2016/02/24 by Jean Bourgain · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Mathematical Analysis and Transform Methods #Advanced Mathematical Physics Problems #Mathematics #Decoupling (probability) #Exponential function #Riemann zeta function #Riemann hypothesis #Exponential growth #Exponential decay #Arithmetic zeta function #Pure mathematics #Riemann Xi function #Mathematical analysis #Physics #Quantum mechanics
paper · pdf · doi:10.1090/jams/860
openalex publication_date 2016/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
We establish a new decoupling inequality for curves in the spirit of earlier work of C. Demeter and the author which implies a new mean value theorem for certain exponential sums crucial to the Bombieri-Iwaniec method as developed further in the work of Huxley. In particular, this leads to an improved bound <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue zeta left-parenthesis one half plus i t right-parenthesis EndAbsoluteValue much-less-than t Superscript 13 slash 84 plus epsilon"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi> ζ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mi>i</mml:mi> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo> ≪ </mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>13</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>84</mml:mn> <mml:mo>+</mml:mo> <mml:mi> ε </mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">|ζ (\frac 12 + it)| ≪ t13/84 + ε </mml:annotation> </mml:semantics> </mml:math> </inline-formula> for the zeta function on the critical line.