Results 1 to 10 of about 86 (74)

Power series with the von Mangoldt function

Functiones Et Approximatio, Commentarii Mathematici, 2012
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Matthias Kunik
exaly   +4 more sources

Correlations of the von Mangoldt and higher divisor functions II: divisor correlations in short ranges [PDF]

Mathematische Annalen, 2019
We study the problem of obtaining asymptotic formulas for the sums $\sum_{X < n \leq 2X} d_k(n) d_l(n+h)$ and $\sum_{X < n \leq 2X} Λ(n) d_k(n+h)$, where $Λ$ is the von Mangoldt function, $d_k$ is the $k^{\operatorname{th}}$ divisor function, $X$ is large and $k \geq l \geq 2$ are real numbers. We show that for almost all $h \in [-H, H]$ with $H =
Terence Tao, Kaisa Matomäki, Maksym Radziwiłł   +2 more
exaly   +5 more sources

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.
Akio Fujii
exaly   +3 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 ...
William Banks
exaly   +3 more sources

AVERAGES OF EXPONENTIAL TWISTS OF THE VON MANGOLDT FUNCTION

Bulletin of the Australian Mathematical Society, 2022
AbstractWe obtain some improved results for the exponential sum $\sum _{x<n\leq 2x}\Lambda (n)e(\alpha k n^{\theta })$ with $\theta \in (0,5/12),$ where $\Lambda (n)$ is the von Mangoldt function. Such exponential sums have relations with the so-called quasi-Riemann hypothesis and were considered by Murty and Srinivas [‘On the uniform ...
XIUMIN REN, WEI ZHANG
openaire   +3 more sources

Quantitative bounds for Gowers uniformity of the Möbius and von Mangoldt functions

Journal of the European Mathematical Society, 2023
We establish quantitative bounds on the U^{k}[N] Gowers norms of the Möbius function  \mu and the von Mangoldt function \Lambda for all k
Tao, Terence, Teräväinen, Joni
openaire   +3 more sources

POLYNOMIAL PATTERNS IN THE PRIMES

Forum of Mathematics, Pi, 2018
Let $P_{1},\ldots ,P_{k}:\mathbb{Z}\rightarrow \mathbb{Z}$ be polynomials of degree at most ...
TERENCE TAO, TAMAR ZIEGLER
doaj   +1 more source

Sums of divisor functions and von Mangoldt convolutions in 𝔽 _q [T] leading to symplectic distributions

Forum Mathematicum, 2022
Abstract In [J. P. Keating, B. Rodgers, E. Roditty-Gershon and Z. Rudnick, Sums of divisor functions in 𝔽 q
Vivian Kuperberg, Matilde Lalín
openaire   +3 more sources

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

On Functional Independence of Beurling Zeta-Functions

Axioms
Let P be a system of generalized prime numbers, and NP the corresponding system of generalized integers.
Antanas Laurinčikas, Darius Šiaučiūnas   +1 more
doaj   +1 more source