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Nguyen has shown that on averaging over a = 1 , . . . , q the 3-fold divisor function has exponent of distribution 2/3, following Banks et al. (Int Math Res Not 1:1–25, 2005). We follow (Blomer, in: Q J Math 59:275–286, 2008) which leads to stronger bounds.

Let f be a Hecke-Maass cusp form for SL(3,ℤ) with Fourier coeffcients Af (m, n), and let ϕ(x) be a C∞-function supported on [1, 2] with derivatives bounded by ϕ(j)(x)≪j 1: We prove an asymptotic formula for the nonlinear \\[\\Sigma_{n\\equiv l \\rm{mod} \\it{q}}\\]Af (m, n)ϕ(n/X)e(3(kn)1/3/q), where e(z) = e2πiz and k ∈ℤ+.

In this note, we shall define the balancing Wieferich prime which is an analogue of the famous Wieferich primes. We prove that, under the abc conjecture for the number field Q([square root of 2]), there are infinitely many balancing non-Wieferich primes. In particular, under the assumption of the abc conjecture for the number field Q([square root of 2]) there are at least O(log x/log log x) such primes p [equivalent to] 1(mod k) for any fixed integer k > 2. Key words: Balancing number; Wieferich prime; arithmetic progression; abc conjecture.

We prove nontrivial bounds for general bilinear forms in hyper-Kloosterman sums when the sizes of both variables may be below the range controlled by Fourier-analytic methods (Pólya-Vinogradov range). We then derive applications to the second moment of cusp forms twisted by characters modulo primes, and to the distribution in arithmetic progressions to large moduli of certain Eisenstein-Hecke coefficients on GL₃. Our main tools are new bounds for certain complete sums in three variables over finite fields, proved using methods from algebraic geometry, especially ℓ-adic cohomology and the Riemann Hypothesis.

Nous montrons que, étant donnésxetq≤x 1/16, toute classe inversibleamoduloqcontient au moins un produit d’exactement trois nombres premiers, chacun étant inférieur ou égal àx 1/3. We prove that, ifxandq≤x 1/16are two parameters, then for any invertible residue classamoduloqthere exists a product of exactly three primes, each one belowx 1/3, that is congruent toamoduloq.

In this paper we give a historical account of the development of Poisson approximation using Stein's method and present some of the main results. We give two recent applications, one on maximal arithmetic progressions and the other on bootstrap percolation. We also discuss generalisations to compound Poisson approximation, Poisson process approximation and multivariate Poisson approximation, and state a few open problems.

We study thresholds for extremal properties of random discrete structures. We determine the threshold for Szemerédi's theorem on arithmetic progressions in random subsets of the integers and its multidimensional extensions, and we determine the threshold for Turán-type problems for random graphs and hypergraphs. In particular, we verify a conjecture of Kohayakawa, Łuczak, and Rödl for Turán-type problems in random graphs. Similar results were obtained independently by Conlon and Gowers.

It is proved that $\\lim_{n\\rightarrow \\infty }inf(p_{n+1}-p_n)\\textless 7\\times 10^7$, where pn is the n-th prime. Our method is a refinement of the recent work of Goldston, Pintz and Yildirim on the small gaps between consecutive primes. A major ingredient of the proof is a stronger version of the Bombieri-Vinogradov theorem that is applicable when the moduli are free from large prime divisors only, but it is adequate for our purpose.

For an integer $b\\geq 2$ , a positive integer is called a b-Niven number if it is a multiple of the sum of the digits in its base-b representation. In this article, we show that every arithmetic progression contains infinitely many b-Niven numbers.

We develop a new technique that allows us to show in a unified way that many well-known combinatorial theorems, including Turan's theorem, Szemerédi's theorem and Ramsey's theorem, hold almost surely inside sparse random sets. For instance, we extend Turán's theorem to the random setting by showing that for every ε > 0 and every positive integer t ≥ 3 there exists a constant C such that, if G is a random graph on n vertices where each edge is chosen independently with probability at least Cn-2/(t+1), then, with probability tending to 1 as n tends to infinity, every subgraph of G with at least $\\left( {1 - \\frac{1}{{t - 1}} + \\in } \\right)$ e (G) edges contains a copy of Kt. This is sharp up to the constant C. We also show how to prove sparse analogues of structural results, giving two main applications, a stability version of the random Turán theorem stated above and a sparse hypergraph removal lemma. Many similar results have recently been obtained independently in a different way by Schacht and by Friedgut, Rödl and Schacht.