Universality and distribution of zeros and poles of some zeta functions
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KRISTIAN SEIP∗ Dedicated to Lawrence Zalcman with admiration −s Abstract. This paper studies zeta functions of the form ∞ n=1 χ(n)n , with χ a completely multiplicative function taking only unimodular values. We denote −s can by σ(χ) the infimum of those α such that the Dirichlet series ∞ n=1 χ(n)n be continued meromorphically to the half-plane Re s > α, and denote by ζχ (s) the corresponding meromorphic function in Re s > σ(χ). We construct ζχ (s) that have σ(χ) ≤ 1/2 and are universal for zero-free analytic functions on the half-critical strip 1/2 < Re s < 1, with zeros and poles at any discrete multisets lying in a strip to the right of Re s = 1/2 and satisfying a density condition that is somewhat stricter than the density hypothesis for the zeros of the Riemann zeta function. On a conceivable version of Cram´er’s conjecture for gaps between primes, the density condition can be relaxed, and zeros and poles can also be placed at β + iγ with β ≤ 1 − λ log log |γ|/ log |γ| when λ > 1. Finally, we show that there exists ζχ (s) with σ(χ) ≤ 1/2 and zeros at any discrete multiset in the strip 1/2 < Re s ≤ 39/40 with no accumulation point in Re s > 1/2; on the Riemann hypothesis, this strip may be replaced by the half-critical strip 1/2 < Re s < 1.
1
Introduction
1.1 Background. This paper centers around Bohr’s approach to the Riemann hypothesis, originating in his discovery [6] that in any sub-strip of 1/2 < Re s < 1, the set of points s at which the Riemann zeta function ζ(s) takes the value a for a fixed complex number a = 0, has positive lower density. In view of the Bohr–Landau theorem [7] on the density of the zeros of ζ(s) to the right of the critical line, this cannot be true for a = 0. Hence, as concluded by Titchmarsh in [25, Ch. 11] , “...the value 0 of ζ(s), if it occurs at all in σ > 1/2, is at any rate quite exceptional, zeros being infinitely rarer than a-values for any value of a other than zero.” It seems that this state of affairs led Bohr and others to believe in the unlikeliness of such “exceptional” zeros and that the Riemann hypothesis could ∗
Research supported in part by Grant 275113 of the Research Council of Norway.
331 ´ JOURNAL D’ANALYSE MATHEMATIQUE, Vol. 141 (2020) DOI 10.1007/s11854-020-0126-3
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K. SEIP
be proved by establishing that ζ(s) is quasi-periodic in an appropriate sense in the strip 1/2 < Re s < 1. While the Riemann hypothesis is indeed equivalent to an assertion about quasi-periodicity, as proved by Bagchi [1] (see Theorem B below), our aim is to show that there exist zeta functions with zeros located essentially anywhere in a strip to the right of Re s = 1/2, subject to a density restriction akin to the density hypothesis for the zeros of the Riemann zeta function, and whose value distribution properties otherwise cannot be easily distinguished from those of ζ(s). The zeta functions that we will consider are of the form (1.1)
ζχ (s) :=
∞ n=1
χ(n)n−s =
p
1 , (1 − χ(p)p−s)
where χ is a completely multiplicative function taking only unimodular values and
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