Sentence:  The <WSD>Riemann zeta-function</WSD> (which has no relation to the Weierstrass function of Chapter 8, and must not be confused with it) was originally of interest because of its connection with number theory. Since then it has served as the model for a proliferation of "zeta-functions" throughout mathematics. Some mention of the <WSD>Riemann zeta-function</WSD>, and treatment of the prime number theorem as an asymptotic result have become a topic treated by writers of introductory texts in complex variables. 
Meaning of the word marked with <WSD> tags:  The function "ζ" defined by the Dirichlet series \textstyle \zeta(s)=\sum_{n=1}^\infty \frac 1 {n^s} = \frac1{1^s} + \frac1{2^s} + \frac1{3^s} + \frac1{4^s} + \cdots, which is summable for points "s" in the complex half-plane with real part > 1; the analytic continuation of said function, being a holomorphic function defined on the complex numbers with pole at 1.