论文标题
双边系列和Ramanujan的径向限制
Bilateral series and Ramanujan's radial limits
论文作者
论文摘要
Ramanujan给Hardy的最后一封信探讨了模块化形式的渐近性能以及某些有趣的$ Q $ series，他称之为\ emph {Mock theta functions}。对于他的模拟theta函数$ f（q）$，他声称作为$ q $的均匀订单$ 2K $ unity $ζ$的根，\ [\ lim_ {q \ toζ} \ big（q） - （q） - （-1）^k（1- q）^k（1- q） o（1），\]，并暗示存在其其他模拟theta功能的类似陈述。 Folsom-ono-rhoades的最新工作为在$ f（q）$的径向限制中的隐含常数提供了一个封闭的公式。在这里，通过不同的方法，我们证明了Ramanujan的所有第五阶模拟theta函数的结果相似。也就是说，我们表明每个第五阶模拟theta函数可能与模块化双侧系列有关，并利用此连接以获得我们的结果。我们进一步探讨了可以应用此方法的其他模拟theta函数。
Ramanujan's last letter to Hardy explored the asymptotic properties of modular forms, as well as those of certain interesting $q$-series which he called \emph{mock theta functions}. For his mock theta function $f(q)$, he claimed that as $q$ approaches an even order $2k$ root of unity $ζ$, \[\lim_{q\to ζ} \big(f(q) - (-1)^k (1-q)(1-q^3)(1-q^5)\cdots (1-2q + 2q^4 - \cdots)\big) = O(1),\] and hinted at the existence of similar statements for his other mock theta functions. Recent work of Folsom-Ono-Rhoades provides a closed formula for the implied constant in this radial limit of $f(q)$. Here, by different methods, we prove similar results for all of Ramanujan's 5th order mock theta functions. Namely, we show that each 5th order mock theta function may be related to a modular bilateral series, and exploit this connection to obtain our results. We further explore other mock theta functions to which this method can be applied.
