Vinogradov estimates

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)