Derivative of zeta function for year 2016

Calculus Level 5

$\mathfrak{A} = \zeta '(-2016)$ If $\mathfrak{A}$ can be represented in the form $a! \dfrac{\zeta(b) }{c^{2017} \pi^{d}}$ where $a$ is maximum, find $a+b+c+d$ .

The answer is 6051.

1 solution

Chew-Seong Cheong
Apr 23, 2016

First derivative of Riemann zeta function of a negative even integer $-2n$ is given by ( see eqn (32) ):

$\begin{aligned} \zeta' (-2n) & = \frac{(-1)^n \zeta (2n+1)(2n)!}{2^{2n+1}\pi^{2n}} \\ \implies \zeta' (-2016) & = \frac{\zeta (2017) 2016!}{2^{2017} \pi^{2016}} \end{aligned}$

$\implies a + b + c + d = 2016+2017+2+2016 = \boxed{6051}$

Derivative of zeta function for year 2016

The answer is 6051.

1 solution

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