2002/05/02 by Michael J. Jacobson, Hugh C. Williams · 2 citations
Mathematics · #Analytic Number Theory Research #History and Theory of Mathematics #Advanced Mathematical Identities #Discriminant #Mathematics #Riemann hypothesis #Combinatorics #Polynomial #Quadratic equation #Prime (order theory) #Order (exchange) #Class number #Delta #Discrete mathematics #Pure mathematics #Mathematical analysis #Geometry #Physics
paper · pdf · doi:10.1090/s0025-5718-02-01418-7
openalex publication_date 2002/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11
Hardy and Littlewood’s Conjecture F implies that the asymptotic density of prime values of the polynomials <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript upper A Baseline left-parenthesis x right-parenthesis equals x squared plus x plus upper A comma"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>A</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">fA(x) = x2 + x + A,</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="upper A element-of double-struck upper Z"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">A ∈ \mathbb Z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , is related to the discriminant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta equals 1 minus 4 upper A"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo> − </mml:mo> <mml:mn>4</mml:mn> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">Δ = 1 - 4A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript upper A Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>A</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">fA(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> via a quantity <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis normal upper Delta right-parenthesis period"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C(Δ ).</mml:annotation> </mml:semantics> </mml:math> </inline-formula> The larger <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis normal upper Delta right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C(Δ )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is, the higher the asymptotic density of prime values for any quadratic polynomial of discriminant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta"> <mml:semantics> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:annotation encoding="application/x-tex">Δ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . A technique of Bach allows one to estimate <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis normal upper Delta right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C(Δ )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> accurately for any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">Δ > 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , given the class number of the imaginary quadratic order with discriminant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta"> <mml:semantics> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:annotation encoding="application/x-tex">Δ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and for any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:m