Mostrando recursos 1 - 20 de 400

  1. Convolutions with probability distributions, zeros of $L$-functions, and the least quadratic nonresidue

    Banks, William D.
    Let ${\tt d}$ be the density of a probability distribution that is compactly supported in the positive semi-axis. Under certain mild conditions we show that $$ \lim_{x\to\infty}x\sum_{n=1}^\infty \frac{{\tt d}^{*n}(x)}{n}=1,\qquad\text{where}\quad {\tt d}^{*n}:=\underbrace{{\tt d} *{\tt d}*\cdots*{\tt d}}_{n\text{~times}}. $$ We also show that if $c>0$ is a given constant for which the function $f(k):=\widehat{\tt d}(k)-1$ does not vanish on the line $\{k\in\mathbb{C}:\Im k=-c\}$, where $\widehat{\tt d}$ is the Fourier transform of ${\tt d}$, then one has the asymptotic expansion $$ \sum_{n=1}^\infty\frac{{\tt d}^{*n}(x)}{n}=\frac{1}{x}\bigg(1+\sum_k m(k) e^{-ikx}+O(e^{-c x})\bigg)\qquad (x\to +\infty), $$ where the sum is taken over those zeros $k$ of $f$ that lie in the strip $\{k\in\mathbb{C}:-c<\Im k<0\}$, $m(k)$ is the multiplicity of any such zero, and the implied constant depends only on $c$....

  2. Functional equation for the Mordell-Tornheim multiple zeta-function

    Okamoto, Takuya; Onozuka, Tomokazu
    We show a relation between the Mordell-Tornheim multiple zeta-function and the confluent hypergeometric function, and using it, we give the functional equation for the Mordell-Tornheim multiple zeta-function. In the double case, the functional equation includes the known functional equation for the Euler-Zagier double zeta-function.

  3. Resolving Grosswald's conjecture on GRH

    McGown, Kevin; Treviño, Enrique; Trudgian, Tim
    In this paper we examine Grosswald's conjecture on $g(p)$, the least primitive root modulo $p$. Assuming the Generalized Riemann Hypothesis (GRH), and building on previous work by Cohen, Oliveira e Silva and Trudgian, we resolve Grosswald's conjecture by showing that $g(p)< \sqrt{p} - 2$ for all $p>409$. Our method also shows that under GRH we have $\hat{g}(p)< \sqrt{p}-2$ for all $p>2791$, where $\hat{g}(p)$ is the least prime primitive root modulo $p$.

  4. Counting lattice points in certain rational polytopes and generalized Dedekind sums

    Kozuka, Kazuhito
    Let ${\mathcal P} \subset {\mathbf R}^n$ be a rational convex polytope with vertices at the origin and on each positive coordinate axes. On the basis of the study on counting lattice points in $t{\mathcal P}$ with positive integer $t$, which is deeply connected with reciprocity laws for generalized Dedekind sums, we study the number of lattice points in the shifted polytope of $t{\cal P}$ by a fixed rational point. Certain generalized multiple Dedekind sums appear naturally in the main result.

  5. An explicit result for primes between cubes

    Dudek, Adrian W.
    We prove that there is a prime between $n^3$ and $(n+1)^3$ for all $n \geq \exp(\exp(33.3))$. This is done by first deriving the Riemann--von Mangoldt explicit formula for the Riemann zeta-function with explicit bounds on the error term. We use this along with other recent explicit estimates regarding the zeroes of the Riemann zeta-function to obtain the result. Furthermore, we show that there is a prime between any two consecutive $m$th powers for $m \geq 5 \times 10^9$. Notably, many of the explicit estimates developed in this paper can also find utility elsewhere in the theory of numbers.

  6. $abc$ triples

    Martin, Greg; Miao, Winnie
    The $abc$ conjecture, one of the most famous open problems in number theory, claims that three relatively prime positive integers $a,b,c$ satisfying $a+b=c$ cannot simultaneously have significant repetition among their prime factors; in particular, the product of the distinct primes dividing the three integers should never be much less than $c$. Triples of relatively prime numbers satisfying $a+b=c$ are called {\em $abc$ triples} if the product of their distinct prime divisors is strictly less than $c$. We catalog what is known about $abc$ triples, both numerical examples found through computation and infinite familes of examples established theoretically. In addition, we...

  7. Polynomials with polar derivatives

    Mir, Abdullah; Wani, Ajaz
    In this paper, we present an integral inequality for the polar derivative of polynomials. Our theorem includes as special cases several interesting generalizations of some Zygmund type inequalities for polynomials.

  8. Small fractional parts of polynomials

    Baker, Roger
    Let $k \ge 6$. Using the recent result of Bourgain, Demeter, and Guth \cite{1586:bdg} on the Vinogradov mean value, we obtain new bounds for small fractional parts of polynomials $\alpha_kn^k + \cdots + \alpha_1n$ and additive forms $\beta_1n_1^k + \cdots + \beta_sn_s^k$. Our results improve earlier theorems of Danicic (1957), Cook (1972), Baker (1982, 2000), Vaughan and Wooley (2000), and Wooley (2013).

  9. On extended Eulerian numbers

    Bayad, Abdelmejid; Hernane, Mohand Ouamar; Togbé, Alain
    In this paper, we will study some properties of the extended Eulerian numbers $H(n,\lambda)$, with a parameter $\lambda$. In fact, for any integer $n$, we investigate the asymptotic behavior, find lower and upper bounds for $H(n,\lambda)$. As application, for a champion number $N$, we obtain asymptotic formulas, lower and upper bounds of the arithmetic functions $\omega(N)$ and $\Omega(N)$.

