黎曼ζ函数分数次幂的希思-布朗恒等式
Heath-Brown identities for fractional powers of $ζ$
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中文总结 AI 辅助
该研究构造了黎曼ζ函数分数次幂的希思-布朗型恒等式，证明了相关指数和的维诺格拉多夫级界，确定了一类和函数的主弧展开、量级及矩的渐近公式，次弧分析避开了L-函数理论。
中文摘要 AI 辅助
我们从算术函数代数中的牛顿二项式级数出发，对所有满足0 < a/b < 1的既约分数a/b，构造了黎曼ζ函数的分数次幂ζ(s)^(±a/b)的有限希思-布朗型恒等式，并利用这些恒等式证明了如下的维诺格拉多夫级别的界：当|α - r/q| ≤ 1/q²且(r, q) = 1时，∑_{n ≤ x} d_{±a/b}(n)e(nα) ≪_{a,b} (x q^{-1/2} + x^{4/5} + x^{1/2} q^{1/2}) (log 2x)^C，其中C = C(a,b) > 0为某个常数。该界没有x^ε损失，且相同的方法给出了默比乌斯函数端点情形的绝对常数。作为应用，我们对实数0 < |z| < 1确定了S_z(x, α) = ∑_{n ≤ x} d_z(n)e(nα)的主弧展开至任意对数精度；对有理数z ∈ (-1,1)，确定了sup_α |S_z(x, α)|的量级，并对每个固定实数s > 2证明了矩∫₀¹ |S_z(x, α)|^s dα的渐近公式。次弧分析完全避开了L-函数理论，除继承自西格尔-瓦尔菲施定理的常数外，所有常数都是有效的。
英文摘要
We construct finite Heath-Brown-type identities for the fractional powers $ζ(s)^{\pm a/b}$ of the Riemann zeta-function, for every reduced fraction $a/b$ with $0 < a/b < 1$, from Newton's binomial series in the algebra of arithmetic functions, and we use them to prove the Vinogradov-quality bound $\sum_{n \le x} d_{\pm a/b}(n)e(nα) \ll_{a,b} ( x q^{-1/2} + x^{4/5} + x^{1/2} q^{1/2} ) (\log 2x)^{C}$, for some constant $C=C(a,b)>0$, whenever $|α- r/q| \le 1/q^2$ with $(r, q) = 1$. The bound carries no $x^{\varepsilon}$ loss, and the same machinery gives the endpoint case of the Möbius function with an absolute constant. As applications we determine the major-arc expansion of $S_z(x, α) = \sum_{n \le x} d_z(n)e(nα)$ to arbitrary logarithmic precision for real $0<|z|<1$. For rational $z \in (-1,1)$, we determine the order of magnitude of $\sup_α |S_z(x, α)|$ and prove an asymptotic formula for the moments $\int_0^1 |S_z(x, α)|^{s}\, dα$ for every fixed real $s > 2$. The minor-arc analysis avoids the theory of $L$-functions entirely, and all constants are effective except those inherited from the Siegel-Walfisz theorem.
