arXiv subjects

Richard R. Hall

Publications and source records attributed to Richard R. Hall.

2 recordsLinked to original sources

arXiv2025

Amplified Fourth Moment of the Riemann Zeta-Function and Applications

The twisted fourth moment of the Riemann zeta-function was established by Hughes and Young [J. Reine Angew. Math. 641 (2010), 203--236] and later improved by Bettin, Bui, Li and Radziwill [J. Eur. Math. Soc. (JEMS) 22 (2020), 3953--3980]. In applications one would often like to take the Dirichlet polynomial to mimic either $1/ζ^r(s)$ (a mollifier) or $ζ(s)^r$ (an amplifier) for some $r>0$. Previous known results include the mean value of the fourth power of $ζ(s)$ times the square or the fourth power of a mollifier, or the square of an amplifier. In this paper we obtain the asymptotic formula for the fourth moment of the Riemann zeta-function times the fourth power of an amplifier. This has various applications to the theory of the Riemann zeta-function, e.g. gaps between zeros of $ζ(s)$ and lower bounds for moments.

math.NT↗

arXiv2018

Switch Functions

We define a switch function to be a function from an interval to $\{1,-1\}$ with a finite number of sign changes. (Special cases are the Walsh functions.) By a topological argument, we prove that, given $n$ real-valued functions, $f_1, \dots, f_n$, in $L^1[0,1]$, there exists a switch function, $σ$, with at most $n$ sign changes that is simultaneously orthogonal to all of them in the sense that $\int_0^1 σ(t)f_i(t)dt=0$, for all $i = 1, \dots , n$. Moreover, we prove that, for each $λ\in (-1,1)$, there exists a unique switch function, $σ$, with $n$ switches such that $\int_0^1 σ(t) p(t) dt = λ\int_0^1 p(t)dt$ for every real polynomial $p$ of degree at most $n-1$. We also prove the same statement holds for every real even polynomial of degree at most $2n-2$. Furthermore, for each of these latter results, we write down, in terms of $λ$ and $n$, a degree $n$ polynomial whose roots are the switch points of $σ$; we are thereby able to compute these switch functions.

math.CA↗