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

Abstract

In this paper, we prove three results on the density, respectively, local density and clustering of zeros of the Beurling zeta function 𝜁(𝑠) close to the one-line 𝜎 ∶= ℜ𝑠 = 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 𝜁, not available in the Beurling context. Our result sharpens results of Kahane, who proved an 𝑂(𝑇) estimate for zeros lying precisely just on a vertical line ℜ𝑠 = 𝑎 in the critical strip. Next we deduce a variant of a well-known theorem of Turán, extending its range of validity even for rectangles of height only 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.

On Value Distribution of Certain Beurling Zeta-Functions

“…From this remark, it follows that P P (∂G ε ) > 0 for at most countably many values of ε, or, in the above terminology, the set G ε is a continuity set of the measure P P for all but at most countably many ε > 0. Thus, Theorem 2 and Theorem 2.1 of [28] with continuity sets show that the limit lim T→∞ P T,P (G ε ) = P P (G ε ) exists, and in view of (17), is positive for all but at most countably many ε > 0. This and the notations for P T,P and G ε give the second assertion of the theorem.…”

“…The paper [17] is devoted to zero-distribution of ζ P (s), where various zero-density results corresponding to those of ζ(s) are given. We stress that in [17], the Beurling prime number theorem [3] was strengthened, and it was proved that asymptotics (3) is implied by the estimate of Cesàro type…”

“…In the present paper, differently from the cited above works, including [14,17], that are devoted to prime number theorem, analytic continuation and zeros of ζ P (s), we focus on the approximation properties of the Beurling zeta-functions. More precisely, we consider the approximation of a set of analytic functions f (s) by shifts ζ P (s + iτ), τ ∈ R, i.e., such τ that, for some compact sets K and ε > 0, sup…”

On Universality of Some Beurling Zeta-Functions

“…It is well known that a function is completely determined by its Fourier transform. Thus, by (16), we have that for almost all ω ∈ Ω P , I A (ω) = c. However, as I A (ω) is the indicator function, it follows that c = 0 or 1. In other words, for almost all ω ∈ Ω P , I A (ω) = 0 or I A (ω) = 1.…”

“…Various authors put much effort into showing that the Beurling zeta-functions have similar properties to classical ones. We mention a recent paper [16] containing deep zero-distribution results for ζ P (s).…”