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 | ||
$$ | $$ | ||
Line 86: | Line 85: | ||
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 ^ {-} | + | 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) |
Vinogradov estimates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vinogradov_estimates&oldid=49146