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Recursive divisor properties

Number theory

The recursive divisor function is found to have a simple generating function, which leads to a number of new Dirichlet convolutions.

Properties of the recursive divisor function

Draft

T. Fink

We recently introduced a new arithmetic function called the recursive divisor function, κx(n)\kappa_x(n). Here we show that its Dirichlet series is ζ(sx)/(2ζ(s))\zeta(s - x)/(2 - \zeta(s)), where ζ(sigma)\zeta(sigma) is the Riemann-zeta function. We relate the recursive divisor function to the ordinary divisor function, κ0×σx=κx×σ0\kappa_0 \times \sigma_x = \kappa_x \times \sigma_0, which yields a number of new Dirichlet convolutions. We also state the equivalent of the Riemann hypothesis in terms of κx\kappa_x.

Draft

T. Fink