Beyond the Basel Problem, Part I

Summary

In the 18th century, the problem of finding the sum of the reciprocals of the squares was known as the Basel problem. It was first solved by Euler, who, in 1735, showed the sum evaluated to ${\pi }^2/6$. We go beyond the Basel problem by evaluating the sum of $1/n^2$ where $n\equiv \pm b\hspace{1em}(\mathrm{mod}a)$, for some integers b and a. In particular, we examine the special cases $n\equiv \pm 1\hspace{1em}(\mathrm{mod}5)$ and $n\equiv \pm 3\hspace{1em}(\mathrm{mod}16)$.

MSC:

• Primary 11M99
• Secondary 40A05

Additional information

Notes on contributors

Kenneth S. Williams

Kenneth S. Williams (MR Author ID: 183165, ORCID 0000-0002-7700-7150) completed his Ph.D. in mathematics at the University of Toronto in 1965. After 36 years as a faculty member at Carleton University in Ottawa, Canada, he retired in 2002 as Professor Emeritus and Distinguished Research Professor. Besides enjoying time with his family, his interests in “retirement” include research in number theory, birding, gardening, bridge, and chess.