Limit Theorems for the Riemann Zeta-Function

Abstract. We show that the set obtained by adding all sufficiently large integers to a fixed quadratic algebraic number is multiplicatively dependent. So also is the set obtained by adding rational numbers to a fixed cubic algebraic number. Similar questions for algebraic numbers of higher degrees are also raised. These are related to the ProuhetTarry-Escott type problems and can be applied to the zero-distribution and universality of some zeta-functions.

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“…More generally, for p = −6t 2 , q = ε − 8t 3 , where t ∈ Q, we found (α + t) 6 (α − 3t) 2 = (α + ε(3t 4 + r)) 2 .…”

Section: Identities For Quartic Algebraic Numbersmentioning

confidence: 68%

Multiplicative dependence of shifted algebraic numbers

Drungilas

Dubickas

2003

Colloq. Math.

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“…Consequently, the first hypothesis of Theorem 6.3.10 of [5] is also satisfied, and we have that the set of all convergent series (5) is dense in H(D V ). As was noted above, this gives the denseness of all convergent series (4).…”

Section: Lemma 7 the Support Of The Random Element ϕ ϕ (S ω; F) Is mentioning

confidence: 72%

“…Hence by Theorem 1.7.10 of [5] the support of the random element ϕ ϕ (s, ω; F) is the closure of the set of all convergent series…”

Section: Lemma 7 the Support Of The Random Element ϕ ϕ (S ω; F) Is mentioning

confidence: 97%

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On the Derivatives of Zeta-Functions of Certain Cusp Forms

Laurinčikas¹

2005

Glasgow Math. J.

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“…The study of the value distribution of ζ(σ + it) for fixed σ and variable t ∈ R was initiated by H. Bohr. Bohr and Courant [BC] have shown that the set {ζ(σ + it) | t ∈ R} is dense in C for any fixed σ ∈ R with 1/2 < σ ≤ 1 (see also [Bo] where m denotes the Lebesgue measure on R.We note that this form is equivalent to that of [La, Theorem 6.5 There are other types of universality theorems, in which parameters other than t as in (1 …”

mentioning

confidence: 96%

Functional distribution of 𝐿(𝑠,𝜒_{𝑑}) with real characters and denseness of quadratic class numbers

Mishou

Nagoshi

2006

Trans. Amer. Math. Soc.

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Abstract. We investigate the functional distribution of L-functions L(s, χ d ) with real primitive characters χ d on the region 1/2 < Re s < 1 as d varies over fundamental discriminants. Actually we establish the so-called universality theorem for L(s, χ d ) in the d-aspect. From this theorem we can, of course, deduce some results concerning the value distribution and the non-vanishing. As another corollary, it follows that for any fixed a, b with 1/2 < a < b < 1 and positive integers r , m, there exist infinitely many d such that for every r = 1, 2, · · · , r the r-th derivative L (r) (s, χ d ) has at least m zeros on the interval [a, b] in the real axis. We also study the value distribution of L(s, χ d ) for fixed s with Re s = 1 and variable d, and obtain the denseness result concerning class numbers of quadratic fields. Before stating our theorems, we recall some related results on the Riemann zetafunction ζ(s). The study of the value distribution of ζ(σ + it) for fixed σ and variable t ∈ R was initiated by H. Bohr. Bohr and Courant [BC] have shown that the set {ζ(σ + it) | t ∈ R} is dense in C for any fixed σ ∈ R with 1/2 < σ ≤ 1 (see also [Bo] where m denotes the Lebesgue measure on R.We note that this form is equivalent to that of [La, Theorem 6.5 There are other types of universality theorems, in which parameters other than t as in (1

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Limit Theorems for the Riemann Zeta-Function

Cited by 309 publications

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Multiplicative dependence of shifted algebraic numbers

Multiplicative dependence of shifted algebraic numbers

On the Derivatives of Zeta-Functions of Certain Cusp Forms

Functional distribution of 𝐿(𝑠,𝜒_{𝑑}) with real characters and denseness of quadratic class numbers

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