TITLE: Could the Riemann zeta function be a solution for a known differential equation?
QUESTION [43 upvotes]: Riemann zeta function is a function of complex variable $s$ that analytically continous the sum of Dirichlet series .defined as :$$\zeta(s)=\sum_{n=1}^{\infty}\displaystyle \frac{1}{n^s} $$ for when the real part is greater than $1$.


My question here is:
    Could the Riemann zeta function be a solution for a known differential equation?


Note: I would like if there is a paper or ref show that zeta function presented a solution for known Differential equation.

REPLY [21 votes]: The Riemann zeta function is "hypertranscendental" in the sense shown HERE
It is not the solution $y(x)$ of a differential equation of the form
$$
F(x,y,y',y'',\dots,y^{(n)})=0
$$
where $F$ is a polynomial (with constant coefficients).

REPLY [20 votes]: The fact that $\zeta$ satisfy no algebraic differential equation is due to its famous relation with the Gamma function which was proved by Hölder not to satisfy such an equation.
Detailed answer can be found here with five (commented and linked) references. 
On the other hand, this function is linked with many other transcendental special functions like polylogarithms which satisfy Fuschian non commutative differential equations.