Year: 2020
Type: preprint
Abstract: Because of its relation to the distribution of prime numbers, the Riemann zeta function ζ (s) is one of the most important functions in mathematics. The zeta function is defined by the following formula for any complex number s with the real component greater than 1.Taking s=2, we see that ζ(2) is equal to the sum of the squares of reciprocals of all positive integers. This leads to the famous problem by Basel in mathematical analysis with important relevance to number theory, solved by Leonhard Euler in 1734. In this paper, we discuss some of the notable proofs given by mathematicians to the basal problem. Most of the theorems are very well known whereas some can be found as proofs of problems present in textbooks.We also give one new proof using the theory of calculus of residues.
Source: arXiv (Cornell University)
Author: Ghosh, Sourangshu
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4394649015