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An additive function on a ring of integers in the imaginary quadratic field \(\mathbb{Q}(\sqrt d)\) with class-number one. (English) Zbl 0840.11037

Various mathematicians, including the author [Kexue Tongbao 29, 1481-1484 (1984) (in Chinese)], have obtained an asymptotic formula for the sum \(\sum_{n\leq x} \beta(n)\), where \(\beta(n)\) is the sum of the prime divisors of \(n\). P. Zarzycki [Acta Arith. 52, 75-90 (1989; Zbl 0682.10032)] generalized the problem by finding an asymptotic formula for \(S(x)= \sum_{N(a)\leq x} {\mathcal B}_\alpha(a)\); here \({\mathcal B}_\alpha(a)= \sum_{p|a} N^\alpha(p)\), the sum being over non-associate prime divisors of the Gaussian integer \(a\), with the norm \(N(u+ iv)= u^2+ v^2\).
The author now considers the function \({\mathcal B}_\alpha(a)\) on the ring of integers in the imaginary quadratic field \(\mathbb{Q}(\sqrt d)\). In the case when the class-number of \(\mathbb{Q}(\sqrt d)\) is one, he establishes an asymptotic formula for \(S(x)\) with an error term \(O(x^{1+ \alpha}/\log^K x)\), where \(K\) is fixed but arbitrary.

MSC:

11N37 Asymptotic results on arithmetic functions
11R11 Quadratic extensions

Citations:

Zbl 0682.10032
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References:

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