Riemann’s zeta function and finite Dirichlet series

2016 ◽  
Vol 27 (6) ◽  
pp. 985-1002 ◽  
Author(s):  
Yu. V. Matiyasevich
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Riemann’S Zeta Function

The Riemann-Siegel expansion for the zeta function: high orders and remainders

1995 ◽  
Vol 450 (1939) ◽  
pp. 439-462 ◽  
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Critical Line ◽  
High Accuracy ◽  
Stokes Line ◽  
Power Dependence ◽  
Multiplier Function ◽  
Riemann’S Zeta Function

On the critical line s ═ ½ + i t ( t real), Riemann’s zeta function can be calculated with high accuracy by the Riemann-Siegel expansion. This is derived here by elementary formal manipulations of the Dirichlet series. It is shown that the expansion is divergent, with the high orders r having the familiar 'factorial' divided by power' dependence, decorated with an unfamiliar slowly varying multiplier function which is calculated explicitly. Terms of the series decrease until r ═ r * ≈ 2π t and then increase. The form of the remainder when the expansion is truncated near r * is determined; it is of order exp(-π t ), indicating that the critical line is a Stokes line for the Riemann-Siegel expansion. These conclusions are supported by computations of the first 50 coefficients in the expansion, and of the remainders as a function of truncation for several values of t .

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