A note on a broken Dirichlet convolution

Emil Daniel Schwab and Barnabás Bede
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 20, 2014, Number 2, Pages 65—73
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Authors and affiliations

Emil Daniel Schwab
Department of Mathematical Sciences, The University of Texas at El Paso
El Paso, TX 79968, USA

Barnabás Bede
Department of Mathematics, DigiPen Institute of Technology
Redmond, WA 98052, USA


The paper deals with a broken Dirichlet convolution ⊗ which is based on using the odd divisors of integers. In addition to presenting characterizations of ⊗-multiplicative functions we also show an analogue of Menon’s identity:
\sum\limits_{\stackrel{a\ (mod\ n)}{(a,n)_{\otimes}=1}}{(a-1,n)}=\phi_{\otimes}(n)\left\tau(n)-\frac{1}{2}\tau_{2}(n)\right
where (a, n) denotes the greatest common odd divisor of a and n, φ(n) is the number of integers a (mod n) such that (a, n) = 1, τ(n) is the number of divisors of n, and τ2(n) is the number of even divisors of n.


  • Dirichlet convolution
  • Möbius function
  • Multiplicative arithmetical functions
  • Menon’s identity

AMS Classification

  • 11A25


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Schwab, E. D. & Bede, B. (2014). A note on a broken Dirichlet convolution. Notes on Number Theory and Discrete Mathematics, 20(2), 65-73.

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