## Additive theory of prime numbers |

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### Contents

Estimates for sums involving the divisor function dn | 11 |

Meanvalue theorems for certain trigonometric sums I | 19 |

Vinogradovs meanvalue theorem and its corollaries | 26 |

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absolutely convergent accordingly obtain Cauchy's inequality chapter congruence congruent solvability convergent convex function Corollary denote the number denotes a sum distinct integers distinct residue classes DIVISOR FUNCTION elementary symmetric function equal equations exponential density formula Fundamental Lemma Further greatest common divisor Hence we obtain hyperellipsoid I. M. Vinogradov integers satisfying integral coefficients integral solutions integral-valued polynomial know by Lemma kth degree least common multiple lemma is obviously lemma is proved lemma is true LEMMAS Lemma Let f(x log log mean-value theorem mod q modp number of distinct number of sets number of solutions obtain f obtain the lemma obviously true polynomial with integral positive integer positive number positive solvability prime factors prime numbers proof of Theorem quadratic form residues mod runs set of distinct set of residues sets of integers Suppose symmetric functions Theorem 11 theory of numbers TRIGONOMETRIC SUMS well-spaced