Remarks on the Mixed Joint Universality for A class of Zeta Functions

Kačinskaitė, Roma; Matsumoto, Kohji

doi:10.1017/s0004972716000733

Cited by 17 publications

(28 citation statements)

References 15 publications

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“…Proof. This is Lemma 5.2 of [9]. In [9] no proof is given, so here we present a proof (which is again standard; see, for example, Theorem 5 of [5]).…”

Section: Proof Of Theorems 3 Andmentioning

confidence: 97%

“…On We prove this theorem in a way whose principle has been already well-developed (see, for example, Theorem 4 of [11] or Lemma 5.1 of [9] as the continuous analogue). However our present situation includes a lot of parameters which may cause a trouble for the readers, so we will explain many details even if they are now standard.…”

Section: A Discrete Limit Theoremmentioning

confidence: 99%

“…In the series of previous works by the authors (see [8], [9], [10], [11] ), the tuple consisting from the wide class of the Matsumoto zeta-functions ϕ(s) and the periodic Hurwitz with coefficients c k such that c k = O k α+β +ε if all prime factors of k are large for every positive ε (for the comment, see [9]).…”

Section: Introductionmentioning

confidence: 99%

“…The novelty is that a wide collection of the periodic Hurwitz zeta-functions ζ (s, α j ; B jl ) will be constructed; here for each α j a collection of sequences B jl is attached. This type of general collection of periodic Hurwitz zeta-functions has been studied in several previous articles (such as [15], [19], [5] and [9]) in the continuous case, but it seems that there is no previous work in the discrete case. In fact, the results in the present article are discrete analogous of Theorem 4.2 from [9], but in [9] we assume the stronger hypothesis, that is the algebraic independence of the parameters α 1 , ..., α r .…”

Section: Introductionmentioning

confidence: 99%

“…Remark. As we already have noted, Theorems 3 and 4 are discrete analogues of Theorem 4.2 from [9], but here, instead of algebraic independence of the parameters α 1 , ..., α r , the linear independence of the elements of the sets L(α, h) and L(α, h) over the set of rational numbers Q are used, respectively.…”

Section: Introductionmentioning

confidence: 99%

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On mixed joint discrete universality for a class of zeta-functions. II

Kačinskaitė

Matsumoto

2019

Lith Math J

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“…Proof. This is Lemma 5.2 of [9]. In [9] no proof is given, so here we present a proof (which is again standard; see, for example, Theorem 5 of [5]).…”

Section: Proof Of Theorems 3 Andmentioning

confidence: 97%

Section: A Discrete Limit Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On mixed joint discrete universality for a class of zeta-functions. II

Kačinskaitė

Matsumoto

2019

Lith Math J

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Joint universality of Hurwitz zeta-functions and nontrivial zeros of the Riemann zeta-function

Macaitienė

Šiaučiūnas

2019

Lith Math J

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On the density function for the value-distribution of automorphic L-functions

Matsumoto

Umegaki

2019

Journal of Number Theory

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The Bohr-Jessen limit theorem is a probabilistic limit theorem on the value-distribution of the Riemann zeta-function in the critical strip. Moreover their limit measure can be written as an integral involving a certain density function. The existence of the limit measure is now known for a quite general class of zeta-functions, but the integral expression has been proved only for some special cases (such as Dedekind zeta-functions). In this paper we give an alternative proof of the existence of the limit measure for a general setting, and then prove the integral expression, with an explicitly constructed density function, for the case of automorphic L-functions attached to primitive forms with respect to congruence subgroups Γ0(N ). 2010 Mathematics Subject Classification. Primary 11F66, Secondary 11M41.This work is a milestone in the value-distribution theory of ζ(s), and various alternative proofs and related results have been pubished; for example, Jessen and Wintner [8], Borchsenius and Jessen [5], Guo [6], and Ihara and the first author [7].An important problem is to consider the generalization of the Bohr-Jessen theorem. The first author [11] proved that the formula (1.2) can be generalized to a fairly general class of zeta-functions with Euler products. However, (1.3) has not yet been generalized to such a general class. The reason is as follows.The original proof of (1.2) and (1.3) by Bohr and Jessen depends on a geometric theory of certain "infinite sums" of convex curves, developed by themselves [3]. In later articles [8] and [5], the effect of the convexity of curves was embodied in a certain inequality due to Jessen and Wintner [8, Theorem 13]. Using this method, the Bohr-Jessen theory was generalized to Dirichlet L-functions (Joyner [9]) and Dedekind zeta-functions of Galois number fields (the first author [12]). These generalizations are possible because these zeta-functions have "convex" Euler products in the sense of [11, Section 5]. But this convexity cannot be expected for more general zetafunctions.In [11], the first author developed a method of proving (1.2) without using any convexity, so succeeded in generalizing the theory. However, the method in [11] cannot give a generalization of (1.3).So far, there is no proof of (1.3) or its analogues without using the convexity, or the Jessen-Wintner type of inequalities. For example, [7] gives a different argument of constructing the density functions for Dirichlet L-functions, but the argument in [7] also depends on the Jessen-Wintner inequality.In [14] [15], the first author obtained certain quantitative results on the value-distribution of Dedekind zeta-functions of non-Galois fields and Hecke L-functions of ideal class characters, whose Euler products are not convex. But in these cases, they are "not so far" from the case of Dedekind zetafunctions of Galois fields. In fact, a simple generalization of the Jessen-Wintner inequality is proved ( [15, Lemma 2]) and is essentially used in the proof.Actually, analyzing the proof of [8, Theorems 12, 13...

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Remarks on the Mixed Joint Universality for A class of Zeta Functions

Cited by 17 publications

References 15 publications

On mixed joint discrete universality for a class of zeta-functions. II

On mixed joint discrete universality for a class of zeta-functions. II

Joint universality of Hurwitz zeta-functions and nontrivial zeros of the Riemann zeta-function

On the density function for the value-distribution of automorphic L-functions

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