Estimates Of Trigonometric Sums Research Articles

The estimates of trigonometric sums and new bounds on a mean value, a sequence and a cryptographic function

The estimates of trigonometric sums and new bounds on a mean value, a sequence and a cryptographic function

Estimates of trigonometric sums over subgroups and some of their applications

In this paper, we obtain new upper bounds for trigonometric sums over subgroups Γ ⊂ Z*p whose size belongs to [p28/95, p182/487]. Using an approach due to Malykhin, we refine estimates of such sums in \(\mathbb{Z}_{{p^r}}^*\) and apply them to the divisibility problem for Fermat quotients.

Inversive congruential generator over a Galois ring of dimension p l

We consider the construction of of an inversive congruential generator over a Galois ring of odd dimension pl, whichwas proposed by Sole and Zinoviev for p = 2. Using the estimates of trigonometric sums on the sequences of pseudorandom numbers, we obtain the estimates of a discrepant function, a generated sequence of pseudorandom numbers, and the associated sequence of two-dimensional “overlapping” points.

On the mean value of general Cochrane sum

In this paper, the general Cochrane sum is defined. The main purpose is to study the mean square value problem of general Cochrane sum by using the estimates for trigonometric sums and the properties of general Kloosterman sum, and finally give a sharp asymptotic formula.

On the 2k-th power mean of % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqipC0xg9qqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaWaaSaaaeaacuWGmbatgaqbaaqaaiabdYeambaarmqr % 1ngBPrgitLxBI9gBaGGbaiab-HcaOiab-fdaXiab-XcaSGGbciab+D % 8aJjab-LcaPaaa!46FC! $$ \frac{{L'}} {L}(1,\chi ) $$ with the weight of Gauss sums

The main purpose of this paper is to study the hybrid mean value of \( \frac{{L'}} {L}(1,\chi ) \) and Gauss sums by using the estimates for trigonometric sums as well as the analytic method. An asymptotic formula for the hybrid mean value \( \sum\limits_{\chi \ne \chi _0 } {|\tau (\chi )||\frac{{L'}} {L}(1,\chi )|^{2k} } \) of \( \frac{{L'}} {L} \) and Gauss sums will be proved using analytic methods and estimates for trigonometric sums.

Frequency characteristics of linear recurrence sequences over Galois rings

The frequencies of occurrences of elements in linear recurrence sequences of vectors over Galois rings are studied. The study of these frequencies is reduced to the study of the corresponding trigonometric sums over Galois rings. Based on estimates for trigonometric sums, nontrivial estimates for the frequencies of occurrence of elements in linear recurrence sequences are obtained, which generalize some known results for sequences over a finite field. These estimates are asymptotically best possible.Bibliography: 25 titles.

Generalized continued fractions and ergodic theory

In this paper a new theory of generalized continued fractions is constructed and applied to numbers, multidimensional vectors belonging to a real space, and infinite-dimensional vectors with integral coordinates. The theory is based on a concept generalizing the procedure for constructing the classical continued fractions and substantially using ergodic theory. One of the versions of the theory is related to differential equations. In the finite-dimensional case the constructions thus introduced are used to solve problems posed by Weyl in analysis and number theory concerning estimates of trigonometric sums and of the remainder in the distribution law for the fractional parts of the values of a polynomial, and also the problem of characterizing algebraic and transcendental numbers with the use of generalized continued fractions. Infinite-dimensional generalized continued fractions are applied to estimate sums of Legendre symbols and to obtain new results in the classical problem of the distribution of quadratic residues and non-residues modulo a prime. In the course of constructing these continued fractions, an investigation is carried out of the ergodic properties of a class of infinite-dimensional dynamical systems which are also of independent interest.

Some estimates of trigonometric sums and their applications

Some estimates of trigonometric sums and their applications

Estimates of trigonometric sums by the third derivative

The well-known Van der Corput estimate of a trigonometric sum by the third derivative is sharpened.