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<p>We obtain a new bound on exponential sums over integers without large prime divisors, improving that of Fouvry and Tenenbaum (1991). For a fixed integer&#xa0;<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\nu \ne 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ν</mi> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, we also obtain new bounds on exponential sums with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\nu \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ν</mi> </math></EquationSource> </InlineEquation>-th powers of such integers. The improvement is based on exploiting more precisely the factorisation of integers without large prime divisors, along with existing Type&#xa0;I and Type&#xa0;II bounds. For&#xa0;<InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\nu =1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ν</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> we use the classical bounds of Vinogradov (1937), while for&#xa0;<InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\nu \ne 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ν</mi> <mo>≠</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> we use bounds of Vaughan (1975) as well as of Fouvry, Kowalski and Michel (2014).</p>

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Exponential sums over integers without large prime divisors

  • 期刊论文

  • 发表日期：

    2025年10月24日

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  • Sary Drappeau,
  • Igor E. Shparlinski

摘要

We obtain a new bound on exponential sums over integers without large prime divisors, improving that of Fouvry and Tenenbaum (1991). For a fixed integer  \(\nu \ne 0\) ν ≠ 0 , we also obtain new bounds on exponential sums with \(\nu \) ν -th powers of such integers. The improvement is based on exploiting more precisely the factorisation of integers without large prime divisors, along with existing Type I and Type II bounds. For  \(\nu =1\) ν = 1 we use the classical bounds of Vinogradov (1937), while for  \(\nu \ne 1\) ν ≠ 1 we use bounds of Vaughan (1975) as well as of Fouvry, Kowalski and Michel (2014).