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Consider the natural exact sequence $1\to\mathcal O^\times\to\mathcal O[\frac 1n]^\times\to\bigoplus_{\mathfrak p|n}\mathfrak p^{\mathbb Z}\to\mathrm{Cl}(K)$ and restrict to norm $\pm 1$ for elements …

EDIT: corrected my response to Q2. Q1: if I understood the question correctly then the answer is negative. Indeed we may assume $(2B,n)=1$ and taking $k=\varphi(n^2)$ we have $n^2|B^k-1,\,n\nmid B^k+1 …

I will show that $\delta_j\le j-1$. I suspect that in fact $\delta_j=j-1$, but I don't have a complete argument for it and it might be wrong. Let $L$ be the lattice spanned by the columns of $A$ and l …

EDIT: Corrected my previous answer, which contained an incorrect reduction of the problem to a result in the literature. The answer is likely negative. By [1, Theorem 1.1], the partial sums of any mul …

Such a bound cannot exist if $K$ has infinitely many units, because for a unit $h_0=0$ but $h_1$ can be large. Hence assume $K=\mathbb Q(\sqrt{-d})\subset\mathbb C$ is imaginary quadratic. Then $h_1^K …

One approach would be as follows: Let $c_i^{(j)},\,j=1,\ldots,\deg q_i$ be the conjugates of $c_i$. Denote $p_{j_0,\ldots,j_n}(x)=\sum_{i=0}^nc_i^{(j_i)}x^i$ and form the product $P(x)=\prod_{j_1,\ldo …

Using the orthogonality relations for Dirichlet characters the problem reduces to estimating sums of the form $\sum_{n\le x}\chi(n)f(n)$. In the case of $d_r^\ell$ it should not be hard to derive an a …

This problem should really be posed over a general finite field (reductions of elements in $\mathcal O_L$ modulo $P$ can be computed in polynomial time). Over any finite field $F$, the equation $ax^2+ …