On divisibility of the class number $h^+$ of the real cyclotomic fields $\mathbb {Q}(\zeta _p+\zeta _p^{-1})$ by primes $q < 10000$

Pavel Trojovský

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Displaying similar documents to “On divisibility of the class number $h^{+}$ of the real cyclotomic fields $ℚ (ζ_{p} + ζ_{p}^{- 1})$ by primes $q < 10000$ ”

On the class number of the maximal real subfields of a family of cyclotomic fields

Mahesh Kumar Ram (2023)

Czechoslovak Mathematical Journal

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For any square-free positive integer $m \equiv 10 (mod 16)$ with $m \geq 26$ , we prove that the class number of the real cyclotomic field $ℚ (ζ_{4 m} + ζ_{4 m}^{- 1})$ is greater than $1$ , where $ζ_{4 m}$ is a primitive $4 m$ th root of unity.

The size of the Lerch zeta-function at places symmetric with respect to the line $ℜ (s) = 1 / 2$

Ramūnas Garunkštis, Andrius Grigutis (2019)

Czechoslovak Mathematical Journal

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Let $ζ (s)$ be the Riemann zeta-function. If $t \geq 6.8$ and $σ > 1 / 2$ , then it is known that the inequality $| ζ (1 - s) | > | ζ (s) |$ is valid except at the zeros of $ζ (s)$ . Here we investigate the Lerch zeta-function $L (λ, α, s)$ which usually has many zeros off the critical line and it is expected that these zeros are asymmetrically distributed with respect to the critical line. However, for equal parameters $λ = α$ it is still possible to obtain a certain version of the inequality $| L (λ, λ, 1 - \bar{s}) | > | L (λ, λ, s) |$ .

Mean values related to the Dedekind zeta-function

Hengcai Tang, Youjun Wang (2024)

Czechoslovak Mathematical Journal

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Let $K / ℚ$ be a nonnormal cubic extension which is given by an irreducible polynomial $g (x) = x^{3} + a x^{2} + b x + c$ . Denote by $ζ_{K} (s)$ the Dedekind zeta-function of the field $K$ and $a_{K} (n)$ the number of integral ideals in $K$ with norm $n$ . In this note, by the higher integral mean values and subconvexity bound of automorphic $L$ -functions, the second and third moment of $a_{K} (n)$ is considered, i.e., $\sum_{n \leq x} a_{K}^{2} (n) = x P_{1} (log x) + O (x^{5 / 7 + ϵ}), \sum_{n \leq x} a_{K}^{3} (n) = x P_{4} (log x) + O (X^{321 / 356 + ϵ}),$ where $P_{1} (t)$ , $P_{4} (t)$ are polynomials of degree 1, 4, respectively, $ϵ > 0$ is an arbitrarily small number.

On some imaginary triquadratic number fields $k$ with ${Cl}_{2} (k) ≃ (2, 4)$ or $(2, 2, 2)$

Abdelmalek Azizi, Mohamed Mahmoud Chems-Eddin, Abdelkader Zekhnini (2021)

Commentationes Mathematicae Universitatis Carolinae

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Let $d$ be a square free integer and $L_{d} : = ℚ (ζ_{8}, \sqrt{d})$ . In the present work we determine all the fields $L_{d}$ such that the $2$ -class group, ${Cl}_{2} (L_{d})$ , of $L_{d}$ is of type $(2, 4)$ or $(2, 2, 2)$ .

On divisibility of $h^{+}$ by the prime $5$

Stanislav Jakubec (1994)

Mathematica Slovaca

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On the $2$ -class group of some number fields with large degree

Mohamed Mahmoud Chems-Eddin, Abdelmalek Azizi, Abdelkader Zekhnini (2021)

Archivum Mathematicum

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Let $d$ be an odd square-free integer, $m \geq 3$ any integer and $L_{m, d} : = ℚ (ζ_{2^{m}}, \sqrt{d})$ . In this paper, we shall determine all the fields $L_{m, d}$ having an odd class number. Furthermore, using the cyclotomic $ℤ_{2}$ -extensions of some number fields, we compute the rank of the $2$ -class group of $L_{m, d}$ whenever the prime divisors of $d$ are congruent to $3$ or $5 (mod 8)$ .

A class number criterion for the equation $(x^{p} - 1) / (x - 1) = p y^{q}$

Benjamin Dupuy (2007)

Acta Arithmetica

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Representation growth of linear groups

Michael Larsen, Alexander Lubotzky (2008)

Journal of the European Mathematical Society

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Let $Γ$ be a group and $r_{n} (Γ)$ the number of its $n$ -dimensional irreducible complex representations. We define and study the associated representation zeta function $𝒵_{Γ} (s) = \sum_{n = 1}^{\infty} r_{n} (Γ) n^{- s}$ . When $Γ$ is an arithmetic group satisfying the congruence subgroup property then $𝒵_{Γ} (s)$ has an “Euler factorization”. The “factor at infinity” is sometimes called the “Witten zeta function” counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite...

Displaying similar documents to “On divisibility of the class number h + of the real cyclotomic fields ℚ ( ζ p + ζ p - 1 ) by primes q < 10000 ”

Displaying similar documents to “On divisibility of the class number $h^{+}$ of the real cyclotomic fields $ℚ (ζ_{p} + ζ_{p}^{- 1})$ by primes $q < 10000$ ”