Density ternary Goldbach for primes in a fixed residue class
Abstract
<jats:p> We prove that if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> is a subset of those primes which are congruent to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mspace width="4.44443pt"/> <mml:mo>(</mml:mo> <mml:mo form="prefix">mod</mml:mo> <mml:mspace width="0.277778em"/> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> such that the relative density of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> in this residue class is larger than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> then every sufficiently large odd integer <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> which satisfies <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≡</mml:mo> <mml:mn>0</mml:mn> <mml:mspace width="4.44443pt"/> <mml:mo>(</mml:mo> <mml:mo form="prefix">mod</mml:mo> <mml:mspace width="0.277778em"/> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> can be written as a sum of three primes from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> . Moreover the threshold of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:math> for the relative density is best possible. </jats:p>