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Renaat Van Malderen

Publications and source records attributed to Renaat Van Malderen.

7 recordsLinked to original sources
arXiv2016

Riemann's Zeta Function: The alternating Xi-Function Xia(s)

As well known, the study of Riemanns zeta function ζ(s) involves the related entire function ξ(s). A close relative of ζ(s) is the alternating zeta function η(s). Similar to ζ(s), also η(s) has a corresponding entire function ξ_a (s). After establishing its definition and a related functional equation, formulas based on incomplete gamma functions are worked out, allowing to numerically evaluate ξ_a (s). Examples verifying the obtained formulas are included.

math.NT↗
arXiv2013

Riemann Zeta Function. Chi(s)via analytic Continuation inside an infinite Loop

The paper deals with the analytic entire function Chi(s) closely related to Riemann Zeta Function Zeta(s). A formula is obtained for Chi(s) essentially within the so-called critical strip. This is achieved by applying Cauchy integral formula to an infinite loop encircling the critical strip. In the obtained formula a remarkable role is played by a special type of complex function known as Incomplete Gamma Function. Numerical examples verifying the obtained formula are included.

math.NT↗
arXiv2012

Riemann Zeta Function. An Attempt to fathom Zeta(3)

Already in 1734 Euler found a short explicit formula for the value of Riemann zeta function Zeta(s) when the argument s equals a positive integer 2n where n=1,2,3,. No such formula exists for odd positive integer arguments of Zeta. The present paper discusses in particular the case of Zeta(3). A formula for Zeta(3) is obtained which in addition to a number of well known constants includes a rapidly converging infinite series, of which each term contains rational numbers and an even power of Pi. An attempt to convert this series into a finite number of terms containing commonly known constants is met with only partial success. The general case for zeta(2n+1) is also worked out.

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arXiv2011

Riemann's Zeta Function. Numerical Evaluation via its Alternating Relative η(s)

The paper describes a method for calculating values of Riemann's Zeta function within the critical strip 0< σ <1 and on its boundary. The approach is based on the "Alternating Zeta function" η(s). The actual Riemann Zeta function ζ(s) is easily obtained from η(s). The obtained accuracy, within certain limits of the described method is discussed. A decent scientific calculator suffices to carry out the involved computations.

math.NT↗
arXiv2009

On the Distribution of the non-trivial Roots of Riemann's Zeta Function. An alternative Approach

Expressing Weierstrass type infinite products in terms of Stieltjes integrals is discussed. The asymptotic behavior of particular types of infinite products is compared against the asymptotic behavior of the entire function Xi(s), well-known in Riemann zeta function theory. An approximate formula for the distribution of the non-trivial roots of Riemann's zeta function is obtained.

math.NT↗
arXiv2005

Explicit formula for even-index Bernoulli numbers

Bernoulli numbers are usually expressed in terms of their lower index numbers (recursive). This paper gives an explicit formula for Bernoulli numbers of even index. The formula contains a remarkable sequence of determinants.

math.NT↗