Is there a way, using the definition, \(\displaystyle\zeta{\left({s}\right)}={\sum_{{{i}={1}}}^{{\infty}}}{i}^{{-{s}}}\) to

Bradley Barron

Bradley Barron

Answered question

2022-04-13

Is there a way, using the definition,
ζ(s)=i=1is
to find this?

Answer & Explanation

mortaraskxnl

mortaraskxnl

Beginner2022-04-14Added 12 answers

Consider the integral
0xtx1et+1dt
=0xtx11+etetdt
=xk=1(1)k10tx1ektdt
=xk=1(1)k1kx0tx1etdt
=xη(x)Γ(x)
(121x)ζ(x)Γ(x+1)

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