The Fourth Moment 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 traditionally called the “trivial” zeros of ζ(s), to distinguish them from the complex zeros of ζ(s), of which the smallest ones (in absolute value) are 12 ± 14.134725 . . . i. It is well-known that all complex zeros of ζ(s) lie in the so-called “critical strip” 0 < σ = Re s < 1, and the Riemann Hypothesis (RH for short) is the conjecture, stated in 1859 by B. Riemann [31], that very likely all complex zeros of ζ(s) have real parts equal to 1 2 . For this reason the line σ = 1 2 is called the “critical line” in the theory of ζ(s). The RH is extensively discussed by many authors, and recently by E. Bombieri [3].