Zeros of Fekete polynomials

Brian Conrey; Andrew Granville; Bjorn Poonen; K. Soundararajan

Displaying similar documents to “Zeros of Fekete polynomials”

Boundedness of Marcinkiewicz functions.

Minako Sakamoto, Kôzô Yabuta (1999)

Studia Mathematica

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The $L^{p}$ boundedness(1 < p < ∞) of Littlewood-Paley’s g-function, Lusin’s S function, Littlewood-Paley’s $g *_{λ}$ -functions, and the Marcinkiewicz function is well known. In a sense, one can regard the Marcinkiewicz function as a variant of Littlewood-Paley’s g-function. In this note, we treat counterparts $μ_{S}^{ϱ}$ and $μ_{λ}^{*, ϱ}$ to S and $g *_{λ}$ . The definition of $μ_{S}^{ϱ} (f)$ is as follows: $μ_{S}^{ϱ} (f) (x) = (ʃ_{| y - x | < t} | 1 / t^{ϱ} ʃ_{| z | \leq t} Ω (z) / {(| z |}^{n - ϱ} {) f (y - z) d z |}^{2} (d y d t) / (t^{n + 1}))^{1 / 2}$ , where Ω(x) is a homogeneous function of degree 0 and Lipschitz continuous of order β (0 < β ≤ 1) on the unit sphere $S^{n - 1}$ , and...

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.

The distribution of the values of a rational function modulo a big prime

Alexandru Zaharescu (2003)

Journal de théorie des nombres de Bordeaux

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Given a large prime number $p$ and a rational function $r (X)$ defined over $𝔽_{p} = ℤ / p ℤ$ , we investigate the size of the set $\{x \in 𝔽_{p} : \tilde{r} (x) > \tilde{r} (x + 1)$ , where $\tilde{r} (x)$ and $\tilde{r} (x + 1)$ denote the least positive representatives of $r (x)$ and $r (x + 1)$ in $ℤ$ modulo $p ℤ$ .

Analytic continuation of fundamental solutions to differential equations with constant coefficients

Christer O. Kiselman (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

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If $P$ is a polynomial in $R^{n}$ such that $1 / P$ integrable, then the inverse Fourier transform of $1 / P$ is a fundamental solution $E_{P}$ to the differential operator $P (D)$ . The purpose of the article is to study the dependence of this fundamental solution on the polynomial $P$ . For $n = 1$ it is shown that $E_{P}$ can be analytically continued to a Riemann space over the set of all polynomials of the same degree as $P$ . The singularities of this extension are studied.

Polydisc slicing in $ℂ^{n}$

Krzysztof Oleszkiewicz, Aleksander Pełczyński (2000)

Studia Mathematica

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Let D be the unit disc in the complex plane ℂ. Then for every complex linear subspace H in $ℂ^{n}$ of codimension 1, $v o l_{2 n - 2} (D^{n - 1}) \leq v o l_{2 n - 2} (H \cap D^{n}) \leq 2 v o l_{2 n - 2} (D^{n - 1})$ . The lower bound is attained if and only if H is orthogonal to the versor $e_{j}$ of the jth coordinate axis for some j = 1,...,n; the upper bound is attained if and only if H is orthogonal to a vector $e_{j} + σ e_{k}$ for some 1 ≤ j < k ≤ n and some σ ∈ ℂ with |σ| = 1. We identify $ℂ^{n}$ with $ℝ^{2 n}$ ; by $v o l_{k} (\cdot)$ we denote the usual k-dimensional volume in $ℝ^{2 n}$ . The result is a complex counterpart of Ball’s [B1]...

The k-functions in multiplicative number theory. II. Uniform distribution of zeta zeros

J. Kaczorowski (1990)

Acta Arithmetica

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On the best constant in the Khinchin-Kahane inequality

Rafał Latała, Krzysztof Oleszkiewicz (1994)

Studia Mathematica

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We prove that if $r_{i}$ is the Rademacher system of functions then ${(ʃ ∥ \sum_{i = 1}^{n} x_{i} r_{i} (t) ∥^{2} d t)}^{1 / 2} \leq \sqrt 2 ʃ ∥ \sum_{i = 1}^{n} x_{i} r_{i} (t) ∥ d t$ for any sequence of vectors $x_{i}$ in any normed linear space F.

Non-vanishing of $n$ -th derivatives of twisted elliptic $L$ -functions in the critical point

Jacek Pomykała (1997)

Journal de théorie des nombres de Bordeaux

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Let $E$ be a modular elliptic curve over $ℚ$ $L^{(n)} (s, E)$ denote the $n$ -th derivative of its Hasse-Weil $L$ -series. We estimate the number of twisted elliptic curves $E_{d}, d \leq D$ such that $L^{(n)} (1, E_{d}) \neq 0$ .

On the exceptional set for the $2 k$ -twin primes problem

Alberto Perelli, János Pintz (1992)

Compositio Mathematica

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