An explicit estimate for the Barban and Vehov weights

Djamel Berkane
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 20, 2014, Number 2, Pages 35—43
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Djamel Berkane
Department of Mathematics, University of Blida, Algeria

Abstract

We show that
\sum_{1\leq n\leq N}\Big(\sum_{\substack{d\mid n}}\lambda_{d}\Big)^2/n\ll \dfrac{\log N}{\log z},$$ where $\lambda_{d}
where λd is a real valued arithmetic function called the Barban and Vehov weight and we give an explicit version of a Theorem of Barban and Vehov which has applications to zero-density theorems.

Keywords

  • Explicit estimates
  • Möbius function
  • Selberg sieve

AMS Classification

  • Primary: 11N37
  • Secondary: 11N35, 11N05

References

  1. Barban, M. B., P. P. Vehov, An extremal problem, Trudy Moskov. Math., Vol. 18, 1968, 83–90.
  2. Bastien, G., M. Rogalski, Convexité, complète monotonie et inégalités sur les fonctions zêta et gamma, sur les fonctions des opérateurs de Baskakov et sur des fonctions arithmétiques, Canad. J. Math., Vol. 54, 2002, No. 5, 916–944.
  3. Berkane, D., O. Bordellès, O. Ramaré, Explicit upper bound for the remainder term in the divisor problem, Math. Comp., Vol. 278(81), 2012, 1025–1051.
  4. Ford, K. Zero-free regions for the Riemann zeta function, Proceeding of the Millenial Conference on Number Theory, Urbana, IL., 2000.
  5. Graham, S. W. An asymptotic estimate related to Selberg’s sieve, J. Number Theory, Vol. 10, 1978, 83–94.
  6. Granville, A., O. Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients, Mathematika, Vol. 43, 1996, No. 1, 1996, 73–107.
  7. Jutila, M. On Linnik’s constant, Math. Scand., Vol. 41, 1977a, No. 1, 45–62.
  8. Motohashi, Y. On a problem in the theory of sieve methods. Res. Inst. Math. Sci. Kyoto Univ. Kökyüroko, Vol. 222 (1974), 9–50 (in Japanese).
  9. Motohashi, Y. On a density theorem of Linnik, Proc. Japan Acad., Vol. 51, 1975, 815–817.
  10. Motohashi, Y. Sieve Methods and Prime Number Theory, Tata Lectures Notes, Vol. 205, 1983.
  11. Ramaré, O. On Snirel’man’s constant, Ann. Scu. Norm. Pisa, Vol. 21, 1995, 645–706.
  12. Rankin, R. A. The difference between consecutive prime numbers, J. Lond. Math. Soc., Vol. 13, 1938, 242–247.

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Cite this paper

APA

Berkane, D. (2014). An explicit estimate for the Barban and Vehov weights. Notes on Number Theory and Discrete Mathematics, 20(2), 35-43.

Chicago

Berkane, Djamel. “An Explicit Estimate for the Barban and Vehov Weights.” Notes on Number Theory and Discrete Mathematics 20, no. 2 (2014): 35-43.

MLA

Berkane, Djamel. “An Explicit Estimate for the Barban and Vehov Weights.” Notes on Number Theory and Discrete Mathematics 20.2 (2014): 35-43. Print.

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