A note on the Lehmer problem over short intervals

Lu, Ya; Yi, Yuan
June 2011
Acta Mathematica Sinica;Jun2011, Vol. 27 Issue 6, p1115
Academic Journal
Let p be an odd prime, c be an integer with ( c, p) = 1, and let N be a positive integer with N ≤ p − 1. Denote by r( N, c; p) the number of integers a satisfying 1 ≤ a ≤ N and 2 ∤ a + ā, where ā is an integer with 1 ≤ ā ≤ p − 1, aā ≡ c (mod p). It is well known that r( N, c; p) = 1/2 N + O( plog p). The main purpose of this paper is to give an asymptotic formula for Σ( r( N, c; p) − 1/2 N).


Related Articles

  • Asymptotics of regularized solutions of a linear nonautonomous system of advanced differential equations. Dolgii, Yu. F.; Surkov, P. G. // Differential Equations;Apr2010, Vol. 46 Issue 4, p470 

    For a nonautonomous system of linear advanced differential equations, we find asymptotic formulas analytically determining how the regular solutions of this system on an interval of the positive half-line depend on the admissible error. The problem is solved in the following setting: additional...

  • Asymptotic behavior of the Verblunsky coefficients for the OPUC with a varying weight. Poplavskyi, M. // Journal of Mathematical Physics;Apr2012, Vol. 53 Issue 4, p043510 

    We present an asymptotic analysis of the Verblunsky coefficients for the polynomials orthogonal on the unit circle with the varying weight e-nV(cos x), assuming that the potential V has four bounded derivatives on [-1, 1] and the equilibrium measure has a one interval support. We obtain the...

  • A priori analysis of initial data for the Riccati equation and asymptotic properties of its solutions. N. A. Chernyavskaya; Jeremy Schiff; L. A. Shuster // Bulletin of the London Mathematical Society;Aug2009, Vol. 41 Issue 4, p723 

    We obtain two main results for the Cauchy problem where x0, y0 ∈ ℝ, r > 0, q ≥ 0, 1/r ∈ L1loc(ℝ), q ∈ L1loc(ℝ) and (1) For given initial data...

  • Strings of congruent primes in short intervals. Freiberg, Tristan // Journal of the London Mathematical Society;Oct2011, Vol. 84 Issue 2, p344 

    Fix ɛ>0, and let p1=2, p2=3, … be the sequence of all primes. We prove that if (q, a)=1, then there are infinitely many pairs pr, pr+1 such that pr ≡ pr+1 ≡ a mod q and pr+1−pr<ɛ log pr. The proof combines the ideas of Shiu and of...

  • On the sum of a prime and a k-th power of prime in short intervals. Wang, Y. // Acta Mathematica Hungarica;May2012, Vol. 135 Issue 3, p248 

    Let $\mathcal{H}_{k}$ denote the set { n∣2| n, $n\not\equiv 1\ (\mathrm{mod}\ p)$ ∀ p>2 with p−1| k}. We prove that when $X^{\frac{11}{20}\left(1-\frac{1}{2k}\right) +\varepsilon}\leqq H\leqq X$, almost all integers $n\in\allowbreak {\mathcal{H}_{k} \cap (X, X+H]}$ can be...

  • A NOTE ON STURM-LIOUVILLE PROBLEMS WHOSE SPECTRUM IS THE SET OF PRIME NUMBERS. MINGARELLI, ANGELO B. // Electronic Journal of Differential Equations;2011, Vol. 2011, Special section p1 

    We show that there is no classical regular Sturm-Liouville problem on a finite interval whose spectrum consists of infinitely many distinct primes numbers. In particular, this answers in the negative a question raised by Zettl in his book, [9]. We also show that there may exist such a problem if...

  • The asymptotic analysis of gaps in the spectrum of a waveguide perturbed with a periodic family of small voids. Nazarov, S. // Journal of Mathematical Sciences;Oct2012, Vol. 186 Issue 2, p247 

    We study the spectrum of the Dirichlet problem for the Laplace operator in a cylindrical waveguide with periodic family of small (of diameter ε > 0) voids. Based on the asymptotic analysis of eigenvalues of the problem in a singularly perturbed periodicity cell, we show that the waveguide...

  • Multidimensional lower bounds for the eigenvalues of Stokes and Dirichlet Laplacian operators. Yolcu, Selma Yıldırım; Yolcu, Türkay // Journal of Mathematical Physics;Apr2012, Vol. 53 Issue 4, p043508 

    This article is concerned with the improvements of certain eigenvalue inequalities of Stokes operator and Dirichlet Laplacian related to the Berezin-Li-Yau type inequalities. The formulas proved extend the earlier works of Melas ['A lower bound for sums of eigenvalues of the Laplacian,' Proc....

  • Availability of a Periodically Inspected System Maintained through Several Minimal Repairs before a Replacement or a Perfect Repair. Jun Yang; Tingting Gang; Yu Zhao // Abstract & Applied Analysis;2013, p1 

    The instantaneous availability of a periodically inspected system with several minimal repairs before a replacement or a perfect repair is studied. First, we investigate two concrete minimal repair models with periodic inspection and constant repair time and give their instantaneous availability...


Read the Article


Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics