Heath-Brown identities for fractional powers of $\zeta$

Abstract

We construct finite Heath-Brown-type identities for the fractional powers $\zeta(s)^{\pm a/b}$ 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 $\sum_{n \le x} d_{\pm a/b}(n)e(n\alpha) \ll_{a,b} ( x q^{-1/2} + x^{4/5} + x^{1/2} q^{1/2} ) (\log 2x)^{C}$, for some constant $C=C(a,b)>0$, whenever $|\alpha - r/q| \le 1/q^2$ with $(r, q) = 1$. The bound carries no $x^{\varepsilon}$ loss, and the same machinery gives the endpoint case of the M\"obius function with an absolute constant. As applications we determine the major-arc expansion of $S_z(x, \alpha) = \sum_{n \le x} d_z(n)e(n\alpha)$ to arbitrary logarithmic precision for real $0<|z|<1$. For rational $z \in (-1,1)$, we determine the order of magnitude of $\sup_{\alpha} |S_z(x, \alpha)|$ and prove an asymptotic formula for the moments $\int_0^1 |S_z(x, \alpha)|^{s}\, d\alpha$ 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.