Hi,

I've run clang-format and found that the code formatting is good.

Thanks for fixing the formatting.

I have separated the numeric calculation from the symbolic in this PR. For this particular PR it is a minor thing, but I think for other number theoretic functions that are slow and difficult to calculate numerically similar separations would come with a number of benefits:

1. The numeric calculation could be used stand alone.
2. The numeric caluculations could be imported and wrapped from other libraries in the future.
3. For functions very slow to calculate like the mertens function we could set a limit on the "auto-calculation", defer the calculation and let a future doit function do the actual calculation. For example M(1000000000000) would take a long time to calculate and could be kept unevaluated until doit is called on the expression. This way expressions like M(1000000000000) * x would be super fast and if x is substituted for 0 in the future the calculation of M would never be needed.
4. The current ntheory.cpp is a mix of pure numeric and "semi-symbolic" functions. It would be easier to find and use the functions if they would be clearly separated.

This is ready for review

What's the name that sympy use? I can't find it in sympy

It is not available in sympy as far as I know. It is in mathematica though (except for the root). See the links above.

Thanks. Can you add these links in the code comments as well?

Sure! Good point.

Sorry, did a bad merge.

Should be good now. Fixed bad merge and squashed commits. Also added function documentation with the references above.