ZEROS OF EISENSTEIN SERIES
ZEROS OF EISENSTEIN SERIES
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艾森斯坦级数的零点
DOI：
10.2206/kyushujm.58.251
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发表时间：
2004
影响因子：
0.4
通讯作者：
W. Kohnen
中科院分区：
文献类型：
--
作者：
W. Kohnen
是关于1的权重为k的归一化爱森斯坦级数：= SL 2（Z），其中求和延伸到整数c和d的所有互质对上，并且H表示复上半平面。在[2]中，兰金和Swinnerton-Dyer令人惊讶地证明了1作用于H的标准基本域中Ek的所有零点都位于单位圆上。这个结果后来被兰金[3]推广到某些庞加莱级数，并被Asai等人[1]推广到函数j − 744及其在通常的Hecke算子下的图像;这里j是经典的模不变量。就我们所知，上述现象似乎还没有被完全理解。本文给出了严格在ρ：= e2πi/3和i之间的单位圆上Ek的零点的一个封闭公式，它是用包含Ek的Fourier系数的无穷级数，即Bernoulli数和幂因子函数表示的.为了得到这个公式，我们将使用[2]的结果和经典的詹森公式，该公式表示全纯函数的对数模在圆内的积分，该函数位于圆内的零点的对数模。注意，与模形式有关的詹森公式似乎是由Rohrlich [4]首先使用的。我们的结果似乎可以推广到[1]和[3]中所研究的模函数。
be the normalized Eisenstein series of weight k with respect to 1 := SL2(Z), where the summation extends over all coprime pairs of integers c and d and H denotes the complex upper half-plane. In [2], Rankin and Swinnerton-Dyer showed that surprisingly all the zeros of Ek in the standard fundamental domain for the action of 1 on H lie on the unit circle. This result was generalized later by Rankin [3] to certain Poincaré series and by Asai et al [1] to the function j − 744 and its images under the usual Hecke operators; here j is the classical modular invariant. The above phenomenon—as far as we can say—does not seem to be yet fully understood. In the present paper, we shall give a closed formula for the zeros of Ek on the unit circle strictly between ρ := e2πi/3 and i, in terms of an infinite series involving the Fourier coefficients of Ek , i.e. Bernoulli numbers and power divisor functions. To obtain this formula, we shall use the results of [2] and the classical Jensen formula which expresses the integral around a circle of the log modulus of a holomorphic function in terms of the log modulus of the zeros of that function lying inside the circle. Note that Jensen’s formula in connection with modular forms seems to have been used first by Rohrlich [4]. It seems possible to generalize our result to the modular functions studied in [1] and [3].