AI 中文总结
本文提出一种基于theta函数泛函方程特殊形式的方法来研究Dirichlet L-函数的Lindelöf假设，通过求解线性方程组估计L函数值，数值实验揭示逆矩阵极限属于实分圆域。
AI 中文摘要
我们考虑了一种可能的方法来处理Dirichlet $L$-函数的Lindelöf假设。该方法基于相应theta函数泛函方程的一种特殊形式。为了估计$L_\chi(0.5+it)$，我们需要求解某些线性方程组。相应矩阵的条目由theta函数级数及其导数的各项构成。数值数据表明，逆矩阵具有深层结构，并使我们能够提出若干猜想。特别是，对于模$q$的特征，当矩阵的大小以步长$2q$的算术级数变化时，逆矩阵中的某些条目似乎趋于有限极限。此外，这些极限属于实分圆域$\mathbb{Q}(\cos(\pi/q))$（在$\sqrt{q}$的缩放因子意义下）。
英文摘要
We consider a possible approach to the Lindelöf hypothesis for Dirichlet $L$-functions. It is based on a special form of the functional equation for the corresponding theta functions. To estimate $L_χ(0.5+it)$ we need to solve certain systems of linear equations. The entries to the corresponding matrices are formed by the summands to the series for theta functions and their derivatives. Numerical data suggest that the inverse matrices have a deep structure and allow us to state a number of conjectures. In particular, it seems that for a character modulo $q$ certain entries to the inverse matrices tend to finite limits when the sizes of the matrices run over arithmetical progressions with step $2q$. Moreover, these limits belong to the real cyclotomic field $\mathbb{Q}(\cos(π/q))$ (up to a scaling factor of $\sqrt{q}$).
Comments27 pages, 35 tables; submitted to Sirius. Mathematical Journal (https://www.mathbooks.ru/Siriusmathjournal). In the second version, minor inaccuracies have been corrected; details of the calculations have been added. It is also reported that one conjecture has been proven by ChatGPT
