On a paper of Dressler and Van de Lune

Abstract

If \(z\in {\mathbb {C}}\) and \(1\le n\) is a natural number then

$$\begin{aligned} \sum _{d_1 d_2 =n} (1-z^{p_1})\cdots (1-z^{p_m}) z^{q_1 e_{1}+\cdots +q_i e_{i} }=1, \end{aligned}$$

where \(d_1=p_1^{r_1}\dots p_m^{r_m }\), \(d_2=q_1^{e_1}\dots q_i^{e_i }\) are the prime decompositions of \(d_1, d_2\). This is one of the identities involving arithmetic functions that we prove using ideas from the paper of Dressler and van de Lune [3].

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References

  1. 1.

    Apostol, T.: Introduction to Analytic Number Theory. Springer, New York (1976)

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  2. 2.

    Apostol, T.: Modular Functions and Dirichlet Series in Number Theory. Springer, New York (1976)

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    Dressler, R.E., van de Lune, J.: Some remarks concerning the number theoretic functions \(\omega (n)\) and \(\Omega (n)\). Proc. Am. Math. Soc. 41(2), 403–406 (1973)

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    Titchmarsh, E.C.: The Theory of the Riemann Zeta-Function, 2nd edn. Oxford Univ. Press, New York (1986). (revised by D. R. Heath-Brown)

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Correspondence to P. A. Panzone.

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Panzone, P.A. On a paper of Dressler and Van de Lune. Bol. Soc. Mat. Mex. 26, 831–839 (2020). https://doi.org/10.1007/s40590-020-00278-z

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Keyword

  • Arithmetic functions
  • Identities
  • Zeta function

Mathematics Subject Classification

  • 11A25
  • 11MXX