Mostrando recursos 1 - 20 de 25

  1. 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.

  2. 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.

  3. 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).

  4. 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.

  5. 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 $SL(m,\mathbb{Z})$.

  6. 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.

  7. 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.

  8. 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.

  9. 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.

  10. The first moment of twisted Hecke $L$-functions with unbounded shifts

    Bettin, Sandro
    We compute the first moment of twisted Hecke $L$-functions of weight $2$ and prime power level going to infinity, uniformly in the conductor of the twist and in the vertical shift.

  11. The first moment of twisted Hecke $L$-functions with unbounded shifts

    Bettin, Sandro
    We compute the first moment of twisted Hecke $L$-functions of weight $2$ and prime power level going to infinity, uniformly in the conductor of the twist and in the vertical shift.

  12. Note on the class number of the $p$th cyclotomic field, III

    Ichimura, Humio
    Let $p=2\ell^f+1$ be a prime number with $f \geq 2$ and an odd prime number $\ell$. For $0 \leq t \leq f$, let $K_t$ be the imaginary subfield of the $p$th cyclotomic field $\mathbb{Q}(\zeta_p)$ with $[K_t : \mathbb{Q}]=2\ell^t$. Denote by $h_{p,t}^-$ the relative class number of $K_t$, and by $h_{p,t}^+$ the class number of the maximal real subfield $K_t^+$. It is known that the ratio $h_{p,f}^-/h_{p,f-1}^-$ is odd (and hence so is $h_{p,f}^+/h_{p,f-1}^+$) whenever $2$ is a primitive root modulo $\ell^2$. We show that $h_{p,f}^+/h_{p,f-1}^+$ is odd under a somewhat milder assumption on $\ell$ and that the ratio $h_{p,f-1}^-/h_{p,f-2}^-$ is always odd when $\ell=3$.

  13. Note on the class number of the $p$th cyclotomic field, III

    Ichimura, Humio
    Let $p=2\ell^f+1$ be a prime number with $f \geq 2$ and an odd prime number $\ell$. For $0 \leq t \leq f$, let $K_t$ be the imaginary subfield of the $p$th cyclotomic field $\mathbb{Q}(\zeta_p)$ with $[K_t : \mathbb{Q}]=2\ell^t$. Denote by $h_{p,t}^-$ the relative class number of $K_t$, and by $h_{p,t}^+$ the class number of the maximal real subfield $K_t^+$. It is known that the ratio $h_{p,f}^-/h_{p,f-1}^-$ is odd (and hence so is $h_{p,f}^+/h_{p,f-1}^+$) whenever $2$ is a primitive root modulo $\ell^2$. We show that $h_{p,f}^+/h_{p,f-1}^+$ is odd under a somewhat milder assumption on $\ell$ and that the ratio $h_{p,f-1}^-/h_{p,f-2}^-$ is always odd when $\ell=3$.

  14. A Bombieri--Vinogradov Theorem with products of Gaussian primes as moduli

    Halupczok, Karin
    We prove a version of the Bombieri--Vinogradov Theorem with certain products of Gaussian primes as moduli, making use of their special form as polynomial expressions in several variables. Adapting Vaughan's proof of the classical Bombieri--Vinogadov Theorem, cp. [10] to this setting, we apply the polynomial large sieve inequality that has been proved in [7] and which includes recent progress in Vinogradov's mean value theorem due to Parsell \emph{et al.} in [9]. From the benefit of these improvements, we obtain an extended range for the variables compared to the range obtained from standard arguments only.

  15. A Bombieri--Vinogradov Theorem with products of Gaussian primes as moduli

    Halupczok, Karin
    We prove a version of the Bombieri--Vinogradov Theorem with certain products of Gaussian primes as moduli, making use of their special form as polynomial expressions in several variables. Adapting Vaughan's proof of the classical Bombieri--Vinogadov Theorem, cp. [10] to this setting, we apply the polynomial large sieve inequality that has been proved in [7] and which includes recent progress in Vinogradov's mean value theorem due to Parsell \emph{et al.} in [9]. From the benefit of these improvements, we obtain an extended range for the variables compared to the range obtained from standard arguments only.

  16. A remark on the conditional estimate for the sum of a prime and a square

    Suzuki, Yuta
    Hardy and Littlewood conjectured that every sufficiently large integer is either a square or the sum of a prime and a square. Let $E(X)$ be the number of positive integers up to $X\ge4$ for which this property does not hold. We prove \[E(X)\ll X^{1/2}(\log X)^A(\log\log X)^4\] with $A=3/2$ under the Generalized Riemann Hypothesis. This is a small improvement on the previous remarks of Mikawa (1993) and Perelli-Zaccagnini (1995) which claim $A=4,3$ respectively.

  17. A remark on the conditional estimate for the sum of a prime and a square

    Suzuki, Yuta
    Hardy and Littlewood conjectured that every sufficiently large integer is either a square or the sum of a prime and a square. Let $E(X)$ be the number of positive integers up to $X\ge4$ for which this property does not hold. We prove \[E(X)\ll X^{1/2}(\log X)^A(\log\log X)^4\] with $A=3/2$ under the Generalized Riemann Hypothesis. This is a small improvement on the previous remarks of Mikawa (1993) and Perelli-Zaccagnini (1995) which claim $A=4,3$ respectively.

  18. A note on a tower by Bassa, Garcia and Stichtenoth

    Anbar, Nurdagül; Beelen, Peter
    In this note, we prove that the tower given by Bassa, Garcia and Stichtenoth in [4] is a subtower of the one given by Anbar, Beelen and Nguyen in [2]. This completes the study initiated in [16,2] to relate all known towers over cubic finite fields meeting Zink's bound with each other.

  19. A note on a tower by Bassa, Garcia and Stichtenoth

    Anbar, Nurdagül; Beelen, Peter
    In this note, we prove that the tower given by Bassa, Garcia and Stichtenoth in [4] is a subtower of the one given by Anbar, Beelen and Nguyen in [2]. This completes the study initiated in [16,2] to relate all known towers over cubic finite fields meeting Zink's bound with each other.

  20. Remarks on the distribution of the primitive roots of a prime

    Chern, Shane
    Let $\mathbb{F}_p$ be a finite field of size $p$ where $p$ is an odd prime. Let $f(x)\in\mathbb{F}_p[x]$ be a~polynomial of positive degree $k$ that is not a $d$-th power in $\mathbb{F}_p[x]$ for all $d\mid p-1$. Furthermore, we require that $f(x)$ and $x$ are coprime. The main purpose of this paper is to give an estimate of the number of pairs $(\xi,\xi^\alpha f(\xi))$ such that both $\xi$ and $\xi^\alpha f(\xi)$ are primitive roots of $p$ where $\alpha$ is a given integer. This answers a question of Han and Zhang.

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