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Register a new userOn the Asymptotics to all Orders of the Riemann Zeta Function and of a Two-Parameter Generalization of the Riemann Zeta Function
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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 $Phi(u,v,beta)$.
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We compute the asymptotics of the fourth moment of the Riemann zeta function times an arbitrary Dirichlet polynomial of length $T^{{1/11} - epsilon}$
In 2008 I thought I found a proof of the Riemann Hypothesis, but there was an error. In the Spring 2020 I believed to have fixed the error, but it cannot be fixed. I describe here where the error was. It took me several days to find the error in a careful checking before a possible submission to a payable review offered by one leading journal. There were three simple lemmas and one simple theorem, all were correct, yet there was an error: what Lemma 2 proved was not exactly what Lemma 3 needed. So, it was the connection of the lemmas. This paper came out empty, but I have found a different proof of the Riemann Hypothesis and it seems so far correct. In the discussion at the end of this paper I raise a matter that I think is of importance to the review process in mathematics.
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 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 $k geq 0$.
We use a spectral theory perspective to reconsider properties of the Riemann zeta function. In particular, new integral representations are derived and used to present its value at odd positive integers.
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