论文标题
莱默对新形式的猜测的变体
Variants of Lehmer's speculation for newforms
论文作者
论文摘要
本着莱默（Lehmer）对Ramanujan tau功能的不变猜测的精神，很自然地询问固定整数是$τ（n）$还是傅立叶系数$ a_f（n）$ a_f（n）$的任何给定的newform $ f（z）$。我们提供了一种方法，该方法适用于具有整数系数和琐碎的残留Mod 2 Galois表示形式的新信息，该方法为奇数整数回答了这个问题。我们确定无限的许多空间，这些空间$ 3 \ leq \ ell \ leq 37 $不是具有整数系数的新形式系数的绝对值。对于带有$ n> 1 $的$τ（n）$，我们证明$$τ（n）\ in \ in \ {\ pm 1，\ pm 3，\ pm 5，\ pm 7，\ pm 7，\ pm 13，\ pm 13，\ pm 17，-19，-19，\ pm 23，\ pm 23，\ pm 37，\ pm 37，\ pm 691 \}，$ $ $ $ $τ（n）\ in \ in \ left \ {\ pm \ ell \ \：\ 41 \ leq \ ell \ el \ leq 97 \ {\ textrm {with}}}} \ \ \ \ \ lest（\ frac {\ ell} {\ ell} -31，-41，-59，-61，-71，-79，-89 \ right \}。 $$我们还获得了这种新型系数的主要因素数量的尖锐下限。在重量方面，对于奇数$ \ ell $的功能，我们证明$ \ pm \ ell^m $不是任何此类新形式的$ f $的系数，重量$ 2K> 2k> m^{\ pm}（\ ell，m）= o _ {\ ell}（\ ell}（\ ell}（m）$ el copl copr and $ el $ el y是$ mmmmmmmmmmmmmmm^\ f \ m）{可计算。
In the spirit of Lehmer's unresolved speculation on the nonvanishing of Ramanujan's tau-function, it is natural to ask whether a fixed integer is a value of $τ(n)$ or is a Fourier coefficient $a_f(n)$ of any given newform $f(z)$. We offer a method, which applies to newforms with integer coefficients and trivial residual mod 2 Galois representation, that answers this question for odd integers. We determine infinitely many spaces for which the primes $3\leq \ell\leq 37$ are not absolute values of coefficients of newforms with integer coefficients. For $τ(n)$ with $n>1$, we prove that $$τ(n)\not \in \{\pm 1, \pm 3, \pm 5, \pm 7, \pm 13, \pm 17, -19, \pm 23, \pm 37, \pm 691\},$$ and assuming GRH we show for primes $\ell$ that $$τ(n)\not \in \left \{ \pm \ell\ : \ 41\leq \ell\leq 97 \ {\textrm{with}}\ \left(\frac{\ell}{5}\right)=-1\right\} \cup \left \{ -11, -29, -31, -41, -59, -61, -71, -79, -89\right\}. $$ We also obtain sharp lower bounds for the number of prime factors of such newform coefficients. In the weight aspect, for powers of odd primes $\ell$, we prove that $\pm \ell^m$ is not a coefficient of any such newform $f$ with weight $2k>M^{\pm}(\ell,m)=O_{\ell}(m)$ and even level coprime to $\ell,$ where $M^{\pm}(\ell,m)$ is effectively computable.
