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Several hypotheses on central problems in [[Analytic number theory|analytic number theory]], advanced by I.M. Vinogradov [[#References|[1]]], [[#References|[2]]] at various times.
 
Several hypotheses on central problems in [[Analytic number theory|analytic number theory]], advanced by I.M. Vinogradov [[#References|[1]]], [[#References|[2]]] at various times.
  
 
==Hypotheses on the distribution of power residues and non-residues.==
 
==Hypotheses on the distribution of power residues and non-residues.==
One of the oldest and best known such hypotheses is that the distance between neighbouring quadratic non-residues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096670/v0966701.png" /> is a magnitude of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096670/v0966702.png" />. (Cf. [[Power residue|Power residue]]; [[Quadratic residue|Quadratic residue]].)
+
One of the oldest and best known such hypotheses is that the distance between neighbouring quadratic non-residues $  \mathop{\rm mod}  p $
 +
is a magnitude of order $  p  ^  \epsilon  $.  
 +
(Cf. [[Power residue|Power residue]]; [[Quadratic residue|Quadratic residue]].)
  
 
==Hypotheses on estimates of trigonometric sums.==
 
==Hypotheses on estimates of trigonometric sums.==
 
One of them is that
 
One of them is that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096670/v0966703.png" /></td> </tr></table>
+
$$
 +
\left | \sum _ {x = 1 } ^ { p }  e ^ {2 \pi i f ( x) } \right |
 +
\ll  P ^ {1- \rho ( n) } ,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096670/v0966704.png" /></td> </tr></table>
+
$$
 +
f ( x)  = \alpha _ {n} x  ^ {n} + \dots + \alpha _ {1} x,\ \
 +
\alpha _ {r}  = {
 +
\frac{a}{q}
 +
} + {
 +
\frac \theta {q}
 +
  ^ {2} } ,\ \
 +
( a, q) = 1,
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096670/v0966705.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096670/v0966706.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096670/v0966707.png" /> is one of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096670/v0966708.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096670/v0966709.png" /> has the order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096670/v09667010.png" />. (Cf. [[Trigonometric sums, method of|Trigonometric sums, method of]]; [[Vinogradov method|Vinogradov method]].)
+
$  | \theta | \leq  1 $,
 +
$  P  ^ {0.25} < q < P ^ {r - 0.25 } $,  
 +
$  r $
 +
is one of the numbers $  2 \dots n $,  
 +
and $  \rho ( n) $
 +
has the order $  n ^ {- 1 - \epsilon } $.  
 +
(Cf. [[Trigonometric sums, method of|Trigonometric sums, method of]]; [[Vinogradov method|Vinogradov method]].)
  
 
==Hypotheses on the number of solutions of Diophantine equations.==
 
==Hypotheses on the number of solutions of Diophantine equations.==
 
One such hypothesis states that the number of solutions of the system of equations
 
One such hypothesis states that the number of solutions of the system of equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096670/v09667011.png" /></td> </tr></table>
+
$$
 +
{x _ {1} ^ {n _ {1} } + \dots + x _ {r} ^ {n _ {1} }  = \
 +
y _ {1} ^ {n _ {1} } + \dots + y _ {r} ^ {n _ {1} } , }
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096670/v09667012.png" /></td> </tr></table>
+
$$
 +
{\dots \dots \dots \dots \dots }
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096670/v09667013.png" /></td> </tr></table>
+
$$
 +
{x _ {1} ^ {n _ {m} } + \dots + x _ {r} ^ {n _ {m} }  = \
 +
y _ {1} ^ {n _ {m} } + \dots + y _ {r} ^ {n _ {m} } , }
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096670/v09667014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096670/v09667015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096670/v09667016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096670/v09667017.png" /> is constant, will be a magnitude of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096670/v09667018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096670/v09667019.png" />, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096670/v09667020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096670/v09667021.png" /> has order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096670/v09667022.png" />. (Cf. [[Diophantine equations|Diophantine equations]].)
+
$  1 \leq  x _ {i} , y _ {i} < P $,
 +
$  i = 1 \dots r $,  
 +
$  1 \leq  n _ {1} < \dots < n _ {m} = n $,  
 +
where $  n $
 +
is constant, will be a magnitude of order $  P  ^ {2r-} k $,  
 +
$  k = n _ {1} + \dots + n _ {m} $,  
 +
for all $  r \geq  r _ {0} $,  
 +
where $  r _ {0} $
 +
has order $  k ^ {1 + \epsilon } $.  
 +
(Cf. [[Diophantine equations|Diophantine equations]].)
  
