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Two Arguments that the Nontrivial Zeros of the Riemann Zeta Function are Irrational

Wolf M.

Computational Methods in Science and Technology

2018

Vol. 24, No. 4

215--220

We have used the first 2600 nontrivial zeros γl of the Riemann zeta function calculated with 1000 digits accuracy and developed them into the continued fractions. We calculated the geometrical means of the denominators of these continued fractions and for all cases we get values close to the Khinchin’s constant, which suggests that γl are irrational. Next we have calculated the n-th square roots of the denominators Qn of the convergents of the continued fractions obtaining values close to the Khinchin-Lévy constant, again supporting the common opinion that γl are irrational.

Remarkable identities

Górowski J., Żabowski J.

Journal of Mathematics and Applications

2014

Vol. 37

39--50

In the paper a number of identities involving even powers of the values of functions tangent, cotangent, secans and cosecans are proved. Namely, the following relations are shown: [wzory] where m, n are positive integers, f is one of the functions: tangent, cotangent, secans or cosecans and wf(x),vf(x),~wf(x) are some polynomials from Q[x]. One of the remarkable identities is the following: [wzór] Some of these identities are used to find, by elementary means, the sums of the series of the form [wzór] , where n is a fixed positive integer. One can also notice that Bernoulli numbers appear in the leading coeficients of the polynomials wf(x),vf(x) and ~ wf(x).

Some remarks on the Báez-Duarte criterion for the Riemann hypothesis

Wolf M.

Computational Methods in Science and Technology

2014

Vol. 20, No. 2

39--47

In this paper we are going to describe the results of the computer experiment, which in principle can rule out validity of the Riemann Hypothesis (RH). We use the sequence ck appearing in the Báez-Duarte criterion for the RH and compare two formulas for these numbers. We describe the mechanism of possible violation of the Riemann Hypothesis. Next we calculate c100000 with a thousand digits of accuracy using two different formulas for ck with the aim to disprove the Riemann Hypothesis in the case these two numbers will differ. We found the discrepancy only on the 996th decimal place (accuracy of 10-996). The computer experiment reported herein can be of interest for developers of Mathematica and PARI/GP.

On a class of integrals appearing in subcritical thermodynamics of superconductors

Gonczarek R., Czerwonko J.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 4

403-406

A formula for the family of integrals I[n] = [...], n [belongs to] N, is proved.