Mostrando recursos 1 - 20 de 36

  1. The minimal number of monochromatic Schur tuples in a cyclic group

    Taczała, Katarzyna
    We discuss a question of Datskovsky [1] about the minimal number of solutions to Schur-type equation $x_1+\ldots+x_{2n-1}=x_{2n}$ in a cyclic group $\mathbb{Z}_N$. We provide lower and upper bounds for this quantity.

  2. Points of order $13$ on elliptic curves

    Kamienny, Sheldon; Newman, Burton
    We study elliptically parametrized families of elliptic curves with a point of order $13$ that do not arise from rational parametrizations. We also show that no elliptic curve over $\mathbb{Q}(\zeta_{13})^+$ can possess a rational point of order $13$.

  3. A short remark on consecutive coincidences of a certain multiplicative function

    Kohnen, Winfried
    We study integral solutions $n$ of the equation $A(n+k)=A(n)$, where $A$ is a certain multiplicative function related to Jordan's totient function.

  4. Euclidean proofs for function fields

    Lachmann, Thomas
    Schur proved the infinitude of primes in arithmetic progressions of the form $\equiv l\mod m$, such that $l^{2}\equiv1\mod m$, with non-analytic methods by ideas inspired from the famous proof Euclid gave for the infinitude of primes. Ram Murty showed that Schur's method has its limits given by the assumption Schur made. We will discuss analogous for the primes in the ring $\mathbb{F}_{q}[T]$.

  5. Tractability of $\mathbb{L}_2$-approximation in hybrid function spaces

    Kritzer, Peter; Laimer, Helene; Pillichshammer, Friedrich
    We consider multivariate $\mathbb{L}_2$-approximation in reproducing kernel Hilbert spaces which are tensor products of weighted Walsh spaces and weighted Korobov spaces. We study the minimal worst-case error $e^{\mathbb{L}_2-\mathrm{app},\Lambda}(N,d)$ of all algorithms that use $N$ information evaluations from the class $\Lambda$ in the $d$-dimensional case. The two classes $\Lambda$ considered in this paper are the class $\Lambda^{{\rm all}}$ consisting of all linear functionals and the class $\Lambda^{{\rm std}}$ consisting only of function evaluations. The focus lies on the dependence of $e^{\mathbb{L}_2-\mathrm{app},\Lambda}(N,d)$ on the dimension $d$. The main results are conditions for weak, polynomial, and strong polynomial tractability.

  6. Representation of a rational number as a sum of ninth or higher odd powers

    Reynya, M.A
    In the present paper, we substantially generalize one of the results obtained in our earlier paper [RM]. We present a solution of a problem of Waring type: if $F(x_1, \dots ,x_N)$ is a~symmetric form of odd degree $n\ge 9$ in $N=16\cdot 2^{n-9}$ variables, then for any $q\in \mathbb{Q}$, $q\neq 0$, the equation $F(x_i)=q$ has rational parametric solutions, that depend on $n-8$ parameters.

  7. An extension theorem for generating new families of non-congruent numbers

    Reinholz, Lindsey; Spearman, Blair K.; Yang, Qiduan
    A technique for generating new families of non-congruent numbers by appending a tail of primes to extend known families of non-congruent numbers is presented. These new non-congruent numbers are comprised of arbitrarily many prime factors belonging to two or three odd congruence classes modulo 8.

  8. On the value distribution of two Dirichlet $\boldsymbol L$-functions

    Laaksonen, Niko; Petridis, Yiannis N.
    Let $\rho$ denote the non-trivial zeros of the Riemann zeta function. We study the relative value distribution of $L(\rho+\sigma,\chi_{1})$ and $L(\rho+\sigma,\chi_{2})$, where $\sigma\in[0,1/2)$ is fixed and $\chi_{1}$, $\chi_{2}$ are two fixed Dirichlet characters to distinct prime moduli. For $\sigma>0$ we prove that a positive proportion of these pairs of values are linearly independent over $\mathbb{R}$, which implies that the arguments of the values are different. For $\sigma=0$ we show that, up to height $T$, the values are different for $cT$ of the Riemann zeros for some positive constant $c$.

