for $g(x$) is$ 1−x−x^2−x^3−···$, so $a_0 = 1$ and $a_1 = −1$.The sum of the reciprocal roots does not equal the ratio $−a_1/a_0$. While this example shows that thereciprocal root sum law cannot be applied blindly to all power series, it does not imply that the lawnever holds. Indeed, the law must hold for the function $\frac{\sin(\sqrt x)}{\sqrt x}$ because we have independentproofs of Euler’s result. Notice the differences between $\frac{\sin(\sqrt x)}{\sqrt x}$ and the $g$ of the counterexample. Thefunction $\frac{\sin(\sqrt x)}{\sqrt x}$ has an infinite number of roots, where $g$ has but one. And $\frac{\sin(\sqrt x)}{\sqrt x}$ has a power series that convergesfor all $x$, where the series for $g$ only converges for $−1 <x< 1$. $\color{red}{\text{Is there a theorem which providesconditions}}$$ \color{red}{\text{under which a power series satisfies the reciprocal root sum law? I don’t know.}}$
Now I tried to search for a this theorem and I couldn't find any thing.
Is there is a theorem like this somewhere ? Is there a proof for a under which a power series satisfies the reciprocal root sum law?