Vinogradov hypotheses

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)