16,659 research outputs found
Large deviations and continuity estimates for the derivative of a random model of on the critical line
In this paper, we study the random field \begin{equation*} X(h) \circeq
\sum_{p \leq T} \frac{\text{Re}(U_p \, p^{-i h})}{p^{1/2}}, \quad h\in [0,1],
\end{equation*} where is an i.i.d. sequence of
uniform random variables on the unit circle in . Harper (2013)
showed that is a good model for the large values of
when is large, if
we assume the Riemann hypothesis. The asymptotics of the maximum were found in
Arguin, Belius & Harper (2017) up to the second order, but the tightness of the
recentered maximum is still an open problem. As a first step, we provide large
deviation estimates and continuity estimates for the field's derivative
. The main result shows that, with probability arbitrarily close to ,
\begin{equation*} \max_{h\in [0,1]} X(h) - \max_{h\in \mathcal{S}} X(h) = O(1),
\end{equation*} where a discrete set containing points.Comment: 7 pages, 0 figur
Simple zeros of modular L-functions
Assuming the generalized Riemann hypothesis, we prove quantitative estimates
for the number of simple zeros on the critical line for the L-functions
attached to classical holomorphic newforms.Comment: 46 page
On Differences of Zeta Values
Finite differences of values of the Riemann zeta function at the integers are
explored. Such quantities, which occur as coefficients in Newton series
representations, have surfaced in works of Maslanka, Coffey, Baez-Duarte, Voros
and others. We apply the theory of Norlund-Rice integrals in conjunction with
the saddle point method and derive precise asymptotic estimates. The method
extends to Dirichlet L-functions and our estimates appear to be partly related
to earlier investigations surrounding Li's criterion for the Riemann
hypothesis.Comment: 18 page
On some reasons for doubting the Riemann hypothesis
Several arguments against the truth of the Riemann hypothesis are extensively
discussed. These include the Lehmer phenomenon, the Davenport-Heilbronn
zeta-function, large and mean values of on the critical line,
and zeros of a class of convolution functions. The first two topics are
classical, and the remaining ones are connected with the author's research.Comment: 30 page
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