Sums of Three or More Primes

Abstract

It has long been known that, under the assumption of the Riemann Hypothesis, one can give upper and lower bounds for the error <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript p less-than-or-equal-to x Endscripts log p minus x"> <mml:semantics> <mml:mrow> <mml:munder> <mml:mo>∑</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> </mml:munder> <mml:mi>log</mml:mi> <mml:mo>⁡</mml:mo> <mml:mi>p</mml:mi> <mml:mo>−</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\sum _{p \le x} \log p - x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the Prime Number Theorem, such bounds being within a factor of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis log x right-parenthesis squared"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>log</mml:mi> <mml:mo>⁡</mml:mo> <mml:mi>x</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">(\log x)^{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of each other and this fact being equivalent to the Riemann Hypothesis. In this paper we show that, provided “Riemann Hypothesis” is replaced by “Generalized Riemann Hypothesis”, results of similar (often greater) precision hold in the case of the corresponding formula for the representation of an integer as the sum of <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> primes for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k greater-than-or-equal-to 4"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k \ge 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and, in a mean square sense, for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k greater-than-or-equal-to 3"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k \ge 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also sharpen, in most cases to best possible form, the original estimates of Hardy and Littlewood which were based on the assumption of a “Quasi-Riemann Hypothesis”. We incidentally give a slight sharpening to a well-known exponential sum estimate of Vinogradov-Vaughan.

Locations

• Transactions of the American Mathematical Society - View - PDF