Many zeta functions and L-functions which are generalizations of the Riemann zeta function play very important roles in modern mathematics (Kummer criterion, class number formula, Weil conjecture, BSD conjecture, Langlands program, Riemann hypothesis,...).

Euler was perhaps the first person to consider the zeta function $\zeta(s)$ ($1\leq s$). Why did Euler study such a function? What was his aim?

Further, though we know their importance well, should we consider that the Riemann zeta function and its generalizations happen to play key roles in modern number theory?