2013/11/18 by Tomás Oliveira e Silva, Siegfried Herzog, Silvio Pardi · 10 citations
Mathematics · #Analytic Number Theory Research #Advanced Mathematical Identities #Algebraic Geometry and Number Theory
paper · pdf · doi:10.1090/s0025-5718-2013-02787-1
This paper describes how the even Goldbach conjecture was confirmed to be true for all even numbers not larger than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4 dot 10 Superscript 18"> <mml:semantics> <mml:mrow> <mml:mn>4</mml:mn> <mml:mo> ⋅ </mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>18</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">4⋅ 1018</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Using a result of Ramaré and Saouter, it follows that the odd Goldbach conjecture is true up to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="8.37 dot 10 Superscript 26"> <mml:semantics> <mml:mrow> <mml:mn>8.37</mml:mn> <mml:mo> ⋅ </mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>26</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">8.37⋅ 1026</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The empirical data collected during this extensive verification effort, namely, counts and first occurrences of so-called minimal Goldbach partitions with a given smallest prime and of gaps between consecutive primes with a given even gap, are used to test several conjectured formulas related to prime numbers. In particular, the counts of minimal Goldbach partitions and of prime gaps are in excellent accord with the predictions made using the prime <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -tuple conjecture of Hardy and Littlewood (with an error that appears to be <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis StartRoot t log log t EndRoot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msqrt> <mml:mi>t</mml:mi> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mi>t</mml:mi> </mml:msqrt> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(√ tlog log t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t"> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding="application/x-tex">t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the true value of the quantity being estimated). Prime gap moments also show excellent agreement with a generalization of a conjecture made in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1982"> <mml:semantics> <mml:mn>1982</mml:mn> <mml:annotation encoding="application/x-tex">1982</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by Heath-Brown.