Another proof of the functional equation for the Riemann zeta function
Let \({\mathcal L}_{\mu, p}\) denote the space of all complex-valued Lebesgue measurable functions \(f\) on \((0,\infty)\) which satisfy \(\int_ 0^ \infty x^{\mu-1} | f(x) |^ p dx<\infty\). Suppose \(k\in {\mathcal L}_{\mu,1}\). Then for any \(f\in {\mathcal L}_{\mu, p}\) \((1\leq p< \infty)\), the integral \(\int_ 0^ \infty k(x/ t) f(t) t^{-1} dt\) exists for almost all positive \(x\) and defines a bounded linear transformation \(T\) from \({\mathcal L}_{\mu, p}\) to itself. Further, if \(1\leq p\leq 2\), then \[ ({\mathcal M} (Tf)) (s)= ({\mathcal M} k) (s)\cdot ({\mathcal M} f) (s) \tag \(*\) \] holds for \(\text{Re } s=\mu\). Here \({\mathcal M} f= \int_ 0^ \infty x^{s-1} f(x) dx\) is the Mellin transformation. Starting with the well-known formula \[ \Gamma(s) \zeta(s)= \int_ 0^ \infty t^{s-1} \biggl( {1\over {e^ t-1}} - {1\over t} \biggr)dt, \qquad 0< \text{Re } s<1, \] the author gives a proof of the functional equation for \(\zeta (s)\) via \((*)\) by choosing \(p=1\), \(0<\mu <1\), \(k(x)= (e^ x -1)^{-1}- x^{-1}\) and \(f(x)= 1-\cos (x^{-1})\). The proof is akin to the fourth proof given in Chapter 2 of \textit{E. C. Titchmarsh}'s classic work [The theory of the Riemann zeta-function, Clarendon Press, Oxford (1951; Zbl 0042.07901); 2nd ed., Clarendon Press, Oxford (1986; Zbl 0601.10026)].
scientific article; zbMATH DE number 737223