Neue Herleitung und explizite Restabschätzung der Riemann-Siegel-Formel

The asymptotic expansion of the function $Z(t)=e^{i\vartheta(t)}\zeta{(1/2+it)}$ for real $t\to+\infty$ where $\vartheta(t)=\Im\log{\Gamma{(1/4+it/2)}}-(t\log{\pi})/2$ – today known as Riemann–Siegel formula – is derived in a new and simpler way. Simplified computation formulas of its coefficients are given as well as explicit error estimates of its first 11 partial sums for $t \ge 200$. In an appendix the first 13 coefficients of the asymptotic series are presented in a form introduced by D. H. Lehmer in 1956. Power series and expansions in terms of Cebysev Polynomials are given for the first 11 coefficients to 50 decimals.

[1]  D. H. Lehmer.  On the roots of the Riemann zeta-function , 1956 .

[2]  Thorsten Gerber,et al.  Handbook Of Mathematical Functions , 2016 .

[3]  J. Arias de Reyna,et al.  Dynamical zeta functions and Kummer congruences , 2003, math/0309190.

[4]  A. Odlyzko.  On the distribution of spacings between zeros of the zeta function , 1987 .

[5]  C. Siegel,et al.  Über Riemanns Nachlaß zur analytischen Zahlentheorie , 1966 .

[6]  R. Brent,et al.  THE FIRST 200,000,001 ZEROS OF RIEMANN'S ZETA FUNCTION , 1982 .

[7]  J. Barkley Rosser,et al.  Rigorous computation and the zeros of the Riemann zeta-function , 1968, IFIP Congress.

[8]  H. Behnke,et al.  Theorie der analytischen Funktionen einer komplexen Veränderlichen , 1965 .

[9]  J. Keating,et al.  A new asymptotic representation for ζ(½ + it) and quantum spectral determinants , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[10]  Arnold Schönhage,et al.  Fast algorithms for multiple evaluations of the riemann zeta function , 1988 .

[11]  L. M. Milne-Thomson.  Methoden der mathematischen Physik , 1944, Nature.

[12]  te Herman Riele,et al.  On the Sign of the Difference ir( x) — li( x) , 2010 .

[13]  T. Apostol.  Introduction to analytic number theory , 1976 .

[14]  Mark J. Christensen.  THE DEFINITE INTEGRAL , 1981 .

[15]  Tony Feng,et al.  Riemann's Zeta Function , 2014 .

[16]  M. Berry,et al.  The Riemann-Siegel expansion for the zeta function: high orders and remainders , 1995, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[17]  Lazhar Fekih-Ahmed,et al.  On the zeros of the Riemann Zeta function , 2010, 1004.4143.

[18]  Juan Arias de Reyna.  High precision computation of Riemann's zeta function by the Riemann-Siegel formula, I , 2011, Math. Comput..

[19]  J. V. deLune,et al.  Rigorous high speed separation of zeros of Riemann's zeta function , 1984 .

[20]  Richard P. Brent,et al.  On the zeros of the Riemann zeta function in the critical strip , 1979 .

[21]  D. H. Lehmer.  Extended computation of the Riemann zeta-function , 1956 .

[22]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[23]  Richard P. Brent,et al.  High Precision Coefficients Related to the Zeta Function. , 1977 .

[24]  D. R. Heath-Brown,et al.  The Theory of the Riemann Zeta-Function , 1987 .

[25]  Jörg Brüdern.  Einführung in die analytische Zahlentheorie , 1995 .

[26]  Jonathan M. Borwein,et al.  Computational strategies for the Riemann zeta function , 2000 .

[27]  Michael V. Berry,et al.  The Riemann Zeros and Eigenvalue Asymptotics , 1999, SIAM Rev..

[28]  Xavier.  The 10 13 first zeros of the Riemann Zeta function , and zeros computation at very large height , 2004 .