Block distribution in random strings

Peter J. Grabner

Displaying similar documents to “Block distribution in random strings”

Non-vanishing of class group $L$ -functions at the central point

Valentin Blomer (2004)

Annales de l’institut Fourier

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Let $K = ℚ (\sqrt{- D})$ be an imaginary quadratic field, and denote by $h$ its class number. It is shown that there is an absolute constant $c > 0$ such that for sufficiently large $D$ at least $c \cdot h \prod_{p ∣ D} (1 - p^{- 1})$ of the $h$ distinct $L$ -functions $L_{K} (s, χ)$ do not vanish at the central point $s = 1 / 2$ .

On some vector balancing problems

Apostolos Giannopoulos (1997)

Studia Mathematica

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Let V be an origin-symmetric convex body in $ℝ^{n}$ , n≥ 2, of Gaussian measure $γ_{n} (V) \geq 1 / 2$ . It is proved that for every choice $u_{1}, . . ., u_{n}$ of vectors in the Euclidean unit ball $B_{n}$ , there exist signs $ε_{j} \in - 1, 1$ with $ε_{1} u_{1} + . . . + ε_{n} u_{n} \in (c l o g n) V$ . The method used can be modified to give simple proofs of several related results of J. Spencer and E. D. Gluskin.

On sums of Hecke series in short intervals

Aleksandar Ivić (2001)

Journal de théorie des nombres de Bordeaux

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We have $\sum_{K - G \leq k_{j} \leq K + G} α_{j} H_{j}^{3} (\frac{1}{2}) ≪_{ϵ} G K^{1 + ϵ}$ for $K^{ϵ} \leq G \leq K, where α_{j} = {|ρ_{j} (1)|}^{2} {(cosh π k_{j})}^{- 1}, and ρ_{j} (1)$ is the first Fourier coefficient of the Maass wave form corresponding to the eigenvalue $λ_{j} = k_{j}^{2} + \frac{1}{4}$ to which the Hecke series $H_{j} (s)$ is attached. This result yields the new bound $H_{j} (\frac{1}{2} ≪_{ϵ} k_{j}^{\frac{1}{3} + ϵ} .$

Oscillatory properties of $M (x) = \sum_{n \leq x} μ (n)$ , III

J. Pintz (1984)

Acta Arithmetica

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The ratio and generating function of cogrowth coefficients of finitely generated groups

Ryszard Szwarc (1998)

Studia Mathematica

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Let G be a group generated by r elements $g_{1}, \dots, g_{r}$ . Among the reduced words in $g_{1}, \dots, g_{r}$ of length n some, say $γ_{n}$ , represent the identity element of the group G. It has been shown in a combinatorial way that the 2nth root of $γ_{2 n}$ has a limit, called the cogrowth exponent with respect to the generators $g_{1}, \dots, g_{r}$ . We show by analytic methods that the numbers $γ_{n}$ vary regularly, i.e. the ratio $γ_{2 n + 2} / γ_{2 n}$ is also convergent. Moreover, we derive new precise information on the domain of holomorphy of γ(z), the generating function...

Oscillatory properties of $M (x) = \sum_{n \leq x} μ (n)$ , I

J. Pintz (1982)

Acta Arithmetica

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On the sequence of numbers of the form $ε ₀ + ε ₁ q + . . . + ε_{n} q^{n}$ , $ε_{i} \in 0, 1$

Paul Erdős, István Joó, Vilmos Komornik (1998)

Acta Arithmetica

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On an approximation property of Pisot numbers II

Toufik Zaïmi (2004)

Journal de Théorie des Nombres de Bordeaux

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Let $q$ be a complex number, $m$ be a positive rational integer and $l_{m} (q) = inf {|P (q)|, P \in ℤ_{m} [X], P (q) \neq 0}$ , where $ℤ_{m} [X]$ denotes the set of polynomials with rational integer coefficients of absolute value $\leq m$ . We determine in this note the maximum of the quantities $l_{m} (q)$ when $q$ runs through the interval $] m, m + 1 [$ . We also show that if $q$ is a non-real number of modulus $> 1$ , then $q$ is a complex Pisot number if and only if $l_{m} (q) > 0$ for all $m$ .