Distribution Modulo One of <i>αp</i><sup>γ</sup> + <i>β</i> for Special Classes of Primes

Distribution Modulo One of <i>αp</i><sup>γ</sup> + <i>β</i> for Special Classes of Primes

Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>,</mo><mi>β</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></...

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Bibliographic Details
Main Authors:	Atanaska Georgieva, Tatiana L. Todorova
Format:	Article
Language:	English
Published:	MDPI AG 2025-07-01
Series:	Axioms
Subjects:	linear sieve almost primes distribution modulo one
Online Access:	https://www.mdpi.com/2075-1680/14/7/532
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Summary:	Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>,</mo><mi>β</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>≠</mo><mn>0</mn></mrow></semantics></math></inline-formula>, and let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>γ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>5</mn><mo>/</mo><mn>6</mn><mo>)</mo></mrow></semantics></math></inline-formula>. Define the set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>M</mi><mn>1</mn></msub></semantics></math></inline-formula> to consist of primes <i>p</i> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>+</mo><mn>2</mn></mrow></semantics></math></inline-formula> is almost prime, and let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>M</mi><mn>2</mn></msub></semantics></math></inline-formula> be the set of primes of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></semantics></math></inline-formula>. We study the distribution of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><msup><mi>p</mi><mi>γ</mi></msup><mtext> </mtext><mo>+</mo><mtext> </mtext><mi>β</mi></mrow></semantics></math></inline-formula> modulo one, as <i>p</i> ranges over the sets <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>M</mi><mn>1</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>M</mi><mn>2</mn></msub></semantics></math></inline-formula>, respectively.
ISSN:	2075-1680