Actions

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

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

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: