Predicting Maximal Gaps in Sets of Primes

• Main Authors: Alexei Kourbatov, Marek Wolf
• Format: Article
• Language: English
• Published: MDPI AG 2019-05-01
• Series: Mathematics
• Subjects: Cramér conjecture; Gumbel distribution; prime gap; prime <i>k</i>-tuple; residue class; Shanks conjecture; totient
• Online Access: https://www.mdpi.com/2227-7390/7/5/400

Let <inline-formula> <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>&gt;</mo> <mi>r</mi> <mo>&#8805;</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> be coprime integers. Let <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="double-struck">P</mi> <mi>c</mi> </msub> <mo>=</mo> <msub> <mi mathvariant="double-struck">P</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> be an increasing sequence of primes <i>p</i> satisfying two conditions: (i) <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>&#8801;</mo> <mi>r</mi> </mrow> </semantics> </math> </inline-formula> (mod <i>q</i>) <i>and</i> (ii) <i>p</i> starts a prime <i>k</i>-tuple with a given pattern <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">H</mi></semantics></math></inline-formula>. Let <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>&#960;</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> be the number of primes in <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="double-struck">P</mi> <mi>c</mi> </msub> </semantics> </math> </inline-formula> not exceeding <i>x</i>. We heuristically derive formulas predicting the growth trend of the maximal gap <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>G</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mo movablelimits="true" form="prefix">max</mo> <mrow> <msup> <mi>p</mi> <mo>&#8242;</mo> </msup> <mo>&#8804;</mo> <mi>x</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mo>&#8242;</mo> </msup> <mo>&#8722;</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> between successive primes <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>,</mo> <msup> <mi>p</mi> <mo>&#8242;</mo> </msup> <mo>&#8712;</mo> <msub> <mi mathvariant="double-struck">P</mi> <mi>c</mi></msub></mrow></semantics></math></inline-formula>. Extensive computations for primes up to <inline-formula> <math display="inline"> <semantics> <msup> <mn>10</mn> <mn>14</mn> </msup> </semantics> </math> </inline-formula> show that a simple trend formula <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>G</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>&#8764;</mo> <mfrac> <mi>x</mi> <mrow> <msub> <mi>&#960;</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mfrac> <mo>&#183;</mo> <mrow> <mo stretchy="false">(</mo> <mi>log</mi> <msub> <mi>&#960;</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msub> <mi>O</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> works well for maximal gaps between initial primes of <i>k</i>-tuples with <inline-formula> <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>&#8805;</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula> (e.g., twin primes, prime triplets, etc.) in residue class <i>r</i> (mod <i>q</i>). For <inline-formula> <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn></mrow></semantics></math></inline-formula>, however, a more sophisticated formula <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>G</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>&#8764;</mo> <mfrac> <mi>x</mi> <mrow> <msub> <mi>&#960;</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mfrac> <mo>&#183;</mo> <mfenced separators="" open="(" close=")"> <mi>log</mi> <mfrac> <mrow> <msubsup> <mi>&#960;</mi> <mrow> <mi>c</mi> </mrow> <mn>2</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mi>x</mi> </mfrac> <mo>+</mo> <mi>O</mi> <mrow> <mo stretchy="false">(</mo> <mi>log</mi> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> </mrow> </semantics> </math> </inline-formula> gives a better prediction of maximal gap sizes. The latter includes the important special case of maximal gaps in the sequence of all primes (<inline-formula><math display="inline"><semantics><mrow><mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>1</mn></mrow></semantics></math></inline-formula>). The distribution of appropriately rescaled maximal gaps <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>G</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> is close to the Gumbel extreme value distribution. Computations suggest that almost all maximal gaps satisfy a generalized strong form of Cram&#233;r&#8217;s conjecture. We also conjecture that the number of maximal gaps between primes in <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="double-struck">P</mi> <mi>c</mi> </msub> </semantics> </math> </inline-formula> below <i>x</i> is <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>O</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>log</mi> <mi>x</mi> <mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>.