The Riemann zeta function ζ(*s*) is given by an infinite sum and an infinite product

for complex numbers *s* with real part greater than 1 [*].

The **infinite** sum is equal to the **infinite** product, but which would give you more accuracy: *N* terms of the sum or *N* terms of the product? We’ll take a look at this question empirically.

The accuracy of the partial sum and the partial product depends on *N* and on *s*. For starters, let’s fix *s* = 3 and look at the approximation error as a function of *N*. Note that the error is plotted on a log scale.

Now let’s fix *N* = 10 and look at the error as a function of *s*.

So our two examples suggest that taking *N* terms of the product gives more accuracy than taking *N* terms of the sum.

In the paper mentioned in the previous post, the author approximates the zeta function with a few terms of the product. That’s what motivated this post. I thought the product might give more accuracy than the sum, and that appears to be the case.

## Related posts

[*] If you evaluate the series for ζ at *s* = -1, despite the fact that the equation does not hold at that point, and evaluate ζ at -1 using analytic continuation, you get the infamous nonsensical result that the positive integers sum to -1/12.

It’s not too surprising… Even at N = 1, the product is winning because it corresponds to the sum over all multiples of 2, not just the single ‘2’.

Similarly N =2 contains all multiples of 2 and 3 (2^i * 2^j), etc.

Small correction: I meant multiples of 2 alone (i.e. powers of 2) for N=1