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Register a new userLower bounds for moments of the derivative of the Riemann zeta function
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Authors
Peng Gao
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We establish in this paper sharp lower bounds for the $2k$-th moment of the derivative of the Riemann zeta function on the critical line for all real $k geq 0$.
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We study the $2k$-th discrete moment of the derivative of the Riemann zeta-function at nontrivial zeros to establish sharp lower bounds for all real $k geq 0$ under the Riemann hypothesis (RH).
We show that as $Tto infty$, for all $tin [T,2T]$ outside of a set of measure $mathrm{o}(T)$, $$ int_{-(log T)^{theta}}^{(log T)^{theta}} |zeta(tfrac 12 + mathrm{i} t + mathrm{i} h)|^{beta} mathrm{d} h = (log T)^{f_{theta}(beta) + mathrm{o}(1)}, $$ for some explicit exponent $f_{theta}(beta)$, where $theta > -1$ and $beta > 0$. This proves an extended version of a conjecture of Fyodorov and Keating (2014). In particular, it shows that, for all $theta > -1$, the moments exhibit a phase transition at a critical exponent $beta_c(theta)$, below which $f_theta(beta)$ is quadratic and above which $f_theta(beta)$ is linear. The form of the exponent $f_theta$ also differs between mesoscopic intervals ($-1<theta<0$) and macroscopic intervals ($theta>0$), a phenomenon that stems from an approximate tree structure for the correlations of zeta. We also prove that, for all $tin [T,2T]$ outside a set of measure $mathrm{o}(T)$, $$ max_{|h| leq (log T)^{theta}} |zeta(tfrac{1}{2} + mathrm{i} t + mathrm{i} h)| = (log T)^{m(theta) + mathrm{o}(1)}, $$ for some explicit $m(theta)$. This generalizes earlier results of Najnudel (2018) and Arguin et al. (2019) for $theta = 0$. The proofs are unconditional, except for the upper bounds when $theta > 3$, where the Riemann hypothesis is assumed.
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 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 Siegel. Using these formulae, we derive explicit representations for the sum $sum_a^b n^{-s}$ for certain ranges of $a$ and $b$. In addition, we present precise estimates relating this sum with the sum $sum_c^d n^{s-1}$ for certain ranges of $a, b, c, d$. We also study a two-parameter generalization of the Riemann zeta function which we denote by $Phi(u,v,beta)$, $uin mathbb{C}$, $vin mathbb{C}$, $beta in mathbb{R}$. Generalizing the methodology used in the study of $zeta(s)$, we derive asymptotic formulae for $Phi(u,v,beta)$.
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.
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