# Analytic number theory functions

Linas Vepstas linasvepstas at gmail.com
Sat Oct 29 22:22:21 UTC 2016

```FYI, this is sort-of an announce:

For the last decade, I've been maintaining a very ad-hoc library of
assorted gmp functions, useful in analytic combinatorics and analytic
number theory. Some/many of these functions are not available in MFPR; they
are all GMP compatible.

Samples include the polylogarithm, the periodic zeta, the topologists' sine
(this occurs in Dirichlet series). the GKW oper, the Minkowski question
mark, the confluent hypergeometric. -- all of these for complex values.

The github account shows that no one has clone the thing, which means that
no one seems to know about it. So, yet another announce:

https://github.com/linas/anant

A more details list of what is in there:

Arbitrary precision constants:
------------------------------
* sqrt(3)/2, log(2)
* e, e^pi
* pi, 2pi, pi/2, sqrt(2pi), log(2pi), 2/pi
* Euler-Mascheroni const
* Riemann zeta(1/2)

Combinatorial functions:
------------------------
* Rising pochhammer symbol (integer)
* Reciprocal factorial
* Sequential binomial coefficient
* Stirling Numbers of the First kind
* Stirling Numbers of the Second kind
* Bernoulli Numbers
* Binomial transform of power sum
* Rising pochhammer symbol (real)    i.e. (s)_n for real s
* Rising pochhammer symbol (complex) i.e. (s)_n for complex s
* Binomial coefficient (complex)     i.e. (s choose n) for complex s

Elementary functions:
---------------------
* pow, exp, log, sine, cosine, tangent for real, complex arguments
* arctan, arctan2 for real argument
* log(1-x) for real, complex x
* sqrt for complex argument

Classical functions:
--------------------
* gamma (factorial) for real, complex argument
* polylogarithm, using multiple algorithms: Borwein-style, Euler-Maclaurin
* polylogarithm on multiple sheets (monodromy)
* Periodic zeta function
* Hurwitz zeta function, using multiple algorithms; complex arguments.
* Riemann zeta function, using multiple algorithms: Borwein, Hasse,
brute-force
for integer, real and complex arguments.
* Confluent hypergeometric function, complex arguments

Number-theoretic functions:
---------------------------
* General complex-valued harmonic number
* Gauss-Kuzmin-Wirsing operator matrix elements
* Minkowski Question Mark function (Stern-Brocot tree), and its inverse
* Taylor's series coefficients for the topologist's sin -- sin(2pi/(1+x))
```