 
==Hypotheses on the number of integer points in domains in the plane and in space.==
 
==Hypotheses on the number of integer points in domains in the plane and in space.==
One such hypothesis states that the number of integer points in the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096670/v09667023.png" /> can be expressed by the formula
+
One such hypothesis states that the number of integer points in the sphere $  x  ^ {2} + y  ^ {2} + z  ^ {2} \leq  R  ^ {2} $
 +
can be expressed by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096670/v09667024.png" /></td> </tr></table>
+
$$
 +
{
 +
\frac{4}{3}
 +
} \pi R  ^ {3} + O ( R ^ {1 + \epsilon } ).
 +
$$
  
 
(Cf. [[Integral points, distribution of|Integral points, distribution of]].)
 
(Cf. [[Integral points, distribution of|Integral points, distribution of]].)

Revision as of 08:28, 6 June 2020


Several hypotheses on central problems in analytic number theory, advanced by I.M. Vinogradov [1], [2] at various times.

Hypotheses on the distribution of power residues and non-residues.

One of the oldest and best known such hypotheses is that the distance between neighbouring quadratic non-residues $ \mathop{\rm mod} p $ is a magnitude of order $ p ^ \epsilon $. (Cf. Power residue; Quadratic residue.)

Hypotheses on estimates of trigonometric sums.

One of them is that

$$ \left | \sum _ {x = 1 } ^ { p } e ^ {2 \pi i f ( x) } \right | \ll P ^ {1- \rho ( n) } , $$

where

$$ f ( x) = \alpha _ {n} x ^ {n} + \dots + \alpha _ {1} x,\ \ \alpha _ {r} = { \frac{a}{q} } + { \frac \theta {q} ^ {2} } ,\ \ ( a, q) = 1, $$

$ | \theta | \leq 1 $, $ P ^ {0.25} < q < P ^ {r - 0.25 } $, $ r $ is one of the numbers $ 2 \dots n $, and $ \rho ( n) $ has the order $ n ^ {- 1 - \epsilon } $. (Cf. Trigonometric sums, method of; Vinogradov method.)

Hypotheses on the number of solutions of Diophantine equations.

One such hypothesis states that the number of solutions of the system of equations

$$ {x _ {1} ^ {n _ {1} } + \dots + x _ {r} ^ {n _ {1} } = \ y _ {1} ^ {n _ {1} } + \dots + y _ {r} ^ {n _ {1} } , } $$

$$ {\dots \dots \dots \dots \dots } $$

$$ {x _ {1} ^ {n _ {m} } + \dots + x _ {r} ^ {n _ {m} } = \ y _ {1} ^ {n _ {m} } + \dots + y _ {r} ^ {n _ {m} } , } $$

$ 1 \leq x _ {i} , y _ {i} < P $, $ i = 1 \dots r $, $ 1 \leq n _ {1} < \dots < n _ {m} = n $, where $ n $ is constant, will be a magnitude of order $ P ^ {2r-} k $, $ k = n _ {1} + \dots + n _ {m} $, for all $ r \geq r _ {0} $, where $ r _ {0} $ has order $ k ^ {1 + \epsilon } $. (Cf. Diophantine equations.)

Hypotheses on the number of integer points in domains in the plane and in space.

One such hypothesis states that the number of integer points in the sphere $ x ^ {2} + y ^ {2} + z ^ {2} \leq R ^ {2} $ can be expressed by the formula

$$ { \frac{4}{3} } \pi R ^ {3} + O ( R ^ {1 + \epsilon } ). $$

(Cf. Integral points, distribution of.)

References

[1] I.M. Vinogradov, "Some problems in analytic number theory" , Proc. 3-rd All-Union Mat. Congress (Moscow, 1956) , 3 , Moscow (1958) pp. 3–13 (In Russian)
[2] I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers" , Interscience (1954) (Translated from Russian)
[3] I.M. Vinogradov, "Selected works" , Springer (1985) (Translated from Russian)
How to Cite This Entry:
Vinogradov hypotheses. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vinogradov_hypotheses&oldid=49147
This article was adapted from an original article by A.A. Karatsuba (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article