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1.
We prove that for any given c, 1 < c < 17/11, almost all natural numbers are representable in the form [x
c] + [p
c], where x is a natural number and p is a prime. 相似文献
2.
Andrew Wisel Irene Studentsova Abdurakhim Muslinkin Ramil Garaev Valeriy Danilov Lidia Schukina 《Phosphorus, sulfur, and silicon and the related elements》2013,188(1)
Abstract The synthesis of dimephosphone (1) was carried out on Pudovik's reaction. 相似文献
3.
Jean Bourgain Moubariz Z. Garaev Sergei V. Konyagin Igor E. Shparlinski 《Proceedings of the Steklov Institute of Mathematics》2013,280(1):61-90
We obtain upper bounds on the number of solutions to congruences of the type (x 1 + s)... (x ν + s) ≡ (y 1 + s)... (x ν + s) ? 0 (mod p) modulo a prime p with variables from some short intervals. We give some applications of our results and in particular improve several recent estimates of J. Cilleruelo and M.Z. Garaev on exponential congruences and on cardinalities of products of short intervals, some double character sum estimates of J. Friedlander and H. Iwaniec and some results of M.-C. Chang and A.A. Karatsuba on character sums twisted with the divisor function. 相似文献
4.
Jean Bourgain Moubariz Z. Garaev Sergei V. Konyagin Igor E. Shparlinski 《Journal d'Analyse Mathématique》2014,124(1):117-147
Recently, several bounds have been obtained on the number of solutions of congruences of the type $$({x_1} + s) \cdots ({x_v} + s) \equiv ({y_1} + s) \cdots ({y_v} + s)\not \equiv 0{\text{ (mod }}p{\text{),}}$$ where p is prime and variables take values in some short interval. Here, for almost all p and all s and also for a fixed p and almost all s, we derive stronger bounds. We also use similar ideas to show that for almost all p, one can always find an element of a large order in any rather short interval. 相似文献
5.
M. Z. Garaev 《Monatshefte für Mathematik》2006,160(1):127-138
We obtain nontrivial estimates of character sums over short intervals for almost all moduli. These bounds and the method of
Karatsuba for solving multiplicative ternary problems are used to prove that for π(X)(1 + o(1)) primes p,p ≤ X, there are p(1 + o(1)) residue classes modulo p of the form xy (mod p), where 1 ≤ x, y ≤ p?(log p)1,087. We also prove that for any prime p there are p(1 + o(1)) residue classes modulo p of the form xy* (mod p), where 1 ≤ x, y ≤ p?(log p)1+o(1) and y* is defined by yy* ≡ 1 (mod p). 相似文献
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8.
For a fixed integer s≥1, we estimate exponential sums with harmonic sums individually and on average, where Hs(n) is computed modulo a prime p. These bounds are used to derive new results about various congruences modulo p involving Hs(n). For example, our estimates imply that for any ɛ>0, the set {Hs(n):n<p1/2+ɛ} is uniformly distributed modulo a sufficiently large p. We also show that every residue class λ can be represented as with max{nν|ν=1,. . . , 7}≤p11/12+ɛ, and we obtain an asymptotic formula for the number of such representations. The same results hold also for the values Bp−r(n) of Bernoulli polynomials where r is fixed, complementing some results of W. L. Fouche.
During the preparation of this paper, F. L. was supported in part by grants SEP-CONACYT 37259-E and 37260-E, and I. S. was
supported in part by ARC grant DP0211459. 相似文献
9.
Summary We provide a general asymptotic formula which permits applications to sums like <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation
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\sum_{x< n\le x+y} \big(d(n)\big)^2, \quad \sum_{x< n\le x+y} d(n^3),\quad \sum_{x< n\le x+y}\big(r(n)\big)^2, \quad \sum_{x<
n\le x+y}r(n^3), $$ where $d(n)$ and $r(n)$ are the usual arithmetic functions (number of divisors, sums of two squares),
and $y$ is small compared to~$x$. 相似文献
10.
I. Kh. Garaev A. V. Kostochko A. I. Petrov E. N. Ibragimov A. E. Golubev 《Russian Journal of General Chemistry》2016,86(6):1459-1468
Statistical polymers and copolymers were firstly synthesized by polymerization of 3,3-bis(azidomethyl) oxetane (BAMO) and 3-azidomethyl-3-methyloxetane (AMMO) in a triisobutylaluminum–water catalytic system, and their physicochemical, physicomechanical, and themochemical properties were studied. It was found that with increasing fraction of poly-AMMO in the BAMO/AMMO copolymers the strength properties of the copolymers enhance and the degree of crystallinity decreases. 相似文献