We prove that $q+1$-regular Morgenstern Ramanujan graphs $X^{q,g}$ (depending on $g\in\mathbb{F}_q[t]$) have diameter at most $\left(\frac{4}{3}+\varepsilon\right)\log_{q}|X^{q,g}|+O_{\varepsilon}(1)$ (at least for odd $q$ and irreducible $g$) provided that a twisted Linnik-Selberg conjecture over $\mathbb{F}_q(t)$ is true. This would break the 30 year-old upper bound of $2\log_{q}|X^{q,g}|+O(1)$, a consequence of a well-known upper bound on the diameter of regular Ramanujan graphs proved by Lubotzky, Phillips, and Sarnak using the Ramanujan bound on Fourier coefficients of modular forms. We also unconditionally construct infinite families of Ramanujan graphs that prove that $\frac{4}{3}$ cannot be improved.

中文翻译：

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拉马努金图和函数域上的指数和

我们证明 $q+1$-regular Morgenstern Ramanujan 图 $X^{q,g}$（取决于 $g\in\mathbb{F}_q[t]$）的直径至多 $\left(\frac {4}{3}+\varepsilon\right)\log_{q}|X^{q,g}|+O_{\varepsilon}(1)$（至少对于奇数 $q$ 和不可约 $g$）假设 $\mathbb{F}_q(t)$ 上的扭曲 Linnik-Selberg 猜想是正确的。这将打破 $2\log_{q}|X^{q,g}|+O(1)$ 的 30 年上限，这是众所周知的常规拉马努金图直径上限的结果由 Lubotzky、Phillips 和 Sarnak 使用模形式的傅立叶系数上的拉马努金界。我们还无条件地构造了无限系列的拉马努金图，证明 $\frac{4}{3}$ 无法改进。