A Note on a Broken Dirichlet Convolution

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: X a (mod n) (a,n)⊗=1 (a − 1, n) = φ⊗(n)[τ (n) − 1 2 τ2(n)], 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