Kloosterman Sum - Short Kloosterman Sums

Short Kloosterman Sums

Short Kloosterman sums are defined as trigonometric sums of the form

where runs through a set of numbers, coprime to, the number of elements in which is essentially smaller than, and the symbol denotes the congruence class, inverse to modulo : .

Up to the early 1990s, estimates for sums of this type were known mainly in the case where the number of summands was greater than . Such estimates were due to H. D. Kloosterman, I. M. Vinogradov, H. Salie, L. Carlitz, S. Uchiyama and A. Weil. The only exceptions were the special modules of the form, where is a fixed prime and the exponent increases to infinity (this case was studied by A.G. Postnikov by means of the method of Ivan Matveyevich Vinogradov).

In the 1990s Anatolii Alexeevitch Karatsuba developed a new method of estimating short Kloosterman sums. Karatsuba's method makes it possible to estimate Kloosterman's sums, the number of summands in which does not exceed, and in some cases even, where is an arbitrarily small fixed number. The last paper of A.A. Karatsuba on this subject was published after his death.

Various aspects of the method of Karatsuba found applications in solving the following problems of analytic number theory:

• finding asymptotics of the sums of fractional parts of the form : : where runs, one after another, through the integers satisfying the condition, and runs through the primes that do not divide the module (A.A.Karatsuba);

• finding the lower bound for the number of solutions of the inequalities of the form : : in the integers, coprime to, (A.A. Karatsuba);

• the precision of approximation of an arbitrary real number in the segment by fractional parts of the form :

: where, (A.A. Karatsuba);

• a more precise constant in the Brun–Titchmarsh theorem :

: where is the number of primes, not exceeding and belonging to the arithmetic progression (J. Friedlander, H. Iwaniec);

• a lower bound for the greatest prime divisor of the product of numbers of the form :

, (D. R. Heath-Brown);

• proving that there are infinitely many primes of the form :

(J. Friedlander, H. Iwaniec);

• combinatorial properties of the set of numbers :

(A.A.Glibichuk).