Functiones et Approximatio Commentarii Mathematici

Power series with the von Mangoldt function

Matthias Kunik and Lutz G Lucht

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Abstract

We study the analytic behavior of a power series with coefficients containing the von Mangoldt function. In particular, we extend an explicit formula of Hardy and Littlewood for related functions and derive further representation formulas in the unit disk that reveal logarithmic singularities on a dense subset of the unit circle. As an essential tool for proving the square integrability of occurring limit functions together with respective error estimates we contribute a new proof of a Ramanujan-like expansion of an arithmetic function consisting of the von Mangoldt function and the Euler function.

Article information

Source
Funct. Approx. Comment. Math., Volume 47, Number 1 (2012), 15-33.

Dates
First available in Project Euclid: 25 September 2012

Permanent link to this document
https://projecteuclid.org/euclid.facm/1348578274

Digital Object Identifier
doi:10.7169/facm/2012.47.1.2

Mathematical Reviews number (MathSciNet)
MR2987108

Zentralblatt MATH identifier
1320.11075

Subjects
Primary: 11L20: Sums over primes
Secondary: 11M06: $\zeta (s)$ and $L(s, \chi)$ 30J99: None of the above, but in this section

Keywords
trigonometric series over primes explicit formulas arithmetic functions Ramanujan sums Hardy spaces

Citation

Kunik, Matthias; Lucht, Lutz G. Power series with the von Mangoldt function. Funct. Approx. Comment. Math. 47 (2012), no. 1, 15--33. doi:10.7169/facm/2012.47.1.2. https://projecteuclid.org/euclid.facm/1348578274


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References

  • G. Andrews, R. Askey and R. Roy, Special functions, Cambridge University Press, 2000.
  • J.B. Conrey and G. Myerson, On the Balazard-Saias criterion for the Riemann hypothesis, Preprint. v1, 2000.
  • J. Dieudonné, Foundations of Modern Analysis, Vol. 1, 2nd ed., Academic Press, New York, 1969.
  • H.M. Edwards, Riemann's zeta function, Dover Publications, Mineola, New York, 2001.
  • G.H. Hardy, Note on Ramanujan's trigonometrical function $c_q(n)$, and certain series of arithmetical functions, Proc. Cambridge Philos. Soc. 20 (1921), 263–271.
  • G.H. Hardy and J.E. Littlewood, Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes, Acta Math. 41 (1916), 119–196.
  • A. Hildebrand, Über die punktweise Konvergenz von Ramanujan-Entwicklungen zahlentheoretischer Funktionen, Acta Arith. 44 (1984), 109–140.
  • K. Hoffman, Banach spaces of analytic functions, Dover Publication, 2007.
  • A.E. Ingham, The distribution of prime numbers, Cambridge University Press, Cambridge, 1990.
  • A.A. Karatsuba and S.M. Voronin, The Riemann zeta-function, Walter de Gruyter, Berlin, New York, 1992.
  • J. Knopfmacher, Abstract Analytic Number Theory, Dover Publ., New York, 1975 (1990).
  • P. Koosis, Introduction to $H_p$-spaces, second edition, Cambridge University Press, Cambridge, 1998.
  • M. Kunik, On the formulas of $\pi(x)$ and $\psi(x)$ of Riemann and von-Mangoldt, Preprint Nr. 09/2005, Otto-von-Guericke Universität Magdeburg, Fakultät für Mathematik, 2005 (http://www-ian.math.uni-magdeburg.de/~kunik/).
  • E. Landau und D. Gaier, Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie, Springer-Verlag, Berlin, Heidelberg, 1986.
  • L. Lucht, Power series with multiplicative coefficients, Math. Z. 177 (1981), 359–374.
  • L.G. Lucht, A survey of Ramanujan expansions, Int. J. Number Theory 6 (2010), 1785–1799.
  • L. Lucht und D. Wolke, Über trigonometrische Reihen mit zahlentheoretischen Koeffizienten, Archiv Math. 57 (1991), 571–580.
  • R. Remmert, Funktionentheorie 2, Springer, Berlin, Heidelberg, New York, 1995.
  • W. Rudin, Real and Complex Analysis, McGraw-Hill, London, 1970.
  • W. Schwarz and J. Spilker, Arithmetical Functions, Lecture Note Series 184, Cambridge Univ. Press, Cambridge, 1994.