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1.
Let τ(n) be the Ramanujan τ-function, x ≥ 10 be an integer parameter. We prove that
We also show that
where ω(n) is the number of distinct prime divisors of n and p denotes prime numbers. These estimates improve several results from [6, 9].
Received: 23 November 2006 相似文献
2.
Siberian Mathematical Journal - We give a simple proof of one assertion used in solving Waring’s problem. 相似文献
3.
We investigate Fuglede's spectral set conjecture in the special case when the set in question is a union of finitely many unit intervals in dimension 1. In this case, the conjecture can be reformulated as a statement about multiplicative properties of roots of associated with the set polynomials with (0,1) coefficients. Let be an N-term polynomial. We say that {θ1,θ2,…,θN−1} is an N-spectrum for A(x) if the θj are all distinct and
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
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Estimates for the Number of Sums and Products and for Exponential Sums in Fields of Prime Order 总被引:4,自引:0,他引:4
Bourgain J.; Glibichuk A. A.; Konyagin S. V. 《Journal London Mathematical Society》2006,73(2):380-398
Our first result is a sum-product theorem forsubsets A of the finite field Fp, p prime, providing a lowerbound on max (|A + A|, |A · A|). The second and mainresult provides new bounds on exponential sums 相似文献
9.
We prove that there exists an absolute constant c > 0 such that for any finite set A ⊆ ℤ with |A| ≥ 2 and any positive integer m < c|A|/ ln |A| there are at most m integers b > 0 satisfying |(A + b) \ A| ≤ m; equivalently, there are at most m positive integers possessing |A| −m (or more) representations as a difference of two elements of A. 相似文献
10.
S.V Konyagin 《Journal of Approximation Theory》1997,90(3):379-384
This paper deals with a negative result concerning a pointwisecomparison of two quite general sequences of linear operatorson the space of continuous functions on the torus or a segment 相似文献