Results 11 to 20 of about 56 (55)

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 Hardy–Littlewood–Chowla conjecture in the presence of a Siegel zero

Journal of the London Mathematical Society, Volume 106, Issue 4, Page 3317-3378, December 2022., 2022
Abstract Assuming that Siegel zeros exist, we prove a hybrid version of the Chowla and Hardy–Littlewood prime tuples conjectures. Thus, for an infinite sequence of natural numbers x$x$, and any distinct integers h1,⋯,hk,h1′,⋯,hℓ′$h_1,\dots ,h_k,h^{\prime }_1,\dots ,h^{\prime }_\ell$, we establish an asymptotic formula for ∑n⩽xΛ(n+h1)⋯Λ(n+hk)λ(n+h1′)⋯λ ...
Terence Tao, Joni Teräväinen
wiley   +1 more source

Simple Barban–Davenport–Halberstam type asymptotics for general sequences

Journal of the London Mathematical Society, Volume 112, Issue 4, October 2025.
Abstract We prove two estimates for the Barban–Davenport–Halberstam type variance of a general complex sequence in arithmetic progressions. The proofs are elementary, and our estimates are capable of yielding an asymptotic for the variance when the sequence is sufficiently nice, and is either somewhat sparse or is sufficiently like the integers in its ...
Adam J. Harper
wiley   +1 more source

Moments of the Riemann zeta function at its local extrema

Mathematika, Volume 71, Issue 4, October 2025.
Abstract Conrey, Ghosh and Gonek studied the first moment of the derivative of the Riemann zeta function evaluated at the non‐trivial zeros of the zeta function, resolving a problem known as Shanks' conjecture. Conrey and Ghosh studied the second moment of the Riemann zeta function evaluated at its local extrema along the critical line to leading order.
Andrew Pearce‐Crump
wiley   +1 more source

Counting primes with a given primitive root, uniformly

Mathematika, Volume 71, Issue 4, October 2025.
Abstract The celebrated Artin conjecture on primitive roots asserts that given any integer g$g$ that is neither −1$-1$ nor a perfect square, there is an explicit constant A(g)>0$A(g)>0$ such that the number Π(x;g)$\Pi (x;g)$ of primes p⩽x$p\leqslant x$ for which g$g$ is a primitive root is asymptotically A(g)π(x)$A(g)\pi (x)$ as x→∞$x\rightarrow \infty$
Kai (Steve) Fan, Paul Pollack
wiley   +1 more source