Results 21 to 30 of about 56 (55)

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

The Carlson‐type zero‐density theorem for the Beurling ζ$\zeta$ function

Journal of the London Mathematical Society, Volume 111, Issue 3, March 2025.
Abstract In a previous paper, we proved a Carlson‐type density theorem for zeroes in the critical strip for the Beurling zeta functions satisfying Axiom A of Knopfmacher. There we needed to invoke two additional conditions: the integrality of the norm (Condition B) and an “average Ramanujan condition” for the arithmetical function counting the number ...
Szilárd Gy. Révész
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

Shifted convolution sums for GL(3)×GL(2)$GL(3)\times GL(2)$ averaged over weighted sets

Mathematika, Volume 70, Issue 2, April 2024.
Abstract Let A(1,m)$A(1,m)$ be the Fourier coefficients of an SL(3,Z)$SL(3,\mathbb {Z})$ Hecke–Maass cusp form F$F$ and λ(m)$\lambda (m)$ be those of an SL(2,Z)$SL(2,\mathbb {Z})$ Hecke holomorphic or Hecke–Maass cusp form g$g$. Let H⊂⟦−X1−ε,X1+ε⟧$\mathcal {H}\subset \llbracket -X^{1-\varepsilon },X^{1+\varepsilon }\rrbracket$ and {a(h)}h∈H⊂C$\lbrace a(
Wing Hong Leung
wiley   +1 more source

Eigenvalues of the Laplace-Beltrami operator and the von-Mangoldt function

Proceedings of the Japan Academy, Series A, Mathematical Sciences, 1993
A relation between the average \(\int_ 0^ X \sum_{n=1}^ y \Lambda(n) dy\) of the von-Mangoldt function \(\Lambda(n)\) and the spectrum of the Laplacian for \(L^ 2 (\Gamma \setminus{\mathcal H})\) with \(\Gamma\) the modular group is proved. The proof is based on a Perron formula for the logarithmic derivative of the Selberg zeta-function.
openaire   +2 more sources