论文标题
1级Hecke特征形式的傅立叶系数
Fourier Coefficients of Level 1 Hecke Eigenforms
论文作者
论文摘要
莱默（Lehmer）1947年关于$τ（n）$消失的猜想仍未解决。在这种情况下，考虑莱默猜想的变体是很自然的。我们确定许多无法为$τ（n）$的整数。例如，在零数$α$中，$ |α| <99 $，我们确定$$τ（n）\ notin \ { - 9，\ pm 15，\ pm 21，-25，-27，-27，-27，-27，-33，-33，\ pm 35，\ pm 35，\ pm 45，\ pm 45，\ pm 49，\ pm 49，-55，pm pm pm pm 63，$ pm 77，\ $ pm 77，-81 \ $。此外，在GRH下，我们有$τ（n）\ neq - |α| $，以及$τ（n）\ notin \ {9,25,27,39，$ 75,81 \}。$我们还考虑1级Hecke Eigenforms在CUSP形式的尺寸1 Hecke Eigenforms中。例如，对于$δe_4= \ sum_ {n = 1}^{\ infty}τ_{16}（n）q^n $，我们表明\ begin {align*}τ_{16}（16}（n） \ neq 33,55,59,67,73,83,89,91 \} \\ quad \ quad \ quad \ quad \ quad \ quad \ quad \ quad \ quad \ quad \ cup \ { - 33，33，-55，-59，-59，-67，-89，-89，-91，-91 \}。 Swinnerton-Dyer排除了其他大型素数，这些素数将特定的Bernoulli数字分开。为了获得这些结果，我们利用了卢卡斯序列的理论，解决高度thue方程的方法，用于求解过椭圆方程的barros的算法以及持续分数的理论。
Lehmer's 1947 conjecture on whether $τ(n)$ vanishes is still unresolved. In this context, it is natural to consider variants of Lehmer's conjecture. We determine many integers that cannot be values of $τ(n)$. For example, among the odd numbers $α$ such that $|α|<99$, we determine that $$τ(n) \notin \{-9, \pm 15, \pm 21, -25, -27, -33, \pm 35, \pm 45, \pm 49, -55, \pm 63, \pm 77, -81, \pm 91 \}.$$ Moreover, under GRH, we have that $τ(n) \neq -|α|$ and that $τ(n) \notin \{9,25,27,39,$ $75,81\}.$ We also consider the level 1 Hecke eigenforms in dimension 1 spaces of cusp forms. For example, for $ΔE_4 = \sum_{n = 1}^{\infty} τ_{16}(n)q^n$, we show that \begin{align*}τ_{16}(n) \notin &\{\pm \ell: 1\leq \ell \leq 99, \ell \text{ is odd}, \ell \neq 33,55,59,67,73,83,89,91\} \\ & \quad \quad \quad \quad \quad \cup \{-33,-55,-59,-67,-89,-91\}.\end{align*} Furthermore, we implement congruences given by Swinnerton-Dyer to rule out additional large primes which divide numerators of specific Bernoulli numbers. To obtain these results, we make use of the theory of Lucas sequences, methods for solving high degree Thue equations, Barros' algorithm for solving hyperelliptic equations, and the theory of continued fractions.
