On the zeros of the Riemann zeta function in the critical strip. II

Richard P. Brent, J. Lune, H. J. J. te Riele, D.T. Winter

Type: Article

Publication Date: 1982-01-01

Citations: 30

DOI: https://doi.org/10.1090/s0025-5718-1982-0669660-1

Abstract

We describe extensive computations which show that Riemann’s zeta function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="zeta left-parenthesis s right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>ζ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\zeta (s)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has exactly 200,000,001 zeros of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma plus"> <mml:semantics> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mo>+</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma +</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the region <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than t greater-than 81 comma 702 comma 130.19"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi>t</mml:mi> <mml:mo>></mml:mo> <mml:mn>81</mml:mn> <mml:mo>,</mml:mo> <mml:mn>702</mml:mn> <mml:mo>,</mml:mo> <mml:mn>130.19</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">0 > t > 81,702,130.19</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; all these zeros are simple and lie on the line <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma equals one half"> <mml:semantics> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma = \frac {1}{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. (This extends a similar result for the first 81,000,001 zeros, established by Brent in <italic>Math. Comp.</italic>, v. 33, 1979, pp. 1361-1372.) Counts of the numbers of Gram blocks of various types and the failures of "Rosser’s rule" are given.

Locations

Mathematics of Computation - View - PDF
CWI's Institutional Repository (Centrum Wiskunde & Informatica) - View - PDF

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