  9. On the Diophantine equation $y^{p} = f(x_{1}, x_{2}, ..., x_{r})$

    Srikanth, Raghavendran; Subburam, Sivanarayanapandian
    In this paper, we study the diophantine equation $$y^{p} = f(x_{1}, x_{2}, ..., x_{r}),$$ where $f(x_{1}, x_{2}, ..., x_{r})$ is a real polynomial in variables $x_{1}, x_{2}, ..., x_{r}$ in $R$, a group of real numbers under the usual addition $+$, having the least element property.

  10. Some remarks on the differences between ordinates of consecutive zeta zeros

    Ivić, Aleksandar
    If $0 < \gamma_1 \le \gamma_2 \le \gamma_3 \le \ldots$ denote ordinates of complex zeros of the Riemann zeta-function $\zeta(s)$, then several results involving the maximal order of $\gamma_{n+1}-\gamma_n$ and the sum $$ \sum_{0<\gamma_n\le T}{(\gamma_{n+1}-\gamma_n)}^k \qquad(k>0) $$ are proved.

  11. Improved explicit bounds for some functions of prime numbers

    Nazardonyavi, Sadegh
    Using recent explicit asymptotic zero--free region and computations of zeros of the Riemann zeta function, obtained by Mossinghoff & Trudgian and Gourdon, respectively, we give an improvement for estimates of some functions related to distribution of primes, such as prime counting function, intervals containing at least one prime, Chebyshev's $\psi$ and $\vartheta$ functions.

  12. On the representation of an even perfect number as the sum of five cubes

    Farhi, Bakir
    The aim of this note is to show that any even perfect number, other than $6$, can be written as the sum of at most five positive integral cubes. We also conjecture that any such number can even be written as the sum of at most three positive integral cubes.

  13. On sums involving Fourier coefficients of Maass forms for $\mathrm{SL}(3,\mathbb Z)$

    Jääsaari, Jesse; Vesalainen, Esa V.
    We derive a truncated Voronoi identity for rationally additively twisted sums of Fourier coefficients of Maass forms for $\mathrm{SL}(3,\mathbb Z)$, and as an application obtain a pointwise estimate and a second moment estimate for the sums in question.

  14. A note on the gaps between zeros of Epstein's zeta-functions on the critical line

    Baier, Stephan; Kotyada, Srinivas; Sangale, Usha Keshav
    It is proved that Epstein's zeta-function $\zeta_{Q}(s)$, related to a positive definite integral binary quadratic form, has a zero $1/2 + i\gamma$ with $ T \leq \gamma \leq T + T^{{3/7} +\varepsilon} $ for sufficiently large positive numbers $T$. This is an improvement of the result by M. Jutila and K. Srinivas (Bull. London Math. Soc. 37 (2005) 45--53).

  15. Theta products and eta quotients of level $24$ and weight $2$

    Alaca, Ayşe; Alaca, Şaban; Aygin, Zafer Selcuk
    We find bases for the spaces $M_2\Big(\Gamma_0(24),(\frac{d}{\cdot})\Big)$ ($d=1,8,12, 24$) of modular forms. We determine the Fourier coefficients of all $35$ theta products $\varphi[a_1,a_2,a_3,a_4](z)$ in these spaces. We then deduce formulas for the number of representations of a positive integer $n$ by diagonal quaternary quadratic forms with coefficients $1$, $2$, $3$ or $6$ in a uniform manner, of which $14$ are Ramanujan's universal quaternary quadratic forms. We also find all the eta quotients in the Eisenstein spaces $E_2\Big(\Gamma_0(24),(\frac{d}{\cdot})\Big)$ ($d=1,8,12, 24$) and give their Fourier coefficients.

