AI 中文总结
在广义黎曼猜想下，研究大整数表示为几乎相等的素数k次幂之和的短区间华林-哥德巴赫问题，证明了分布尺度指数可大于1/2，并给出解数渐近公式，基本解决了该问题。
AI 中文摘要
设 N 为足够大的奇数。我们研究将一个大整数 m N^k 表示为 m 个几乎相等的素数 k 次幂之和的问题，其中共同分布尺度至少为 N^theta。在假设广义黎曼猜想（GRH）成立的前提下，我们证明：对于任意 epsilon > 0，可取 theta > 1/2，只要 m < k(k+1)/2 且 m、k 为足够大的奇数。这一结果是通过采用一个与早期工作密切相关的新工具而获得的。对于足够大的 m，通过改进对次要弧段的讨论，我们进一步证明：在 GRH 下，当 m > 16k log k + 4k + 2 时，该方程有解。此外，我们给出了解数的一个渐近公式，其主项涉及 N^{2k theta (1 - (1 - 1/k)^{m_1}) + theta(m-1) + 1 - k}，其中 m_1 = 4k。在 GRH 下且对于足够大的 m、k，短区间中的华林-哥德巴赫问题基本得到解决。
英文摘要
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.
Comments18 pages
