A computational technique for evaluating 𝐿(1,𝜒) and the class number of a real quadratic field

Abstract

A description is given of a method for estimating <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L left-parenthesis 1 comma chi right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi> χ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L(1,\chi )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to sufficient accuracy to determine the class number of a real quadratic field. This algorithm was implemented on an IBM/370-158 computer and the class number, regulator, and value of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L left-parenthesis 1 comma chi right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi> χ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L(1,\chi )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> were obtained for each real quadratic field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q left-parenthesis StartRoot upper D EndRoot right-parenthesis left-parenthesis upper D equals 2 comma 3 comma ellipsis comma 149999 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msqrt> <mml:mi>D</mml:mi> </mml:msqrt> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="thickmathspace"/> <mml:mo stretchy="false">(</mml:mo> <mml:mi>D</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:mn>149999</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Q(\sqrt D )\;(D = 2,3, \ldots ,149999)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Several tables, summarizing various results of these computations, are also presented.