Results 21 to 30 of about 86 (74)

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

Odd moments and adding fractions

Proceedings of the London Mathematical Society, Volume 131, Issue 1, July 2025.
Abstract We prove near‐optimal upper bounds for the odd moments of the distribution of coprime residues in short intervals, confirming a conjecture of Montgomery and Vaughan. As an application, we prove near‐optimal upper bounds for the average of the refined singular series in the Hardy–Littlewood conjectures concerning the number of prime k$k$‐tuples
Thomas F. Bloom, Vivian Kuperberg
wiley   +1 more source

Smallest totient in a residue class

Bulletin of the London Mathematical Society, Volume 57, Issue 6, Page 1908-1917, June 2025.
Abstract We obtain a totient analogue for Linnik's theorem in arithmetic progressions. Specifically, for any coprime pair of positive integers (m,a)$(m,a)$ such that m$m$ is odd, there exists n⩽m2+o(1)$n\leqslant m^{2+o(1)}$ such that φ(n)≡a(modm)$\varphi (n)\equiv a\ (\mathrm{mod}\ m)$.
Abhishek Jha
wiley   +1 more source

A discrete mean value of the Riemann zeta function

Mathematika, Volume 70, Issue 4, October 2024.
Abstract In this work, we estimate the sum ∑0<ℑ(ρ)⩽Tζ(ρ+α)X(ρ)Y(1−ρ)$$\begin{align*} \sum _{0 < \Im (\rho) \leqslant T} \zeta (\rho +\alpha)X(\rho) Y(1\!-\! \rho) \end{align*}$$over the nontrivial zeros ρ$\rho$ of the Riemann zeta function where α$\alpha$ is a complex number with α≪1/logT$\alpha \ll 1/\log T$ and X(·)$X(\cdot)$ and Y(·)$Y(\cdot)$ are ...
Kübra Benli, Ertan Elma, Nathan Ng
wiley   +1 more source

Negative discrete moments of the derivative of the Riemann zeta‐function

Bulletin of the London Mathematical Society, Volume 56, Issue 8, Page 2680-2703, August 2024.
Abstract We obtain conditional upper bounds for negative discrete moments of the derivative of the Riemann zeta‐function averaged over a subfamily of zeros of the zeta function that is expected to be arbitrarily close to full density inside the set of all zeros.
Hung M. Bui, Alexandra Florea, Micah B. Milinovich   +2 more
wiley   +1 more source

Equidistribution of high traces of random matrices over finite fields and cancellation in character sums of high conductor

Bulletin of the London Mathematical Society, Volume 56, Issue 7, Page 2315-2337, July 2024.
Abstract Let g$g$ be a random matrix distributed according to uniform probability measure on the finite general linear group GLn(Fq)$\mathrm{GL}_n(\mathbb {F}_q)$. We show that Tr(gk)$\mathrm{Tr}(g^k)$ equidistributes on Fq$\mathbb {F}_q$ as n→∞$n \rightarrow \infty$ as long as logk=o(n2)$\log k=o(n^2)$ and that this range is sharp.
Ofir Gorodetsky, Valeriya Kovaleva
wiley   +1 more source

Large deviations of the argument of the Riemann zeta function

Mathematika, Volume 70, Issue 3, July 2024.
Abstract Let S(t)=1πImlogζ12+it$S(t) = \frac{1}{\pi }\operatorname{Im}\log \zeta \left(\frac{1}{2}+it\right)$. We prove an unconditional lower bound on the measure of the sets {t∈[T,2T]:S(t)⩾V}$\lbrace t\in [T,2T] \colon S(t) \geqslant V\rbrace$ for loglogT⩽V≪logTloglogT1/3$\sqrt {\log \log T} \leqslant V \ll \left(\frac{\log T}{\log \log T}\right)^{1 ...
Alexander Dobner
wiley   +1 more source

On Popov's formula involving the Von Mangoldt function

, 2017
We offer a generalization of a formula of Popov involving the Von Mangoldt function. Some commentary on its relation to other results in analytic number theory is mentioned as well as an analogue involving the m$\ddot{o}$bius function.
openaire   +2 more sources