  10. The Picard groups of the stacks $\mathscr{Y}_0(2)$ and $\mathscr{Y}_0(3)$

    Niles, Andrew
    We compute the Picard group of the stack of elliptic curves equipped with a cyclic subgroup of order two, and of the stack of elliptic curves equipped with a cyclic subgroup of order three, over any base scheme on which $6$ is invertible. This generalizes a result of Fulton-Olsson, who computed the Picard group of the stack of elliptic curves (with no level structure) over a~wide variety of base schemes.

  11. Exact divisors of polynomials with prime variable

    Scourfield, Eira J.
    In 1952 Paul Erdős obtained upper and lower bounds of the same order of magnitude for the number $N(x)$ of divisors of an irreducible polynomial $f(n)$ with integer coefficients for $n$ up to $x$; an asymptotic formula for $N(x)$ when $f$ has degree at least $3$ has not yet been established. However progress has been made in the corresponding problem when the divisors of $f(n)$ are restricted in some way and $f$ is not necessarily irreducible. In this paper we consider a~polynomial $f$ $ $with integer coefficients that may not be irreducible or squarefree. Our aim is to obtain an asymptotic formula for the number of exact divisors...

  12. Arithmetic local constants for abelian varieties with extra endomorphisms

    Chetty, Sunil
    This work generalizes the theory of arithmetic local constants, introduced by Mazur and Rubin, to better address abelian varieties with a larger endomorphism ring than $\mathbb{Z}$. We then study the growth of the $p^\infty$-Selmer rank of our abelian variety, and we address the problem of extending the results of Mazur and Rubin to dihedral towers $k\subset K\subset F$ in which $[F:K]$ is not a $p$-power extension.

  13. Nonvanishing and central critical values of twisted $L$-functions of cusp forms on average

    Schwagenscheidt, Markus
    Let $f$ be a cusp form of integral weight $k \geq 4$ for $\Gamma_{0}(N)$ with nebentypus $\psi$. Generalising work of Kohnen we construct a kernel function for the $L$-function $L(f,\chi,s)$ of $f$ twisted by a primitive Dirichlet character $\chi$ and use it to show that the average $\sum_{f \in S_{k}(N,\psi)}\frac{L(f,\chi,s)}{\langle f,f\rangle}\overline{a_{f}(1)}$ over an orthogonal basis of $S_{k}(N,\psi)$ does not vanish on certain rectangles inside the critical strip if the weight $k$ or the level $N$ is big enough. As another application of the kernel function we prove an averaged version of Waldspurger's Theorem.

  14. Mean square formula for the double zeta-function

    Kiuchi, Isao; Minamide, Makoto
    We prove the mean square formula of the Euler--Zagier type double zeta-function $\zeta_{2}(s_{1},s_{2})$ and provide an improvement on the $\Omega$ results of Kiuchi, Tanigawa, and Zhai. We also calculate the double integral $\int_{2}^{N}\int_{2}^{T}|\zeta_{2}(s_{1},s_{2})|^{2}dt_{1} dt_{2}$ under certain conditions.

  15. Images of polynomial maps on ample fields

    Kosters, Michiel
    In this article we study the following problem. Let $k$ be an infinite field and let $f \in k[x]$. Consider the evaluation map $f_k: k \to k$. Assume that $f_k$ is not surjective. Is $k \smallsetminus f_k(k)$ infinite? We give a positive answer to this question when $k$ is a perfect ample field. In fact, we prove that $|k \smallsetminus f_k(k)|=|k|$. This conclusion follows from a similar statement about finite morphisms between normal projective curves over perfect ample fields.

  16. Characterization of Sporadic perfect polynomials over $\mathbb{F}_2$

    Gallardo, Luis H.; Rahavandrainy, Olivier
    We complete, in this paper, the characterization of all known even perfect polynomials over the prime field $\mathbb{F}_2$. In particular, we prove that the last two of the eleven known ``sporadic'' perfect polynomials over $\mathbb{F}_2$ are the unique of them of the form $x^a(x+1)^b M^{2h} \sigma(M^{2h})$, where $M$ is a Mersenne prime and $a,b, h \in \mathbb{N}^*$.

  17. Level stripping for vector-valued Siegel modular forms of genus 2

    Keaton, Rodney
    In this paper, we present a method by which one can strip primes from the level of a vector-valued genus 2 Siegel modular form while preserving a congruence modulo this prime. An application of this result to four-dimensional Galois representations will also be presented.

  18. Comparing local constants of ordinary elliptic curves in dihedral extensions

    Chetty, Sunil
    We establish, for a substantial class of elliptic curves, that the arithmetic local constants introduced by Mazur and Rubin agree with quotients of analytic root numbers.

  19. Classes logarithmiques et capitulation

    Jaulent, Jean-François
    We study a logarithmic version of the classical result of Artin-Furwängler on principalization of ideal classes in the Hilbert class-field by applying the group theoretic description of the transfert map to logarithmic class-groups of degree 0.

  20. The amplification method in the context of $GL(n)$ automorphic forms

    Ricotta, Guillaume
    In [SV] and [BMb], the authors proved the existence of a so-called higher rank amplifier and in [HRRa], the authors described an explicit version of a $GL(3)$ amplifier. This article provides, for $n\geq 4$, a totally explicit $GL(n)$ amplifier and gives all the results required to use it effectively.

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