Algebra & Number Theory· 2015Q1
An averaged form of Chowla’s conjecture
- 135citations
- Q1SCImago
- 2015year
Short summary
An averaged version of Chowla's conjecture, which states that the product of Liouville functions [math] is small on average, is proven for [math] distinct natural numbers [math] and [math] up to [math].
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Key points
- An averaged form of Chowla's conjecture is proven for [math] distinct natural numbers [math] and a fixed [math].
- The sum [math] is shown to be [math] as [math] goes to infinity.
- A related exponential sum estimate is provided: [math] for [math] and [math] as [math] uniformly.
- The methods extend to provide an averaged form of Elliott's conjecture for general bounded multiplicative functions.
AI-generated from the title and abstract; the full text is not read.
Abstract
Let [math] denote the Liouville function. A well-known conjecture of Chowla asserts that, for any distinct natural numbers [math] , one has ¶\n<math display="block">\n<mrow> <munder class="msub"><mrow><mo mathsize="big">∑</mo> </mrow><mrow><mn>1</mn><mo class="MathClass-rel">≤</mo><mi>n</mi><mo class="MathClass-rel">≤</mo><mi>X</mi></mrow></munder><mspace class="thinspace" width="0.3em"/><mi>λ</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>n</mi> <mo class="MathClass-bin">+</mo> <msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mo class="MathClass-close">)</mo></mrow><mo class="MathClass-rel">⋯</mo><mi>λ</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>n</mi> <mo class="MathClass-bin">+</mo> <msub><mrow><mi>h</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mo class="MathClass-close">)</mo></mrow> <mo class="MathClass-rel">=</mo> <mi>o</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>X</mi></mrow><mo class="MathClass-close">)</mo></mrow> </mrow>\n</math>\n¶ as [math] . This conjecture remains unproven for any [math] with [math] . Using the recent results of Matomäki and Radziwiłł on mean values of multiplicative functions in short intervals, combined with an argument of Kátai and Bourgain, Sarnak, and Ziegler, we establish an averaged version of this conjecture, namely ¶\n<math display="block">\n<mrow> <munder class="msub"><mrow><mo mathsize="big">∑</mo> </mrow><mrow><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo class="MathClass-punc">,</mo><mo>…</mo><mo class="MathClass-punc">,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>k</mi></mrow></msub><mo class="MathClass-rel">≤</mo><mi>H</mi></mrow></munder><mrow><mo class="MathClass-open" fence="true" mathsize="2.03em">|</mo><mrow><munder class="msub"><mrow><mo mathsize="big">∑</mo> </mrow><mrow><mn>1</mn><mo class="MathClass-rel">≤</mo><mi>n</mi><mo class="MathClass-rel">≤</mo><mi>X</mi></mrow></munder><mspace class="thinspace" width="0.3em"/><mi>λ</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>n</mi> <mo class="MathClass-bin">+</mo> <msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mo class="MathClass-close">)</mo></mrow><mo class="MathClass-rel">⋯</mo><mi>λ</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>n</mi> <mo class="MathClass-bin">+</mo> <msub><mrow><mi>h</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mo class="MathClass-close">)</mo></mrow></mrow><mo class="MathClass-close" fence="true" mathsize="2.03em">|</mo></mrow> <mo class="MathClass-rel">=</mo> <mi>o</mi><mrow><mo class="MathClass-open">(</mo><mrow><msup><mrow><mi>H</mi></mrow><mrow><mi>k</mi></mrow></msup><mi>X</mi></mrow><mo class="MathClass-close">)</mo></mrow> </mrow>\n</math>\n¶ as [math] , whenever [math] goes to infinity as [math] and [math] is fixed. Related to this, we give the exponential sum estimate ¶\n<math display="block">\n<mrow> <msubsup><mrow><mo class="MathClass-op">∫ </mo></mrow><mrow><mn>0</mn></mrow><mrow><mi>X</mi></mrow></msubsup><mrow><mo class="MathClass-open" fence="true" mathsize="2.03em">|</mo><mrow><munder class="msub"><mrow><mo mathsize="big">∑</mo> </mrow><mrow><mi>x</mi><mo class="MathClass-rel">≤</mo><mi>n</mi><mo class="MathClass-rel">≤</mo><mi>x</mi><mo class="MathClass-bin">+</mo><mi>H</mi></mrow></munder><mspace class="thinspace" width="0.3em"/><mspace class="thinspace" width="0.3em"/><mspace class="thinspace" width="0.3em"/><mi>λ</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>n</mi></mrow><mo class="MathClass-close">)</mo></mrow><mi>e</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>α</mi><mi>n</mi></mrow><mo class="MathClass-close">)</mo></mrow></mrow><mo class="MathClass-close" fence="true" mathsize="2.03em">|</mo></mrow><mspace class="thinspace" width="0.3em"/><mi>d</mi><mi>x</mi> <mo class="MathClass-rel">=</mo> <mi>o</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>H</mi><mi>“</mi><mi>X</mi></mrow><mo class="MathClass-close">)</mo></mrow> </mrow>\n</math>\n¶ as [math] uniformly for all [math] , with [math] as before. Our arguments in fact give quantitative bounds on the decay rate (roughly on the order of [math] ) and extend to more general bounded multiplicative functions than the Liouville function, yielding an averaged form of a (corrected) conjecture of Elliott.
The authors' abstract, as published at the source. Algebra & Number Theory, 2015 · DOI ↗
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Field: Algebra and Number Theory
Algebra and Number TheoryMathematics