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pro vyhledávání: '"Banks, William A"'
Autor:
Banks, William D.
For a primitive Dirichlet character $X$, a new hypothesis $RH_{sim}^\dagger[X]$ is introduced, which asserts that (1) all simple zeros of $L(s,X)$ in the critical strip are located on the critical line, and (2) these zeros satisfy some specific condi
Externí odkaz:
http://arxiv.org/abs/2410.11605
Autor:
Banks, William D.
Let $y\ne 0$ and $C>0$. Under the Riemann Hypothesis, there is a number $T_*>0$ $($depending on $y$ and $C)$ such that for every $T\ge T_*$, both \[ \zeta(\tfrac12+i\gamma)=0 \quad\text{and}\quad\zeta(\tfrac12+i(\gamma+y))\ne 0 \] hold for at least o
Externí odkaz:
http://arxiv.org/abs/2405.11084
Autor:
Banks, William D.
For an infinite set M of natural numbers, let FS(M) be the set of all nonzero finite sums of distinct numbers in M. An IP set is any set of the form FS(M). Let p_n denote the n-th prime number for each $n \ge 1$. A de Polignac number is any number m
Externí odkaz:
http://arxiv.org/abs/2403.10637
Autor:
Banks, William D.
For each primitive Dirichlet character $\chi$, a hypothesis ${\rm GRH}^\dagger[\chi]$ is formulated in terms of zeros of the associated $L$-function $L(s,\chi)$. It is shown that for any such character, ${\rm GRH}^\dagger[\chi]$ is equivalent to the
Externí odkaz:
http://arxiv.org/abs/2309.03817
The type $\tau$($\alpha$) of an irrational number $\alpha$ measures the extent to which rational numbers can closely approximate $\alpha$. More precisely, $\tau$($\alpha$) is the infimum over those t$\in$R for which |$\alpha$--h/k|
Externí odkaz:
http://arxiv.org/abs/2307.05965
Autor:
Banks, William D.
We show that the Generalized Riemann Hypothesis for all Dirichlet L-functions is a consequence of certain conjectural properties of the zeros of the Riemann zeta function. Conversely, we prove that the zeros of $\zeta(s)$ satisfy those properties und
Externí odkaz:
http://arxiv.org/abs/2303.09510
Autor:
Banks, William D.
For any $\theta>\frac13$, we show that there are constants $c_1,c_2>0$ that depend only on $\theta$ for which the following property holds. If $\chi_1,\chi_2$ are two distinct primitive Dirichlet characters modulo $q$, and $T\ge c_1q^\theta$, then $L
Externí odkaz:
http://arxiv.org/abs/2302.07073
Autor:
Banks, William D., Sinha, Saloni
Gonek, Graham, and Lee have shown recently that the Riemann Hypothesis (RH) can be reformulated in terms of certain asymptotic estimates for twisted sums with von Mangoldt function $\Lambda$. Building on their ideas, for each $k\in\mathbb{N}$, we stu
Externí odkaz:
http://arxiv.org/abs/2209.11768
Autor:
Banks, William D.
For any real $\beta_0\in[\tfrac12,1)$, let ${\rm GRH}[\beta_0]$ be the assertion that for every Dirichlet character $\chi$ and all zeros $\rho=\beta+i\gamma$ of $L(s,\chi)$, one has $\beta\le\beta_0$ (in particular, ${\rm GRH}[\frac12]$ is the Genera
Externí odkaz:
http://arxiv.org/abs/2205.04576
Autor:
Banks, William, Shparlinski, Igor E.
Motivated by a question of V. Bergelson and F. K. Richter (2017), we obtain asymptotic formulas for the number of relatively prime tuples composed of positive integers $n\le N$ and integer parts of polynomials evaluated at $n$. The error terms in our
Externí odkaz:
http://arxiv.org/abs/2205.00253