Subscribe to the gold package and get unlimited access to Shamra Academy
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
168
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 $Phi(u,v,beta)$.
rate research
Read More
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
