Generating series of q-adic valuations are non-holonomic, non-algebraic over characteristic 0 fields, and take transcendental values at infinitely many rational and algebraic irrational points.
On the representation of number-theoretic functions by arithmetic terms
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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.
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On non-holonomicity, transcendence and $p$-adic valuations
Generating series of q-adic valuations are non-holonomic, non-algebraic over characteristic 0 fields, and take transcendental values at infinitely many rational and algebraic irrational points.