论文标题
渐近身份，用于分隔数的加性卷积
Asymptotic identities for additive convolutions of sums of divisors
论文作者
论文摘要
在1916年的论文中，Ramanujan研究了添加剂卷积$ s_ {a，b}（n）$ of-distivisors函数的$σ_a（n）$和$σ_b（n）$，并在$ a $和$ b $是积极的奇数整体时证明了它的渐近公式。他还推测，他的渐近公式应适用于所有正面真正的$ a $ $ $ $ $ $。拉马努扬的猜想随后由英厄姆（Ingham）证明，然后由哈尔伯斯塔姆（Halberstam）提出了节省误差术语。 在本文中，我们给出了Ramanujan的猜想的新证明，该猜想在大多数参数范围内在渐近学中获得了较低的术语。我们还描述了与第二作者论文中研究的几何拓扑问题的联系，并作为我们研究这一款项的最初动机。
In a 1916 paper, Ramanujan studied the additive convolution $S_{a, b}(n)$ of sum-of-divisors functions $σ_a(n)$ and $σ_b(n)$, and proved an asymptotic formula for it when $a$ and $b$ are positive odd integers. He also conjectured that his asymptotic formula should hold for all positive real $a$ and $b$. Ramanujan's conjecture was subsequently proved by Ingham, and then by Halberstam with a power saving error term. In this paper, we give a new proof of Ramanujan's conjecture that obtains lower order terms in the asymptotics for most ranges of the parameters. We also describe a connection to a counting problem in geometric topology that was studied in the second author's thesis and which served as our initial motivation in studying this sum.
