Limit involving the zeta function.

Calculus Level 3

The zeta function is a function of complex argument defined as: ζ ( s ) : = ∑ n = 1 ∞ 1 n s . \zeta(s):=\sum_{n=1}^\infty \dfrac1{n^s}. What is the following limit equal to? lim ⁡ s → ∞ ζ ( π s ) ζ ( s ln ⁡ s ) . \lim\limits_{s\to\infty}\dfrac{\zeta(\pi^s)}{\zeta(s\ln s)}.


The answer is 1.

حكيم الفيلسوف الضائع by حكيم الفيلسوف الضائع , 35, Morocco

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1 solution

A known result is: lim ⁡ s → ∞ ζ ( s ) = 1. \lim_{s\to\infty}\zeta(s)=1. And since π s \pi^s and s ln ⁡ s s\ln s are all increasing functions we have: lim ⁡ s → ∞ ζ ( π 2 ) ζ ( s ln ⁡ s ) = 1 / 1 = 1 . \lim_{s\to\infty}\dfrac{\zeta(\pi^2)}{\zeta(s\ln s)}=1/1=\boxed1.

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