On Karatsuba Conjecture and the Lindelöf Hypothesis

On large values of the Riemann zeta-function on short segments of the critical line

“…In this paper, we prove the existence of positive and negative values of the function S(t) whose moduli exceed 3, on each segment of length H = 0.8 ln ln ln t + c 0 (see . For comparison, we note that it appears in the process of calculation of first 200 billions zeros of ζ(s) on the critical line (S. Wedeniwski [17], 2003) that Since the function S(t) is "responsible" for the irregularity in the distribution of zeros of ζ(s), Theorems 3 and 4 imply some conditional results related the distribution of Gram's intervals G n = (t n−1 , t n ] which contain an "abnormal" (that is, unequal to 1) number of ordinates of zeros of ζ(s) (see Theorems 5,6).…”

“…Since the function S(t) is "responsible" for the irregularity in the distribution of zeros of ζ(s), Theorems 3 and 4 imply some conditional results related the distribution of Gram's intervals G n = (t n−1 , t n ] which contain an "abnormal" (that is, unequal to 1) number of ordinates of zeros of ζ(s) (see Theorems 5,6).…”

On the multiplicities of zeros of ζ(s) and its values over short intervals

“…Shao-Ji Feng [6] proved that the LH implies Conjecture 1 with an arbitrary constant A > 0. Other relevant works on this subject include the papers of M.E.…”

Lower bounds for the Riemann zeta function on short intervals of the critical line

“…Garaev [6] also showed Conjecture 3 is true on the Riemann Hypothesis, and Feng [4] has shown that Conjecture 1 is true on the Lindelöf hypothesis. We also note that Karatsuba [9] has shown that slightly more general versions of the above conjectures imply new bounds on the multiplicity of the zeros of ζ(s) inside the critical strip.…”