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Difference between revisions of "Vinogradov estimates"

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2) Vinogradov's estimate for Weyl sums (cf. [[Weyl sum|Weyl sum]]). Let  $  n \geq  12 $
 
2) Vinogradov's estimate for Weyl sums (cf. [[Weyl sum|Weyl sum]]). Let  $  n \geq  12 $
 
be a constant and let  $  \nu = 1/n $.  
 
be a constant and let  $  \nu = 1/n $.  
Furthermore, let the points of  $  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
dimensional space be subdivided into two classes — class 1 and class 2. A point in class 1 is a point
 
  
 
$$  
 
$$  
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3) Vinogradov's estimates for trigonometric sums with prime numbers. Let  $  \epsilon \leq  0.001 $.  
 
3) Vinogradov's estimates for trigonometric sums with prime numbers. Let  $  \epsilon \leq  0.001 $.  
Also, let the points of  $  n $-
+
Also, let the points of  $  n $-dimensional space be subdivided into classes, in accordance with the notation of theorem 2), as follows.
dimensional space be subdivided into classes, in accordance with the notation of theorem 2), as follows.
 
  
 
Class 1a comprises those points satisfying the condition
 
Class 1a comprises those points satisfying the condition
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$$
 
$$
  
if  $  m \leq  \Delta  ^ {-} 2 $.
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if  $  m \leq  \Delta  ^ {- 2} $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Vinogradov,  "The method of trigonometric sums in the theory of numbers" , Interscience  (1954)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Vinogradov,  "The method of trigonometric sums in the theory of numbers" , Interscience  (1954)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR></table>

Latest revision as of 13:30, 14 May 2022


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)
How to Cite This Entry:
Vinogradov estimates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vinogradov_estimates&oldid=49146
This article was adapted from an original article by A.A. Karatsuba (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article