We determine the order of magnitude of H(x, y, z), the number of integers n ≤ x having a divisor in (y, z], for all x, y and z. We also study H r (x, y, z), the number of integers n ≤ x having exactly r divisors in (y, z]. When r = 1 we establish the order of magnitude of H 1 (x, y, z) for all x, y, z satisfying z ≤ x 1/2−ε . For every r ≥ 2, C > 1 and ε > 0, we determine the order of magnitude of H r (x, y, z) uniformly for y large and y + y/(log y) log 4−1−ε ≤ z ≤ min(y C , x 1/2−ε ). As a consequence of these bounds, we settle a 1960 conjecture of Erdős and some conjectures of Tenenbaum. One key element of the proofs is a new result on the distribution of uniform order statistics.
Abstract. We give a new explicit construction of n × N matrices satisfying the Restricted Isometry Property (RIP). Namely, for some ε > 0, large N and any n satisfying N 1−ε ≤ n ≤ N , we construct RIP matrices of order k ≥ n 1/2+ε and constant δ = n −ε . This overcomes the natural barrier k = O(n 1/2 ) for proofs based on small coherence, which are used in all previous explicit constructions of RIP matrices. Key ingredients in our proof are new estimates for sumsets in product sets and for exponential sums with the products of sets possessing special additive structure. We also give a construction of sets of n complex numbers whose k-th moments are uniformly small for 1 ≤ k ≤ N (Turán's power sum problem), which improves upon known explicit constructions when (log N ) 1+o(1) ≤ n ≤ (log N ) 4+o(1) . This latter construction produces elementary explicit examples of n × N matrices that satisfy RIP and whose columns constitute a new spherical code; for those problems the parameters closely match those of existing constructions in the range (log N ) 1+o(1) ≤ n ≤ (log N ) 5/2+o(1) .
Let pn denote the n-th prime. We prove that max p n+1 X (pn+1 − pn) ≫ log X log log X log log log log X log log log X for sufficiently large X, improving upon recent bounds of the first three and fifth authors and of the fourth author. Our main new ingredient is a generalization of a hypergraph covering theorem of Pippenger and Spencer, proven using the Rödl nibble method.
kevin ford 1 < x < P ea 1 x . . . a k x k 2 s eÀa 1 h 1 À . . . À a k h k da < J s; k P; 0; . . . ; 0 J s; k P:Hence, writing Q P, we obtainAlso, counting only the solutions of (1.4) with x i y i for each i gives J s; k P > Q s . Therefore J s; k P > max2s À k P 2 s À kk 1 = 2 ; P s : 1:5 kevin fordUpper bounds take the form of J s; k P < Ds; kP 2 s À kk 1 = 2 hs; k ; 1:6where hs; k > 0 and Ds; k is independent of P. Stechkin in 1975 [24] proved (1.6) with hrk; k 1 2 k 2 1 À 1=k r ; Drk; k expfC minr; kk 2 log kg for an absolute constant C. The constant factor was improved by Wooley [31]. Small improvements to the exponents of P were subsequently made by Arkhipov and Karatsuba [1] and Tyrina [26] (signi®cant for s p k 2 ). Also signi®cant is Wooley's result [32] when s p k 3 = 2 À « , which is very close to the`ideal' bounds Ck; sP s in that range of s. For our purposes, the most important improvement comes from Wooley [30], who improved the exponents substantially in a wide range of s, showing that (1.6) holds with hk; s < 1 2 k 2 e 1 = 2 À 2 s = k 2 valid for s p k 2 log k (see [5, Lemma 5.2]). In Theorem 3 below, we combine Wooley's method with the main idea from [1] to improve this to hk; s < 3 8 k 2 e 1 = 2 À 2 s = k 2 . In the application to bounding the Riemann zeta function, we will take s to be of order k 2 , so this small improvement is signi®cant.Theorem 3. Let k and s be integers with k > 1000 and 2k 2 < s < 1 2 k 2 1 2 log 3 8 k. Then J s; k P < k 2:055 k 3 À 5:91 k 2 3 s 1:06 sk 2 s 2 = k À 9:7278 k 3 P 2 s À kk 1 = 2 D s P > 1;where D s 3 8 k 2 e 1 = 2 À 2 s = k 2 1:7 = k : Further, if k > 129, there is an integer s < rk 2 such that for P > 1, J s; k P < k vk 3 P 2 s À kk 1 = 2 0:001 k 2
We show that for a prime p the smallest a with a p−1 ≡ 1 (mod p 2 ) does not exceed (log p) 463/252+o(1) which improves the previous bound O((log p) 2 ) obtained by H. W. Lenstra in 1979. We also show that for almost all primes p the bound can be improved as (log p) 5/3+o(1) .
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