Stop the Zeta function from evaluating

Question

I have a function F whose power series i want to find using Series.

It gives me the result I want, except there is a term of $\pi^4/10800$, and i want it to tell me if the result is Zeta[2]^2/300 or Zeta[4]/120. Is there a way to force the Zeta function to not evaluate?

F[x] = Integrate[Binomial[x, k], {k, 0, x}];
Series[F[x], {x, 0, 3}] // TeXForm

$x+\frac{\pi ^2 x^3}{36}+O\left(x^4\right)$

I want the output to be $x+x^3\frac{\zeta(2)}{6}$ instead.

Answer 1

You could Block Zeta so that it doesn't evaluate:

Block[{Zeta=Inactive[Zeta]},
    Series[F[x],{x,0,6}]
] //TeXForm

$x+\frac{1}{6} x^3 \operatorname{Zeta}(2,1)-\frac{1}{6} x^4 \operatorname{Zeta}(3)+\frac{1}{60} x^5 \left(\operatorname{Zeta}(2,1)^2+9 \operatorname{Zeta}(4,1)\right)+x^6 \left(-\frac{1}{30} \operatorname{Zeta}(3) \operatorname{Zeta}(2,1)-\frac{2 \operatorname{Zeta}(5)}{15}\right)+O\left(x^7\right)$