Small class numbers and extreme values of 𝐿-functions of quadratic fields

Abstract

The table of class numbers <italic>h</italic> of imaginary quadratic fields described in [1] was placed on magnetic tape. This tape was then processed to find the occurrences of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h less-than-or-slanted-equals 125"> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>⩽<!-- ⩽ --></mml:mo> <mml:mn>125</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">h \leqslant 125</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and to find the successive extreme values of the Dirichlet <italic>L</italic>-functions <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 Subscript negative upper D Baseline 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:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>χ<!-- χ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mi>D</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L(1,{\chi _{ - D}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="chi Subscript negative upper D"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>χ<!-- χ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mi>D</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\chi _{ - D}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the Kronecker symbol of the field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q left-parenthesis StartRoot negative upper D EndRoot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msqrt> <mml:mo>−<!-- − --></mml:mo> <mml:mi>D</mml:mi> </mml:msqrt> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Q(\sqrt { - D} )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of discriminant - <italic>D</italic>. A comparison was made between the observed extrema and the bounds obtained for the <italic>L</italic>-functions by Littlewood [5] assuming Riemann hypotheses.

Locations

• Mathematics of Computation - View - PDF