Abstract For 1 ≤ i ≤ r , let χ i be primitive Dirichlet characters modulo q i and Z ( t , χ i ) be the Z-function corresponding to the Dirichlet L-series L ( s , χ i ) . Let Ω ( t ) be a real linear combination of Z ( t , χ i ) . Since Z ( t , χ i ) is real for real t, Ω ( t ) is real for real t. In this paper, we show that the Lebesgue measure of the set, where the functional values of Ω ( t ) is positive or negative in the interval [ T , 2 T ] is at least T r 2 . We also study the Lebesgue measure of the set that the certain complex linear combinations of Z ( t , χ i ) takes positive or negative values respectively. In particular, we study the distribution of signs of the Z-function correspond to the Davenport-Heilbronn function. Moreover, we prove that for sufficiently large T, the generalized Davenport-Heilbronn function has at least H ( log T ) 2 φ ( q ) − ϵ odd order zeros along the critical line on the interval [ T , T + H ] .

中文翻译：

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Karatsuba 和广义 Davenport-Heilbronn Z 函数的符号分布

摘要 对于 1 ≤ i ≤ r ，设 χ i 为以 qi 为模的原始狄利克雷字符，Z ( t , χ i ) 为对应于狄利克雷 L 级数 L ( s , χ i ) 的 Z 函数。令Ω (t) 是Z (t, χ i) 的实数线性组合。由于 Z ( t , χ i ) 对于实数 t 是实数，Ω ( t ) 对于实数 t 是实数。在本文中，我们展示了该集合的 Lebesgue 测度，其中 Ω ( t ) 在区间 [ T , 2 T ] 中为正或为负的函数值至少为 T r 2 。我们还研究了 Z ( t , χ i ) 的某些复线性组合分别取正值或负值的集合的 Lebesgue 测度。特别是，我们研究了对应于 Davenport-Heilbronn 函数的 Z 函数的符号分布。此外，我们证明对于足够大的 T，