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References [1] Daboussi , H. 1996 Effective estimates of exponential sums over primes Berndt , B. C. (eds.) et al. Analytic

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. [2] Graham , S. W. , Kolesnik , G. 1991 Van der Corput’s method of exponential sums University Press Cambridge 10.1017/CBO9780511661976 . [3] Huxley , M. N

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Abstract  

We obtain a new lower bound on the number of prime divisors of integers whose g-ary expansion contains a fixed number of nonzero digits.

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Abstract  

We determine, up to a constant factor, the L 1 mean of the exponential sum formed with the r-free integers. This improves earlier results of Brdern, Granville, Perelli, Vaughan and Wooley. As an application, we improve the known bound for the L 1 norm of the exponential sum defined with the Mbius function.

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. [3] Davenport , H. 1933 On certain exponential sums, I Reine Angew. Math. 169 158 – 176 . [4

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Abstract  

We continue the study of sums of the form

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sum\limits_{m_j p \leqq x} {Y_{mj} X_p e(\alpha m_j p)} ,$$ \end{document}
begun by Indlekofer and Kátai. Here |Y n|,|X p| ≦ 1 and α is irrational. We prove one conjecture of Kátai, disprove another by both authors, and give what may be a close to best possible result valid for all irrational α.

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