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The name of a number of theorems of I.M. Vinogradov. The following ones are the best known.

1) Vinogradov's estimate for character sums (cf. Dirichlet character). If $\chi$ is a non-principal character mod $D$, then if $N > 0$, $M \geq 1$,

$$\left | \sum _ {n = N + 1 } ^ { {n } + M } \chi ( n) \right | \leq \sqrt D \mathop{\rm log} D.$$

2) Vinogradov's estimate for Weyl sums (cf. Weyl sum). Let $n \geq 12$ be a constant and let $\nu = 1/n$. Furthermore, let the points of $n$-dimensional space be subdivided into two classes — class 1 and class 2. A point in class 1 is a point

$$\left ( \frac{a _ {n} }{q _ {n} } + z _ {n} \dots \frac{a _ {1} }{q _ {1} } + z _ {1} \right )$$

where the first terms are rational irreducible fractions with positive denumerators, with lowest common multiple $Q$ which is not larger than $p ^ \nu$, while the second term satisfies the condition

$$| z _ {s} | \leq p ^ {- s + \nu } .$$

A point in class 2 is a point not belonging to class 1. Then, putting

$$\rho = \ { \frac{1}{8 n ^ {2} ( \mathop{\rm log} n + 0.5 \mathop{\rm log} \mathop{\rm log} n + 1.3) } } ,$$

for points in class 2,

$$| T _ {m} | = \ \left | \sum _ {1 \leq x \leq P } e ^ {2 \pi i m ( \alpha _ {n} x ^ {n} + \dots + \alpha _ {1} x) } \right | \ll P ^ {1- \rho }$$

if $m \leq P ^ {2 \rho }$. If, on the other hand, one puts

$$\delta _ {s} = z _ {s} p ^ {s} ,\ \ \delta _ {0} = \max ( | \delta _ {n} | \dots | \delta _ {1} | ),$$

then, if $m \leq P ^ {4 \nu ^ {2} }$, for points of class 1,

$$| T _ {m} | \ll P ( m, Q) ^ \nu Q ^ {- \nu + \epsilon }$$

or even

$$| T _ {m} | \ll PQ ^ {- \nu + \epsilon } \delta _ {0} ^ {- \nu } \ \textrm{ if } \delta _ {0} \geq 1.$$

3) Vinogradov's estimates for trigonometric sums with prime numbers. Let $\epsilon \leq 0.001$. Also, let the points of $n$-dimensional space be subdivided into classes, in accordance with the notation of theorem 2), as follows.

Class 1a comprises those points satisfying the condition

$$Q \leq e ^ {u} ^ \epsilon ,\ \ \delta _ {0} \leq e ^ {u} ^ \epsilon ,\ \ \textrm{ where } u = \mathop{\rm log} P.$$

Class 1b comprises those points not in class 1a and satisfying the condition

$$Q \leq P ^ {0.2 \nu } ,\ \ \delta \leq P ^ \nu .$$

Finally, all other points belong to class 2.

For points in class 1a one sets

$$\Delta = u ^ {9 \epsilon } Q ^ {- 0.5 \nu + \epsilon ^ \prime } ,\ \ \mu = ( m, Q) ^ {0.5 \nu } ,$$

or even

$$\Delta = u ^ {9 \epsilon } \delta _ {0} ^ {- 0.5 \nu } ,\ \ \mu = m ^ {- 0.5 \nu } \ \ \textrm{ if } \delta _ {0} \geq 1.$$

For points in class 1b, setting $\epsilon = 2 \epsilon ^ \prime$, one defines

$$\Delta = Q ^ {- 0.5 + \epsilon _ {3} } ,\ \ \mu = ( m, Q) ^ {0.5 \nu } \ \ \textrm{ if } Q > e ^ {u ^ \epsilon } ,$$

$$\Delta = Q ^ {- 0.5 \nu + \epsilon _ {3} } \delta _ {0} ^ {- 0.5 \nu + \epsilon _ {3} } ,\ \mu = 1 \ \textrm{ if } \delta _ {0} > e ^ {u ^ \epsilon }$$

(if $Q > e ^ {u ^ \epsilon }$, $\delta _ {0} > e ^ {u ^ \epsilon }$, any of the above pairs of values of $\Delta$ and $\mu$ may be taken). Finally, one sets

$$\Delta = P ^ {- \rho _ {1} } ,\ \ \rho _ {1} = \ { \frac{1}{17 n ^ {2} ( 2 \mathop{\rm log} n + \mathop{\rm log} \mathop{\rm log} n + 2.9) } } ,\ \ \mu = 1$$

for points in class 2. Then

$$\left | \sum _ {p \leq P } e ^ {2 \pi i m ( \alpha _ {n} p ^ {n} + \dots + \alpha _ {1} p) } \right | \ll \begin{array}{c} P \\ u \end{array} \Delta \mu$$

if $m \leq \Delta ^ {- 2}$.

#### References

 [1] I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers" , Interscience (1954) (Translated from Russian) [2] L.-K. Hua, "Abschätzungen von Exponentialsummen und ihre Anwendung in der Zahlentheorie" , Enzyklopaedie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen , 1 : 2 (1959) (Heft 13, Teil 1)
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