Codes for the paper "AT LEAST TWO OF ζ(5), ζ(7), . . . , ζ(35) ARE IRRATIONAL" Li Lai, Li Zhou.
The paper could be downloaded from arxiv: https://arxiv.org/abs/2103.00904.
main_zeta35.m is the code for the proof of main Theorem 1.2 : At least two of ζ(5), ζ(7), ... , ζ(35) are irrational.
For the purpose of consistency, the names of variables are the same as those in the paper.
Take the parameters as follows:
It takes almost 1 minute to get the results as follows :
x0 = 2.894938334463284,
C1 = 1.677993128952727e+04,
C2 = 1.677928264107618e+04.
main_beta10.m is the code for the proof of Theorem 6.1: At least one of β(2), β(4), β(6), β(8), β(10) is irrational.
For
It takes a few seconds to get the results as follows:
We provide more details in functionvalue.txt，this text records the explicit expression of the step function on the interval [0,1).
In an early version, the parameters we used are
the results are:
C1 = -905.2339772598 and C2 = 903.4669134561.
One of the referees provided two more collections of parameters with smaller :
C1 = -167.1117519102 and C2 = 166.9923927370,
C1 = -169.4835461763 and C2 = 168.9817093310.
This motivated us to run a more extensive MATLAB program, we did not exhaust all parameters with , which is impractical. But we checked more than 10^5 collections of parameters such that the ratio
is close to [94:32:32:32:32:33:34:35:36:37:38:39]. It is likely that
is the minimal possible choice. During the process, we found five more collections of parameters as follows:
C1 = -165.5986604745 and C2 = 164.3470880379,
C1 =-173.9203291400 and C2 = 172.8309128877,
C1 = -137.7413056968 and C2 = 136.8284928437,
C1 = -143.0851483016 and C2 = 142.1598888402,
C1 = -184.8701856342 and C2 = 183.8905111835.