Negative Values of the Riemann Zeta Function on the Critical Line

Abstract

We investigate the intersections of the curve R ∋ t → ζ( 1 2 + it) with the real axis. We show unconditionally that the zeta-function takes arbitrarily large positive and negative values on the critical line.

Lower bounds of discrete moments of the derivatives of the Riemann zeta-function on the critical line

“…Theorem 1.1 generalizes a result of Kalpokas, Korolev and Steuding [11], who obtained the lower bound for S k,l (T, φ) in the case l = 0.…”

Small values of the Riemann zeta function on the critical line

“…We follow the proof of [KKS,Proposition 10]. We will use the following formulas (see [I, formula 2.17]):…”

“…where we have used the asymptotic T 2π log T + O(T ) for the number of t n (φ) ≤ T (see [KS,Theorem 1]). Since the points s = 1/2 + it n (φ) are the roots of the function χ(s) − e 2iφ , the sum in question can be rewritten as a contour integral:…”

Gram's law in the theory of the Riemann zeta-function. Part 1

“…[27], [28] и [29]). Менее очевидной оказывается ситуация в случае, когда вместо максимума значений дзета-функции на соответствующем множестве точек критической прямой рассматривается минимум.…”