From explicit estimates for primes to explicit estimates for the Möbius function
Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic.
In the earlier paper [Proc. Amer. Math. Soc. 135 (2007)], we studied solutions g: ℕ → ℂ to convolution equations of the form , where are given arithmetic functions associated with Dirichlet series which converge on some right half plane, and also g is required to be such a function. In this article, we extend our previous results to multidimensional general Dirichlet series of the form (), where is an additive subsemigroup. If X is discrete and a certain solvability criterion is satisfied,...