  16. Oscillations of Fourier coefficients of $GL(m)$ Hecke-Maass forms and nonlinear exponential functions at primes

    Jiang, Yujiao; Lü, Guangshi
    Let $F(z)$ be a Hecke-Maass form for $SL(m,\mathbb{Z})$ and $A_F(n,1, \dots, 1)$ be the coefficients of $L$-function attached to $F.$ We study the cancellation of $A_F(n,1, \dots, 1)$ for twisted with a nonlinear exponential function at primes, namely the sum \begin{equation*} \sum_{n \leq N} \Lambda (n)A_F(n,1, \dots, 1)e ( \alpha n^\theta ), \end{equation*} where $0<\theta<2/m$. We also strengthen the corresponding previous results for holomorphic cusp forms for $SL(2,\mathbb{Z}),$ and improve the estimates of Ren-Ye on the resonance of exponential sums involving Fourier coefficients of a Maass form for $SL(m,\mathbb{Z})$.

  17. A higher rank Selberg sieve with an additive twist and applications

    Murty, M. Ram; Vatwani, Akshaa
    We develop an axiomatic formulation of the higher rank version of the classical Selberg sieve with an ``additive twist'' and provide asymptotic formulas for the same. As an application of this higher rank sieve, we obtain improvements of results of Heath-Brown and Ho-Tsang on almost prime $k$-tuples.

  18. On some results of M.A. Malik concerning polynomials

    Mir, Abdullah; Hussain, Imtiaz
    If $P(z)$ is a polynomial of degree $n$ having all its zeros in $|z|\le k, k\le 1,$ Rather, Gulzar and Ahangar [8] proved that for every $\alpha \in \mathbb{C}$ with $|\alpha|\ge k$ and $\gamma >0,$ \[ n(|\alpha|-k)\Bigg\{\int_0^{2\pi}\Big|\frac{P(e^{i\theta})}{D_{\alpha}P(e^{i\theta})}\Big|^\gamma d\theta\Bigg\}^\frac{1}{\gamma}&\leq \Bigg\{\int_0^{2\pi}\Big|1+ke^{i\theta}\Big|^\gamma d\theta\Bigg\}^\frac{1}{\gamma}. \] In this paper, we shall obtain a result which generalizes and sharpens the above inequality by obtaining a bound that depends upon the location of all the zeros of $P(z)$ rather than just on the location of the zero of largest modulus.

  19. Bounding the least prime ideal in the Chebotarev Density Theorem

    Zaman, Asif
    Let $K$ be a number field and suppose $L/K$ is a finite Galois extension. We establish a bound for the least prime ideal occurring in the Chebotarev Density Theorem. Namely, for every conjugacy class $C$ of $\mathrm{Gal}(L/K)$, there exists a prime ideal $\mathfrak{p}$ of $K$ unramified in $L$, for which its Artin symbol $\big[ \frac{L/K}{\mathfrak{p}} \big] = C$, for which its norm $N^K_{\mathbb{Q}}\mathfrak{p}$ is a rational prime, and which satisfies \[ N^K_{\mathbb{Q}} \mathfrak{p} \ll d_L^{40}, \] where $d_L = |\mathrm{disc}(L/\mathbb{Q})|$. All implicit constants are effective and absolute.

  20. Bounding the least prime ideal in the Chebotarev Density Theorem

    Zaman, Asif
    Let $K$ be a number field and suppose $L/K$ is a finite Galois extension. We establish a bound for the least prime ideal occurring in the Chebotarev Density Theorem. Namely, for every conjugacy class $C$ of $\mathrm{Gal}(L/K)$, there exists a prime ideal $\mathfrak{p}$ of $K$ unramified in $L$, for which its Artin symbol $\big[ \frac{L/K}{\mathfrak{p}} \big] = C$, for which its norm $N^K_{\mathbb{Q}}\mathfrak{p}$ is a rational prime, and which satisfies \[ N^K_{\mathbb{Q}} \mathfrak{p} \ll d_L^{40}, \] where $d_L = |\mathrm{disc}(L/\mathbb{Q})|$. All implicit constants are effective and absolute.

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