论文标题
Beurling Zeta函数的Beurling Primes和零的分布I.
Distribution of Beurling primes and zeroes of the Beurling zeta function I. Distribution of the zeroes of the zeta function of Beurling
论文作者
论文摘要
我们证明了密度resp的三个结果。 Beurling Zeta函数零的局部密度和聚类$ζ（S）$接近单行$σ：= \ re s = 1 $。这里的分析带来了一些新闻，有时甚至是Riemann Zeta功能的经典案例。定理4提供了零密度估计值，这是Selberg类的已知结果的补充。请注意，Selberg类的密度结果取决于使用$ζ$的功能方程，我们在Beurling上下文中不假定。在定理5中，我们推断出特拉的著名定理的一种变体，即使仅针对高度的矩形$ h = 2 $，也扩展了其有效性范围。在定理6中，我们将从Riemann Zeta案例中扩展Ramachandra的零聚类结果。较弱的结果 - 另一方面，这是蒙哥马利经典书籍\ cite {Mont}的平均结果的强烈锐化 - 由Diamond，Montgomery和Vorhauer弄清楚。在这里，我们表明，Ramachandra纸的晦涩的技术（例如，具有$ 10^8 $等系数的多项式）可以摆脱，从而提供了这种集群现象的有效性的更透明的证明。
We prove three results on the density resp. local density and clustering of zeros of the Beurling zeta function $ζ(s)$ close to the one-line $σ:=\Re s=1$. The analysis here brings about some news, sometimes even for the classical case of the Riemann zeta function. Theorem 4 provides a zero density estimate, which is a complement to known results for the Selberg class. Note that density results for the Selberg class rely on use of the functional equation of $ζ$, which we do not assume in the Beurling context. In Theorem 5 we deduce a variant of a well-known theorem of Turán, extending its range of validity even for rectangles of height only $h=2$. In Theorem 6 we will extend a zero clustering result of Ramachandra from the Riemann zeta case. A weaker result -- which, on the other hand, is a strong sharpening of the average result from the classic book \cite{Mont} of Montgomery -- was worked out by Diamond, Montgomery and Vorhauer. Here we show that the obscure technicalities of the Ramachandra paper (like a polynomial with coefficients like $10^8$) can be gotten rid of, providing a more transparent proof of the validity of this clustering phenomenon.
