# Kloostermania turns 1.0

Since the Pocket Kloostermania has remained unchanged for almost five years, an update is probably welcome! This is now available, in time to honor N. Katz on the occasion of this 71st birthday (which I will help celebrate in China next week).

Important changes in the new version:

(1) It was recompiled — hence a new modern look instead of a style reminiscent of the dark ages –, and the resulting binary should work on any Android system with version 4.0.3 or later;

(2) To keep up with recent progress, the program now displays Kloosterman sums and/or Birch sums, instead of Kloosterman and/or Salié sums. On the other hand, the moduli used are now restricted to primes, to simplify things a bit, and only one parameter is used: the sums displayed are
$\sum_{x\in\mathbf{F}_p^{\times}}e\Bigl(\frac{ax+\bar{x}}{p}\Bigr)$
and
$\sum_{x\in\mathbf{F}_p}e\Bigl(\frac{ax+x^3}{p}\Bigr)$

(3) In addition to being able to change the modulus by swiping horizontally, a vertical swipe on the screen scrolls among the values of the parameter $a$. As was the case in the previous version (and even more because phones are faster now), the scrolling is usually too fast for a single step, so tapping once close to one of the edges of the window displaying the sum will perform just one step of the corresponding move (e.g., tapping close to the right of the screen goes to the next prime modulus). The value of the sum and its parameters and then displayed quickly.

(4) The plots are presented in orthonormal configurations, to give a more faithful representation of the paths in the plane;

(5) In the “About” dialog, a single click will display (if a PDF viewer is installed) the paper of W. Sawin and myself that explains the limiting distribution of Kloosterman (and Birch) paths…

(6) It is possible to save (or rather to “share” in the usual Android way) a picture of the sums currently displayed, as a PNG file.

(6) And the launcher icon is better.

The installation file is available right now on the updated Kloostermania page!