On mean values of multivariable complex valued multiplicative functions and applications. (English) Zbl 1471.11250

Essouabri, Driss

Mishou, Hidehiko (ed.) et al., Various aspects of multiple zeta functions – in honor of Professor Kohji Matsumoto’s 60th birthday. Proceedings of the international conference, Nagoya University, Nagoya, Japan August 21–25, 2020. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 84, 23-64 (2020).

Let \(\{n, d\}\subseteq\mathbb N,\;P(\mathbf{x})\in\mathbb R_{+}[\mathbf{x}],\;\mathbf{x}:=(x_{1}, \ldots, x_{n})\), let \(P\) be a homogeneous polynomial of degree \(d\), and suppose that \(P(\mathbb R_{+}^{n}\setminus \{0\})\neq 0\). Building on his previous work [Contemp. Math. 566, 65–98 (2012; Zbl 1279.11068)], the author further examines the analytic behaviour of the function \[s\mapsto Z(f, P; s),\;s\in{\mathbb{C}},\;Z(f, P; s):=\sum_{\mathbf{m}\in\mathbb N^{n}}f(\mathbf{m})P(\mathbf{m})^{-s/d},\] where \(f(\mathbf{m})\) is a complex valued multiplicative function, and thereby obtains new asymptotic formulae for the sums \[\sum_{\mathbf{m}\in\mathbb N^{n}}, P(\mathbf{m})<tf(\mathbf{m})\] as \(t\rightarrow\infty .\) Several arithmetic applications of that work are given.
For the entire collection see [Zbl 1446.11004].

Reviewer: B. Z. Moroz (Bonn)

MSC:

11N37	Asymptotic results on arithmetic functions
11M32	Multiple Dirichlet series and zeta functions and multizeta values
11M41	Other Dirichlet series and zeta functions
11R42	Zeta functions and \(L\)-functions of number fields
11R52	Quaternion and other division algebras: arithmetic, zeta functions
11S40	Zeta functions and \(L\)-functions
11S45	Algebras and orders, and their zeta functions
14G10	Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11E45	Analytic theory (Epstein zeta functions; relations with automorphic forms and functions)
11F66	Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F70	Representation-theoretic methods; automorphic representations over local and global fields
11F72	Spectral theory; trace formulas (e.g., that of Selberg)
11P21	Lattice points in specified regions

Keywords:

mean values of multivariable arithmetic functions; multiplicative functions; lattice points; zeta functions; meromorphic continuation; Newton polyhedron

Citations:

Zbl 1279.11068

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