Recent zbMATH articles in MSC 11T23https://zbmath.org/atom/cc/11T232024-03-13T18:33:02.981707ZUnknown authorWerkzeugGroup actions and harmonic analysis in number theory. Abstracts from the workshop held May 7--12, 2023https://zbmath.org/1528.110012024-03-13T18:33:02.981707ZSummary: This workshop focuses on new problems and new methods at the interface of harmonic analysis (taken in a very broad sense) and ergodic theory, with applications focused on number theory. Special emphasis is put on equidistribution problems on arithmetic symmetric spaces, effective methods in homogeneous dynamics, periods of automorphic forms, families of \(L\)-functions over number fields and function fields, and applications of Fourier uniqueness.Triple correlations of ternary divisor functions. IIhttps://zbmath.org/1528.110982024-03-13T18:33:02.981707Z"Lou, Miao"https://zbmath.org/authors/?q=ai:lou.miaoFor the arithmetic functions \(f,a,b\), let
\[
\mathcal{T}(f,a,b;X,H)=\sum_{|h|\le H}\left(1-\frac{|h|}{H}\right)\sum_{X\le n\le 2X}f(n)\,a(n-h)\,b(n+h).
\]
In this paper, the author gives asymptotic estimates for \(\mathcal{T}(f,d_2,d_3;X,H)\) and \(\mathcal{T}(f,d_3,d_3;X,H)\) with \(f\) to be any sequence of complex numbers, and the \(k\)-th divisor function \(d_k\) represents the number of ways \(n\) can be written as a product of \(k\) factors. As a corollary, for \(A>0\), \(\varepsilon\in(0,1/6)\), \(H\in[X^{2/3+\varepsilon},X^{1-\varepsilon}]\) for some \(X>2\), the author proves that
\[
\mathcal{T}(d_3, d_3, d_3; X, H)=HX\,P_6(\log X)+O(HX/(\log X)^A),
\]
where \(P_6\) is a polynomial of degree \(6\).
For Part I see [the author and \textit{G. Lü}, J. Number Theory 198, 318--345 (2019; Zbl 1461.11128)].
Reviewer: Mehdi Hassani (Zanjan)