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| On the Number of Integral Ideals in Two Different Quadratic Number Fields |
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Citation： |
Zhishan YANG.On the Number of Integral Ideals in Two Different Quadratic Number Fields[J].Chinese Annals of Mathematics B,2016,37(4):595~606 |
| Page view： 1863
Net amount： 1399 |
Authors： |
Zhishan YANG; |
Foundation： |
This work was supported by the Fundamental Research
Funds for the Central Universities (No.14QNJJ004). |
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| Abstract： |
Let $K$ be an algebraic number field of finite degree over the
rational field $\mathbb{Q}$, and $a_K(n)$ the number of integral
ideals in $K$ with norm $n$. When $K$ is a Galois extension over
$\mathbb{Q}$, many authors contribute to the integral power sums of
$a_K(n)$,
\begin{align*}
\sum_{n \leq x}a_K(n)^l, \quad l=1,2,3,\cdots.
\end{align*}
This paper is interested in the distribution of integral ideals
concerning different number fields. The author is able to establish
asymptotic formulae for the convolution sum
\begin{align*}
\sum_{n\leq x}a_{K_{1}}(n^j)^la_{K_{2}}(n^j)^l, \quad j=1,2,l=2,3,\cdots,
\end{align*}
where $K_1$ and $K_2$ are two different quadratic fields. |
Keywords： |
Asymptotic formula, Integral ideal, Number field |
Classification： |
17B40, 17B50 |
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