Hello everybody, I have composed a program (in Python using the mpmath library) to compute to high precision zeta(s) and Z(t) (the Riemann-Siegel function) by means of the Riemann-Siegel expansion.
I have obtained, I think that for the first time, rigorous bounds for the terms and rest of the Riemann-Siegel expansion even for points off the critical line. The program is running well and is comparable with Mathematica, even sometimes it runs faster. I even have found cases where Mathematica gives wrong results and my program doesn't. For example: Z(100000*^(2**20)) with 9 decimal digits cpu Mathematica 1.719 cpu mpmath 11.12 (but the value given by Mathematica is erroneous -0.00112995 the value given by mpmath 2.6302645) Mathematica when computing to 30 digits 2.63026450661456012104256623412 ===== Z(100000*^(2**20)) with 180 decimal digits cpu Mathematica 25.406 cpu mpmath 31.15 (cpu on zetasum 28.62) ===== Z(10 000 000) with 600 decimal digits Mathematica gives an error message and computes 505 decimal digits in cpu time 314.891 sec. Augmenting the internal variable $MaxExtraPrecision Mathematica computes 600 digits in 1008.19 additional sec. cpu time of mpmath 837.08 (giving the 600 correct digits) ===== Z(9460455379268814.253184) with 100 decimal digits cpu time Math 4334.89 sec cpu time of mpmath 6259.4623333 of which 6257.3257419 spent in the zetasum (this is the biggest value known of Z(t) = 937.1985242635 6253567375 .... ) ==== When I ask for moderate precision usually we have cpu time Math < cpu time mpmath < 2*cpu time Math But much of the time my program spent in computing the zetasum that almost surely Mathematica also sums. ====== But I have some fears since I am not a programmer. I do not knew nothing about Python even the name when I started in February of 2009 to write the program. I knew the Sage, and my first intention was to make it for Sage. There I read for the first time of Python. Then I saw mpmath. That was a revelation. As I use Windows my installation of Sage were not accessible. The programs of mpmath were accessible for me, they were my school for Python. So, my fear is that my program although running, must have many mistakes of a beginner. Also I think the program can be made faster with some modifications, for example: using a C-implementation to compute the zeta sum. Fredrik Johansson advised me to post this message on this list. I would like to see my program (or some adequate modification) included in the mpmath (or sage). I pretended to send with this message my program and a pdf file with some benchmarks. But I do not know how to do it. I will send it to you if you are interested. Thanks for your attention, Juan Arias de Reyna -- To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
