Approximation of Riemann’s Zeta Function by Finite Dirichlet Series: A Multiprecision Numerical Approach

Abstract

The finite Dirichlet series of the title are defined by the condition that they vanish at as many initial zeros of the zeta function as possible. It turns out that such series can produce extremely good approximations to the values of Riemann’s zeta function inside the critical strip. In addition, the coefficients of these series have remarkable number-theoretical properties discovered in large-scale high-precision numerical experiments. So far, we have found no theoretical explanation for the observed phenomena.

2000 AMS Subject Classification::

• 11M26
• 11M06
• 11-04
• 65F40
• 33F05
• 65D99

Keywords:

• L-function
• Riemann hypothesis
• zeta function

Notes

¹Further details can be found at the website [Matiyasevich 14].

²The Arprec library can be found at http://crd-legacy.lbl.gov/dhbailey/mpdist/; accessed June 10, 2013. The GMP package is available at http://gmplib.org">http://gmplib.org">http://gmplib.org; accessed June 10, 2013. Arb is available at http://fredrikj.net/arb/.