On some representation numbers by $a\sum x_i^2+b\sum x_ix_j$ representing one

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发表日期：

2024年1月13日

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Ick Sun Eum

摘要

Let $Q=a\sum x_i^2+b\sum x_ix_j$ $Q = a \sum x_{i}^{2} + b \sum x_{i} x_{j}$ be an integral positive definite quadratic form of level N in $r(\ge 2)$ $r (\geq 2)$ variables and $r_Q(n)$ $r_{Q} (n)$ the representation number by Q for nonnegative integers n. First, we provide a necessary and sufficient condition that Q represents one and find the exact value of $r_Q(1)$ $r_{Q} (1)$ . Second, for such forms, we show that $r_Q(n)$ $r_{Q} (n)$ satisfies a certain congruence relation for infinitely many n. Finally, when $r(\ge 4)$ $r (\geq 4)$ is even and $(-1)^{r/2}N$ ${(- 1)}^{r / 2} N$ is a fundamental discriminant, we classify the quadratic forms Q, which represent one, whose representation numbers $r_Q(n)$ $r_{Q} (n)$ satisfy a partially multiplicative relation $r_Q(p^2n)r_Q(1)=r_Q(p^2)r_Q(n)$ $r_{Q} (p^{2} n) r_{Q} (1) = r_{Q} (p^{2}) r_{Q} (n)$ and provide closed formulas for $r_Q(n)$ $r_{Q} (n)$ .

On some representation numbers by \(a\sum x_i^2+b\sum x_ix_j\) representing one

摘要