Functional Equation of the Zeta Function, Hecke’s Proof

Abstract

Let f be a function on R ⁿ. We shall say that f tends to 0 rapidly at infinity if for each positive integer m the functionEquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiablA % AiHjaacIcacaaIXaGaey4kaSYaaqWaaeaacaWG4baacaGLhWUaayjc % SdGaaiykamaaCaaaleqabaGaamyBaaaakiaadAgacaGGOaGaamiEai % aacMcacaGGSaGaaGjcVlaadIhacqGHiiIZcaWGsbWaaWbaaSqabeaa % caWGUbaaaOGaaiilaaaa!4B9C!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$x \mapsto {(1 + \left| x \right|)^m}f(x),{\kern 1pt} x \in {R^n},$$ is bounded for |x| sufficiently large. Here as in the rest of this chapter, |x| is the Euclidean norm of x. Equivalently, the preceding condition can be formulated by saying that for every polypomial P (in n variables) the function Pf is bounded, or that the functionEquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiablA % AiHnaaemaabaGaamiEaaGaay5bSlaawIa7amaaCaaaleqabaGaamyB % aaaakiaadAgacaGGOaGaamiEaiaacMcaaaa!4133!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$x \mapsto {\left| x \right|^m}f(x)$$ is bounded, for x sufficiently large (i.e. |x| sufficiently large).