For any natural number and chosen base p, the number admits a unique expression of the form $a_np^n + ... + a_2p^2 + a_1p^1 + a_0$, where $a_k < p$ for all k. This property is effectively what makes our Hindu-Arabic positional numeral notation system possible.
When you multiply two polynomials (or power series) together, you're essentially taking the discrete convolution of their ordered sequences of coefficients. This is essentially what we're doing when we multiply numbers in a positional numeral system, except that there's an additional "carry" step involved to put the number back into its canonical representation.
Another useful type of infinite series is a Dirichlet series, which looks like $ ... + a_3 3^p + a_2 2^p + a_1$; p is now the exponent rather than the base. Could this sort of series also be the foundation for a numeral system?
For instance, is there a way to canonically represent any number as a finite sum $a_n n^p + ... + a_3 3^p + a_2 2^p + a_1$? When multiplying two numbers in this format, would multiplication resemble Dirichlet convolution rather than ordinary convolution, with some other type of "carry" to make the whole thing work out?
I'm interested to see how this would work - or if it would fail, and if so, why!