Note on Type L-Functions of Euler Product of Second Degree

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DOI: https://doi.org/10.28924/2291-8639-21-2023-135
Published: Dec 8, 2023
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Ali H. Hakami, Note on Type L-Functions of Euler Product of Second Degree, Int. J. Anal. Appl., 21 (2023), 135.

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Ali H. Hakami

Abstract

In this paper we are concerned with a special case of Euler functions. We shall study the Euler product of degree two (Type of L-Euler functions) and give some results. More precisely, we shall deal with some Dirichlet series associated with a class of arithmetic {an} under the condition that apapk = apk+1 + pαapk−1, provided p is prime, k≥1, and α is a fixed complex number. We will demonstrate that there is an Euler’s product for the Dirichlet series Σnan/ns. This result is important in analysis, especially in analytic number theory.

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References

  1. T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976.
  2. H. Cohn, Advanced Number Theory, Dover Publication, New York, 1962.
  3. H. Davenport, Multiplicative Number Theory, Lectures in Advanced Mathematics, Vol. 1, Markham, Chicago, 1967.
  4. P.G.L. Dirichlet, Lectures on Number Theory, History of Mathematics, Vol. 16, American Mathematical Society, Providence, R.I., 1999.
  5. N. Elkies, Introduction to Analytic Number Theory, Course Notes, Dover, 1962.
  6. G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, Oxford, 1998.
  7. A.A. Karatsuba, Basic Analytic Number Theory, Springer, Berlin, 1993.
  8. K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Springer, New York, 1982.
  9. A.E. Ingham, The Distribution of Prime Numbers, Cambridge University Press, Cambridge, 1932.
  10. I. Niven, H.S. Zuckerman, H.L. Montgomery, An Introduction to the Theory of Numbers, Wiley, New York, 1991.
  11. A. Strömbergsson, Analytic Number Theory, Lecture Notes, Uppsala universitet, 2008.
  12. H.L. Montgomery, R.C. Vaughan, Multiplicative Number Theory I. Cambridge University Press, Cambridge, 2007.
  13. D. Zagier, Elliptic Modular Forms and Their Applications, in: K. Ranestad (Ed.), The 1-2-3 of Modular Forms, Springer, Berlin, Heidelberg, 2008: pp. 1–103. https://doi.org/10.1007/978-3-540-74119-0_1.
  14. L. Rousu, Modular Forms and Related Topics, PhD Thesis, Uppsala University, 2021.
  15. A. Cardoso, Arithmetical functions and Dirichlet series, Master Thesis, Universiade Do Porto, 2022.
  16. K. Matsumoto, H. Tsumura, Double Dirichlet Series Associated With Arithmetic Functions II, Kodai Math. J. 46 (2023), 10–30. https://doi.org/10.2996/kmj46102.
  17. R.Taylor, Galois Representations, In: Proceedings of ICM 2002, Vol. III, 1-3, 2002.
  18. T. Sunada, L-Functions in Geometry and Some Applications, in: K. Shiohama, T. Sakai, T. Sunada (Eds.), Curvature and Topology of Riemannian Manifolds, Springer Berlin Heidelberg, Berlin, Heidelberg, 1986: pp. 266– 284. https://doi.org/10.1007/BFb0075662.
  19. J. Cremona, The L-Functions and Modular Forms Database Project, Found. Comput. Math. 16 (2016), 1541–1553. https://doi.org/10.1007/s10208-016-9306-z.