On the representation of number-theoretic functions by arithmetic terms

TL;DR

This paper derives elementary closed-form formulas for key number-theoretic functions, providing new explicit representations that involve only basic operations, which were previously unknown.

Contribution

It introduces novel elementary closed-form expressions for fundamental number-theoretic functions, expanding the tools available for their analysis.

Findings

Formulas for p-adic valuation and divisor functions derived

Explicit expressions for Euler's totient and modular inverse obtained

New closed forms involve only elementary operations

Abstract

We present closed forms for several functions that are fundamental in number theory and we explain the method used to obtain them. Concretely, we find formulas for the p-adic valuation, the number-of-divisors function, the sum-of-divisors function, Euler's totient function, the modular inverse, the integer part of the root, the integer part of the logarithm, the multiplicative order and the discrete logarithm. Although these are very complicated, they only involve elementary operations, and to our knowledge no other closed form of this kind is known for the aforementioned functions.