Category Archives: mpmath

Testing Li’s criterion

Posted on March 28, 2013 by Fredrik

In this post, I will look at testing the Riemann hypothesis via Li’s criterion and numerical evaluation. Li’s criterion involves a property of Riemann xi function $$\xi(s) = (s-1) \pi^{-s/2} \Gamma\left(1+\tfrac{1}{2} s\right) \zeta(s),$$ so-defined because it puts the functional equation … Continue reading →

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Coping with a big argument

Posted on March 26, 2013 by Fredrik

Implementing special functions so that they work properly on the entirety of their (representable) domain is a very hard problem. Take for example the digamma function: it behaves like $\psi(z) \sim \log z$, so you would never really expect it … Continue reading →

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Machin-like formulas for logarithms

Posted on March 21, 2013 by Fredrik

Machin-like formulas express $\pi$ as an integer combination of arctangents evaluated at reciprocals of integers. The most famous is $$\pi = 16 \arctan \frac{1}{5} – 4 \arctan \frac{1}{239}$$ which historically has been used for many record computations of $\pi$, including … Continue reading →

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An even speedier gamma function

Posted on February 27, 2013 by Fredrik

I’ve just pushed some new code to the Arb git repository for computing numerical rising factorials ($x (x+1) (x+2)\cdots (x+n-1)$) faster. It works by expanding subproducts as symbolic polynomials, and evaluating them using the rectangular splitting algorithm mentioned in the … Continue reading →

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