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Addition is O(log n), so the naive version is O(n log n).

Gauss would have used a pre-[1960](https://en.wikipedia.org/wiki/Karatsuba_algorithm) multiplication algorithm, with complexity at best O((log n) ^ 2).

Using an algorithm discovered in 2019 (cf. Harvey and van der Hoeven), it's possible to multiply in O(log n * log log n), if you have a lot of time. This is presumably optimal.

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Sure, or you just define the input (the n) to be the number of digits in (the logarithm of) the number you’re summing up to. That’s generally how people talk about big O notation when it comes to basic arithmetic operations, mostly because we’re usually talking about computers with constant-time instructions for basic arithmetic operations (granted, only up to the maximum integer or float supported by the software).

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