Approximation of Riemann's zeta function by finite Dirichlet series: multiprecision numerical approach

Abstract

The finite Dirichlet series from the title are defined by the condition that they vanish at as many initial zeroes of the zeta function as possible. It turned 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 accuracy numerical experiments. So far no theoretical explanation to the observed phenomena was found.

References in corpus (2)

• A Parallel Algorithm for Calculation of Large Determinants with High Accuracy for GPUs and MPI clusters
• Zeros of partial sums of the Riemann zeta-function

Cited by in corpus (7)

• A Parallel Algorithm for Calculation of Large Determinants with High Accuracy for GPUs and MPI clusters
• Discretized Keiper/Li approach to the Riemann Hypothesis
• Two arguments that the nontrivial zeros of the Riemann zeta function are irrational
• On the nontrivial zeros of the Dirichlet eta function
• The Mangoldt function and the non-trivial zeros of the Riemann zeta function
• Arb: Efficient Arbitrary-Precision Midpoint-Radius Interval Arithmetic
• Generalized coefficients of the Dirichlet series