Distribution of Beurling primes and zeroes of the Beurling zeta function I. Distribution of the zeroes of the zeta function of Beurling

We prove three results on the density resp. local density and clustering of zeros of the Beurling zeta function $\zeta (s)$ close to the one-line $\sigma :=\mathrm{\Re }s=1$. 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 $\zeta$, which we do not assume in the Beurling context. In Theorem 5 we deduce a variant of a well-known theorem of Tur'an, 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.