On Some Recent Results in the Theory of the Zeta-function

which is valid for any complex s, it follows that ζ(s) has zeros at s = −2,−4, . . . . These zeros are called the “trivial” zeros of ζ(s), to distinguish them from the complex zeros of ζ(s). The zeta-function has also an infinity of complex zeros. It is well-known that all complex zeros of ζ(s) lie in the so-called “critical strip” 0 < σ = R s < 1, and if N(T ) denotes the number of zeros ρ = β + iγ (β, γ real) of ζ(s) for which 0 < γ ≤ T , then (f = O(g) and f ≪ g both mean that |f(x)| ≤ Cg(x) for some C > 0 and x ≥ x0)