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We make explicit an argument of Heath-Brown concerning large and small gaps between nontrivial zeroes of the Riemann zeta-function, $zeta(s)$. In particular, we provide the first unconditional results on gaps (large and small) which hold for a positive proportion of zeroes. To do this we prove explicit bounds on the second and fourth power moments of $S(t+h)-S(t)$, where $S(t)$ denotes the argument of $zeta(s)$ on the critical line and $h ll 1 / log T$. We also use these moments to prove explicit results on the density of the nontrivial zeroes of $zeta(s)$ of a given multiplicity.
We present several formulae for the large $t$ asymptotics of the Riemann zeta function $zeta(s)$, $s=sigma+i t$, $0leq sigma leq 1$, $t>0$, which are valid to all orders. A particular case of these results coincides with the classical results of Sieg
This is a review of some of the interesting properties of the Riemann Zeta Function.
A recent paper by Agelas [Generalized Riemann Hypothesis, 2019, hal-00747680v3] claims to prove the Generalized Riemann Hypothesis (GRH) and, as a special case, the Riemann Hypothesis (RH). We show that the proof given by Agelas contains an error. In
We compute the asymptotics of the fourth moment of the Riemann zeta function times an arbitrary Dirichlet polynomial of length $T^{{1/11} - epsilon}$
We show that as $Tto infty$, for all $tin [T,2T]$ outside of a set of measure $mathrm{o}(T)$, $$ int_{-(log T)^{theta}}^{(log T)^{theta}} |zeta(tfrac 12 + mathrm{i} t + mathrm{i} h)|^{beta} mathrm{d} h = (log T)^{f_{theta}(beta) + mathrm{o}(1)}, $$ f