Closed form for \(\displaystyle{\sum_{{{n}={1}}}^{\infty}}{\frac{{{\cos{{\left(\pi{\log{{n}}}\right)}}}}}{{{n}^{{2}}}}}\)

Joey Rodgers

Joey Rodgers

Answered question

2022-03-17

Closed form for n=1cos(πlogn)n2

Answer & Explanation

Luis Melton

Luis Melton

Beginner2022-03-18Added 2 answers

Of course there is. The fastest way to obtain it is to heuristically write
cos(πlnn) as 12(niπ+niπ). The answer is then given by
n=1cos(πlnn)n2=ζ(2+iπ)+ζ(2iπ)2.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?