关于割线狄利克雷级数和Eisenstein级数的Eichler积分
On a secant Dirichlet series and Eichler integrals of Eisenstein series
摘要：
我们认为，甚至$s$，割线狄利克雷级数$ψ_s (τ) = \sum_{n = 1}^{\infty} \frac{\sec (πn τ)}{n^s}$，最近介绍和研究的Lal\'ın，Rodrigue和罗杰斯。特别是，我们表明，作为解释和部分证明Lal\'ın，Rodrigue和罗杰斯，值$ψ_{2 m} ( \sqrt{r})$，与$r > 0$理性，是合理的倍数$π^{2 m}$。然后，我们把割线狄利克雷级数的性质纳入上下文，表明他们是Eichler积分的奇数重量Eisenstein级数的水平$4$。这使我们考虑一般爱森斯坦级数的艾希勒积分，并确定其周期多项式。在水平$1$的情况下，这些多项式最近被Murty，Smyth和Wang证明了它们的大部分根在单位圆上。我们提供的证据表明，这种现象延伸到更高级别的情况下。这一观察补充了最近的结果由康利，农民和Imamoglu以及埃尔-Guindy和Raji的零点的Hecke特征型的周期多项式的水平$1$的情况下。最后，我们简要回顾了拉马努金的作品中类似类型的结果。
英文摘要：
We consider, for even $s$, the secant Dirichlet series $ψ_s (τ) = \sum_{n = 1}^{\infty} \frac{\sec (πn τ)}{n^s}$, recently introduced and studied by Lal\'ın, Rodrigue and Rogers. In particular, we show, as conjectured and partially proven by Lal\'ın, Rodrigue and Rogers, that the values $ψ_{2 m} ( \sqrt{r})$, with $r > 0$ rational, are rational multiples of $π^{2 m}$. We then put the properties of the secant Dirichlet series into context by showing that they are Eichler integrals of odd weight Eisenstein series of level $4$. This leads us to consider Eichler integrals of general Eisenstein series and to determine their period polynomials. In the level $1$ case, these polynomials were recently shown by Murty, Smyth and Wang to have most of their roots on the unit circle. We provide evidence that this phenomenon extends to the higher level case. This observation complements recent results by Conrey, Farmer and Imamoglu as well as El-Guindy and Raji on zeros of period polynomials of Hecke eigenforms in the level $1$ case. Finally, we briefly revisit results of a similar type in the works of Ramanujan.
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