Value distribution for the derivatives of the logarithm of $L$-functions from the Selberg class in the half-plane of absolute convergence
Value distribution for the derivatives of the logarithm of $L$-functions from the Selberg class in the half-plane of absolute convergence
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Selberg 类 $L$ 函数的对数导数在绝对收敛半平面上的值分布
DOI：
10.1016/j.jmaa.2015.08.003
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发表时间：
2016
影响因子：
1.3
通讯作者：
Lukasz Pankowski
中科院分区：
文献类型：
--
作者：
Takashi Nakamura;Lukasz Pankowski
在本文中，我们证明了对于每一个δ> 0，当m∈N∪{0}且L (s):=∑N = 1∞a (N) N−s是Selberg类s的一个元素时，函数(log (L) L (s))(m)在条带1< Re (s)< 1+ δ上取任意无穷常值，只要∑p≤x| a (p)| 2 ~ κ π (x)对于某个κ> 0。特别地，L (s)在条带1< Re (s)< 1+ δ中取任意非零值，且L (s)的一阶导数在半平面Re (s)> 1中有无限多个零。
In the present paper, we show that, for every δ> 0, the function (log L (s))(m), where m∈ N∪{0} and L (s):=∑ n= 1∞ a (n) n− s is an element of the Selberg class S, takes any value infinitely often in the strip 1< Re (s)< 1+ δ, provided∑ p≤ x| a (p)| 2∼ κ π (x) for some κ> 0. In particular, L (s) takes any non-zero value infinitely often in the strip 1< Re (s)< 1+ δ, and the first derivative of L (s) has infinitely many zeros in the half-plane Re (s)> 1.