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Some remarks on trigonometric sums

Authors:

K -H Indlekofer I Kátai

Institution:

1.University of Paderborn,Paderborn,Germany;2.Department of Computer Algebra,E?tv?s Loránd University,Budapest,Hungary

Abstract:

Let

$S(x|\alpha ;Y_m ,X_p ) = \sum\limits_{m_j p \leqq x} {Y_{m_j } X_p e^{2\pi i\alpha m_j p} } ,$

where m ₁ < m ₂ < … < m _t ≦ $x^{\delta _x }$ , δ _x → 0, p runs over the primes p ≧ $\sqrt x ,\left| {Y_{m_j } } \right|$ ≦ 1, |X _p| ≦ 1. It is assumed that m _v, $Y_{m_v }$ , X _p may depend on x. Assume that $\nu _x : = \sum\limits_{j = 1}^t {1/m_j \to \infty (x \to \infty )}$ . It is proved that

$\mathop {\max }\limits_{Y_m ,X_p } \left| {S(x|\alpha ;Y_m ,X_p )} \right| = o_x (1)\sum\limits_{j = 1}^t \pi \left( {\frac{x} {{m_j }}} \right)$

for almost all irrational α, π(x) = number of primes up to x. Research supported by the Applied Number Theory Research Group of the Hungarian Academy of Science and by a grant from OTKA T46993.

Keywords:

and phrases" target="_blank"> and phrases exponential sums

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