Effective estimates of exponential sums over primes

@inproceedings{Daboussi1996EffectiveEO,
  title={Effective estimates of exponential sums over primes},
  author={Hedi Daboussi},
  year={1996}
}
The asymptotic behavior of the sum $$S(x,\alpha ) = \sum\limits_{n \leqslant x} {\pi (n)e(n\alpha ),}$$ where α is real, e(α) = e2πiα, and Λ is the von Mangoldt function, has been extensively studied by many authors. It plays a central role in Vinogradov’s solution of the 3-primes conjecture [10]. It is also a main tool in the study of the equidistribution of the sequence {pα, p prime} modulo 1. 

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