Results 11 to 20 of about 2,061 (84)

Extreme values of derivatives of zeta and L-functions. [PDF]

Bull Lond Math Soc
Abstract It is proved that as T→∞$T \rightarrow \infty$, uniformly for all positive integers ℓ⩽(log3T)/(log4T)$\ell \leqslant (\log _3 T) / (\log _4 T)$, we have maxT⩽t⩽2Tζ(ℓ)1+it⩾Yℓ+o1log2Tℓ+1,$$\begin{equation*} \hspace*{1.5pc}\max _{T\leqslant t\leqslant 2T}{\left|\zeta ^{(\ell )}{\left(1+it\right)}\right|} \geqslant {\left({\mathbf {Y}_{\ell }}+ o{\
Yang D.
europepmc   +2 more sources

Aspects of the screw function corresponding to the Riemann zeta‐function

Journal of the London Mathematical Society, Volume 108, Issue 4, Page 1448-1487, October 2023., 2023
Abstract We introduce a screw function corresponding to the Riemann zeta‐function and study its properties from various aspects. Typical results are several equivalent conditions for the Riemann hypothesis in terms of the screw function. One of them can be considered an analog of so‐called Weil's positivity or Li's criterion.
Masatoshi Suzuki
wiley   +1 more source

Lower bounds for negative moments of ζ′(ρ)$\zeta ^{\prime }(\rho )$

Mathematika, Volume 69, Issue 4, Page 1081-1103, October 2023., 2023
Abstract We establish lower bounds for the discrete 2kth moment of the derivative of the Riemann zeta function at nontrivial zeros for all k<0$k<0$ under the Riemann hypothesis and the assumption that all zeros of ζ(s)$\zeta (s)$ are simple.
Peng Gao, Liangyi Zhao
wiley   +1 more source

Multiplicative functions in short arithmetic progressions

Proceedings of the London Mathematical Society, Volume 127, Issue 2, Page 366-446, August 2023., 2023
Abstract We study for bounded multiplicative functions f$f$ sums of the form ∑n⩽xn≡a(modq)f(n),$$\begin{align*} \hspace*{7pc}\sum _{\substack{n\leqslant x\\ n\equiv a\ (\mathrm{mod}\ q)}}f(n), \end{align*}$$establishing that their variance over residue classes a(modq)$a \ (\mathrm{mod}\ q)$ is small as soon as q=o(x)$q=o(x)$, for almost all moduli q$q$,
Oleksiy Klurman, Alexander P. Mangerel, Joni Teräväinen   +2 more
wiley   +1 more source

The elliptic sieve and Brauer groups

Proceedings of the London Mathematical Society, Volume 126, Issue 6, Page 1884-1922, June 2023., 2023
Abstract A theorem of Serre states that almost all plane conics over Q${{\mathbb {Q}}}$ have no rational point. We prove an analogue of this for families of conics parametrised by elliptic curves using elliptic divisibility sequences and a version of the Selberg sieve for elliptic curves.
Subham Bhakta, Daniel Loughran, Simon L. Rydin Myerson, Masahiro Nakahara   +3 more
wiley   +1 more source

Discorrelation of multiplicative functions with nilsequences and its application on coefficients of automorphic L‐functions

Mathematika, Volume 69, Issue 1, Page 250-285, January 2023., 2023
Abstract We introduce a class of multiplicative functions in which each function satisfies some statistic conditions, and then prove that the above functions are not correlated with finite degree polynomial nilsequences. Besides, we give two applications of this result. One is that the twisting of coefficients of automorphic L‐function on GLm(m⩾2)$GL_m
Xiaoguang He, Mengdi Wang
wiley   +1 more source

The Fourier transform of the non-trivial zeros of the zeta function

, 2017
The non-trivial zeros of the Riemann zeta function and the prime numbers can be plotted by a modified von Mangoldt function. The series of non-trivial zeta zeros and prime numbers can be given explicitly by superposition of harmonic waves.
Csoka, Levente
core   +1 more source

On Universality of Some Beurling Zeta-Functions

Axioms
Let P be the set of generalized prime numbers, and ζP(s), s=σ+it, denote the Beurling zeta-function associated with P. In the paper, we consider the approximation of analytic functions by using shifts ζP(s+iτ), τ∈R. We assume the classical axioms for the
Andrius Geštautas, Antanas Laurinčikas
doaj   +1 more source

Mean values of Dirichlet polynomials and applications to linear equations with prime variables

, 2004
We prove a new mean-value theorem for Dirichlet polynomials with coefficients given by the von Mangoldt function. We then use our theorem to derive new estimates for certain exponential sums over primes.
Angel V. Kumchev, Choi, Stephen Kwok-kwong   +2 more
core   +2 more sources

The Riemann Hypothesis via the generalizedvon Mangoldt function

Functiones et Approximatio Commentarii Mathematici
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 $Λ$. Building on their ideas, for each $k\in\mathbb{N}$, we study twisted sums with the \emph{generalized von Mangoldt function} $$ Λ_k(n):=\sum_{d\,\mid\,n}μ(d)\Big(\log ...
Banks, William, Sinha, Saloni
openaire   +2 more sources