Generations of Correlation Averages

Abstract

We give a general link between weighted Selberg integrals of any arithmetic function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>f</mml:mi></mml:mrow></mml:math> and averages of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mi>f</mml:mi></mml:mrow></mml:math> correlations in short intervals, proved by the elementary dispersion method (our version of Linnik’s method). We formulate conjectural bounds for the so-called modified Selberg integral of the divisor functions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, gauged by the Cesaro weight in the short interval <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mfenced open="[" close="]" separators="|"><mml:mrow><mml:mi>x</mml:mi><mml:mo>-</mml:mo><mml:mi>H</mml:mi><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mo>+</mml:mo><mml:mi>H</mml:mi></mml:mrow></mml:mfenced></mml:math> and improved by these some recent results by Ivić. The same link provides, also, an unconditional improvement. Then, some remarkable conditional implications on the 2<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math>th moments of Riemann zeta function on the critical line are derived. We also give general requirements on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mrow><mml:mi>f</mml:mi></mml:mrow></mml:math> that allow our treatment for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:mrow><mml:mi>f</mml:mi></mml:mrow></mml:math> weighted Selberg integrals.

Locations

• Journal of Numbers - View - PDF
• arXiv (Cornell University) - View - PDF
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