Density estimates for the zeros of the Beurling zeta function in the critical strip

In this paper, we prove three results on the density, respectively, local density and clustering of zeros of the Beurling zeta function zeta(s)$\zeta (s)$ close to the one-line sigma:=Rs=1$\sigma :=\Re s=1$. The analysis here brings about some news, sometimes even for the classical case of the Riemann zeta function. As a complement to known results for the Selberg class, first we prove a Carlson type zero density estimate. Note that density results for the Selberg class rely on use of the functional equation of zeta, not available in the Beurling context. Our result sharpens results of Kahane, who proved an O(T)$O(T)$ estimate for zeros lying precisely just on a vertical line Rs=a$\Re s=a$ in the critical strip. Next we deduce a variant of a well-known theorem of Turan, extending its range of validity even for rectangles of height only h=2$h=2$. Finally, we 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 of Montgomery - was worked out by Diamond, Montgomery and Vorhauer. On our way, we show that some obscure technicalities of the Ramachandra paper can be avoided