Appropriate values of zeta functions

Question

What are some approximate values for $\zeta(3)$ and $\zeta (1.5)$ (upto three decimal places)? Looked at the paper "Chebyshev approximations for Riemann zeta functions" but I don't think those approximations are easily computable (at least for me).

Answer 1

Let $Z(x, M) = \sum_{n=1}^M n^{-x}$ be a truncated approximation of $\zeta(x)$. Simply using the naïve summation

def Z(x, M):
    return sum(n ** (-x) for n in range(1, M + 1))

The approximation of $\zeta(3) \approx 1.202056903150321$ converges fairly quickly, with maximum accuracy achieved after $10^6$ iterations.

• $Z(3, 10) \approx 1.197531985674193$
• $Z(3, 10^2) \approx 1.2020074006596781$
• $Z(3, 10^3) \approx 1.2020564036593433$
• $Z(3, 10^4) \approx 1.202056898160098$
• $Z(3, 10^5) \approx 1.2020569031097323$
• $Z(3, 10^6) \approx 1.202056903150321$
• $Z(3, 10^7) \approx 1.202056903150321$
• $Z(3, 10^8) \approx 1.202056903150321$
• $Z(3, 10^9) \approx 1.202056903150321$

OTOH, the approximation of $\zeta(1.5)$ converges more slowly, with the result still not being settled after $10^9$ iterations.

• $Z(1.5, 10) \approx 1.9953364933456017$
• $Z(1.5, 10^2) \approx 2.412874098703719$
• $Z(1.5, 10^3) \approx 2.5491456029175756$
• $Z(1.5, 10^4) \approx 2.5923758486729866$
• $Z(1.5, 10^5) \approx 2.6060508091764736$
• $Z(1.5, 10^6) \approx 2.6103753491852295$
• $Z(1.5, 10^7) \approx 2.611742893169012$
• $Z(1.5, 10^8) \approx 2.6121753486854478$
• $Z(1.5, 10^9) \approx 2.6123121030481857$

But if 3 decimal places is all you need, just say $\zeta(1.5) \approx 2.612$.

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Or, you could simply follow @Conrad's advice and consult the great oracle of mathematical knowledge, WolframAlpha, which gives:

• Zeta(3) = 1.2020569031595942853997381615114499907649862923404988817922715553...
• Zeta(1.5) = 2.6123753486854883433485675679240716305708006524000634075733282488...