Ritoprovo Roy: p-adic numbers, p-adic analysis and zeta functions

The field of $p$-adic numbers was first used by Hensel in 1897 followed by the proof of Ostrowski classifying the so called $p$-adic norm in $\mathbb{Q}$. This means that the rational numbers should not be thought of as merely a subset of the real numbers but rather as a subset of a spectrum of topological fields obtained by completing the rational number field with respect to each of the possible norms. In the latter half of the 20th century Kubota and Leopoldt, followed by Kenkichi Iwasawa expanded the horizons and applied the $p$-adic incarnations of the classical $\zeta$ and $L$-functions. This study is of relatively recent origin and has been a useful motif in the study of special values of various $L$-functions and their arithmetic significance. I attempt to study the $p$-adic analysis with its applications to $\zeta$ and $L$-functions. I am following the literature of Neal Koblitz on $p$-adic numbers, $p$-adic analysis and $\zeta$ functions and wish to present the concepts explained above.