Let N be a sufficiently large odd integer. We study the representation of a large integer m N^k as a sum of m almost equal k-th powers of primes, where the common distribution scale is at least N^theta. Assuming the Generalized Riemann Hypothesis (GRH), we prove that for any epsilon>0 one may take theta>1/2, provided that m<k(k+1)/2 and that m, k are odd integers sufficiently large. This is achieved by employing a new tool closely related to earlier work. For sufficiently large m, by refining the discussion of the minor arcs, we further show that under GRH the equation has a solution provided m>16k log k + 4k + 2. Moreover, we give an asymptotic formula for the number of solutions, with main term involving N^{2k theta (1 - (1 - 1/k)^{m_1}) + theta(m-1) + 1 - k}, where m_1 = 4k. Under GRH and for sufficiently large m, k, the short-interval Waring-Goldbach problem is essentially settled.
Let p(n,k) denote the number of partitions of n into exactly k parts. In 1942 Auluck, Chowla and Gupta conjectured that for every n the sequence p(n,1), p(n,2), ..., p(n,n) is unimodal. Szekeres proved this for all sufficiently large n, but with a threshold that was never made explicit. We prove the conjecture for ever...
Grishkov and Logachev have asked (Problem 1.4 of their recent survey) whether, for a fixed finite field F_q, the analytic ranks of the twists of the Carlitz module are bounded. We answer this in the negative for every q>= 3. For q = p^s and n>= 1 put P_{q,n} = (theta^(q^n) - theta)^(q-2). We show that the directed grap...
We prove the LeBrun--Salamon conjecture for positive quaternionic K\"ahler manifolds of quaternionic dimension n<= 14. To do so, we first show that the isometry group Isom(M) of any minimal counterexample M has rank at most one, hence dimension at most three. On the other hand, we consider a characteristic number which...
Let H be a subgroup of a finite group G, and put n = [G:H]>1. If p is the least prime divisor of n, we prove that the number of subgroups K with H<= K<= G is less than c(p) n^((log_p n)/4). Here c(p) is the product of (1 - p^(-j))^(-1) over all positive integers j, multiplied by the sum of p^(-z^2) over all integers z....
Motivated by the questions posed by W. V. Li and A. Wei and the conjecture of E. Lundberg and A. Thomack, we study the expected number of zeros of random harmonic polynomials $H_{n,m}(z)=p_n(z)+\overline{q_m(z)}$ with independently and identically distributed Gaussian coefficients. In this paper we verify the conjectur...
Da-Wei Lu, Yu-Chen Wang· Proceedings of the American...· 0 citations
We prove an asymptotic formula for correlations of divisor sums with quadratic cutoff functions shifted independently by \(s,u\in[0,1/2]\). The error is \(O_g(Y/(\log Y)^2)\), uniformly for \(Y^{1/5}\leq R\leq Y^{1/3}\), including coincident shifts. The main term is an explicit bilinear form in the first and second der...
Yi-Xiu Xiao, Hong-Ze Li· 1 citation· ⚡1
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