Heath-Brown identities for fractional powers of $ζ$

Technical Summary: Heath-Brown Identities for Fractional Powers of $\zeta$

Problem Statement
The paper addresses the estimation of exponential sums involving fractional powers of the Riemann zeta-function, specifically $S_z(x, \alpha) = \sum_{n \le x} d_z(n) e(n\alpha)$, where $d_z$ is the multiplicative function defined by the Dirichlet coefficients of $\zeta(s)^z$. While the behavior of these sums is well-understood for integer exponents (e.g., the divisor function $\tau_k = d_k$ and the Möbius function $\mu = d_{-1}$), the fractional case $z \in (-1, 1) \setminus \mathbb{Z}$ presents a gap. Classical methods relying on factorization (like $\tau_k = 1^{*k}$) fail for non-integers, and existing identities (such as Vaughan's or Heath-Brown's) do not directly apply without modification. Previous partial results, such as the bound for $d_{1/2}$ in [MV, Exercise 9], yielded a suboptimal middle exponent of $6/7$ in the minor-arc estimate, whereas the integer cases achieve the "Vinogradov quality" benchmark of $4/5$. The central problem is to construct a finite combinatorial identity for $\zeta(s)^{\pm a/b}$ that allows for the derivation of bounds with the optimal $4/5$ exponent and purely logarithmic losses, without the $x^\varepsilon$ factor.

Methodology
The author employs a two-layered approach: algebraic construction followed by analytic estimation.

1. Algebraic Construction (Heath-Brown-type Identities):
   The core innovation is the construction of finite identities for $\zeta(s)^{\pm a/b}$ using Newton's binomial series within the algebra of arithmetic functions.

   • For a fractional power $\zeta(s)^{-h/b}$, the author defines a truncated Dirichlet polynomial $F(s) = \sum_{n \le V} d_{-h/b}(n)n^{-s}$.
   • By considering the remainder $H = 1 - \zeta(s)^h F(s)^b$, the author utilizes the property that $H$ has support on integers $> V$. Consequently, $H^K$ has support on integers $> V^K$.
   • Substituting $H$ into the truncated Newton binomial expansion of $(1-H)^{-1/b}$ yields an exact identity for $\zeta(s)^{-h/b}$ valid for coefficients $n \le V^K$.
   • The paper selects depth $K=5$ and $V = x^{1/5}$. This ensures the identity holds for all $n \le x$. The resulting identity expresses $\zeta(s)^{-h/b}$ as a finite linear combination of convolutions involving $F$ and powers of $\zeta$. Crucially, every "long" factor in these convolutions is a positive power of $\zeta$ (the constant function 1), ensuring smoothness, while "short" factors are bounded by 1.
   • For positive exponents $z = h/b$, the identity is derived indirectly via $\zeta(s)^{h/b} = \zeta(s) \cdot \zeta(s)^{-(b-h)/b}$, preserving the smoothness of the long factors.
   • The endpoint case $z = -1$ (the Möbius function) is recovered as a special case where the identity degenerates into the classical Heath-Brown identity.

2. Analytic Estimation:
   The constructed identities reduce the exponential sum $S_z(x, \alpha)$ to a sum of multilinear forms $T_{J,L}(x, \alpha)$.

   • Dyadic Decomposition: Variables are localized to dyadic intervals.
   • Combinatorial Dichotomy: A "window-or-long-variable" lemma is applied. For any dyadic configuration with product $P \in (x^{4/5}, x]$, either a subset of variables can be grouped into the "Type II window" $[x^{2/5}, x^{3/5}]$, or a single variable is long ($> x^{3/10}$) while the product of the rest is short ($< x^{7/10}$).
   • Estimates:
     • Type II: Bilinear estimates are applied when the window condition is met, yielding the $x^{4/5}$ term.
     • Type I: When a long smooth variable exists, the sum is treated as a Type I sum. The author avoids the false assumption that divisor-weighted Type I sums admit only logarithmic losses. Instead, a two-range strategy is used:
       • For $q \le x^{3/10}$ and $q \ge x^{7/10}$, a mean-square Type I estimate is used.
       • For the intermediate range $x^{3/10} < q < x^{7/10}$, a pointwise estimate with an $x^{o(1)}$ loss is used. The gap between $x^{7/10}$ and the target $x^{4/5}$ absorbs this loss, ensuring the final bound retains the $4/5$ exponent with only logarithmic factors.

Key Results

1. Minor-Arc Bounds (Theorem 1.1 & 1.3):
   For any reduced fraction $a/b$ with $0 < a/b < 1$, and for $z = \pm a/b$, the paper proves:
   $$\sum_{n \le x} d_{\pm a/b}(n) e(n\alpha) \ll_{a,b} \left( \frac{x}{\sqrt{q}} + x^{4/5} + \sqrt{xq} \right) (\log 2x)^C$$
   for $|\alpha - r/q| \le 1/q^2$.

   • This bound holds with no $x^\varepsilon$ loss, matching the shape of Vinogradov's estimate for the von Mangoldt function and the known bounds for integer powers.
   • Theorem 1.3 extends this to the endpoint $z = -1$ (Möbius function) with an absolute constant, refining previous estimates by removing the $\varepsilon$.

2. Major-Arc Expansion (Theorem 1.5):
   For real $0 < |z| < 1$, the paper provides a major-arc expansion of $S_z(x, \alpha)$ to arbitrary logarithmic precision. The main term involves local factors $g_z(q)$ (generalizing the singular series) and coefficients derived from the Selberg-Delange method. The error term is $O(x (\log x)^{-A})$ for any $A > 0$, though the implied constant is ineffective due to the reliance on the Siegel-Walfisz theorem.

3. Uniform Bounds and Moments (Theorems 1.6 & 1.7):

   • Supremum: The paper determines the order of magnitude of $\sup_\alpha |S_z(x, \alpha)|$. For $z > 0$, the supremum is attained at $\alpha=0$ and is asymptotic to the mean value. For $z < 0$, cancellation reduces the supremum by a factor of $(\log x)^{1+|z|}$ compared to the $\ell^1$ norm.
   • Moments: For rational $z \in (-1, 1)$ and real $s > 2$, the paper proves an asymptotic formula for the moments $\int_0^1 |S_z(x, \alpha)|^s d\alpha$. The leading term is determined by a singular series involving $g_z(q)$, analogous to Pandey's result for integer divisor functions. The restriction $s > 2$ is necessary as the singular series diverges at $s=2$.

Significance and Claims
The paper claims to resolve the gap between integer and fractional powers of the zeta-function in the context of exponential sums. Its primary significance lies in:

• Optimality: Achieving the $x^{4/5}$ exponent for fractional powers, which was previously an open question and only known for specific cases with weaker bounds.
• Efficiency: Eliminating the $x^\varepsilon$ loss, which is crucial for applications requiring precise error terms (e.g., in the circle method).
• Algebraic Novelty: The construction of finite Heath-Brown identities for fractional powers via Newton's binomial series and support arguments, providing a new algebraic tool for handling non-integer exponents in the algebra of arithmetic functions.
• Effectivity: The minor-arc bounds and the construction of the identities are fully effective. The ineffectivity in the major-arc applications is explicitly attributed to the Siegel-Walfisz theorem, a standard limitation in this domain, rather than the new machinery.

The author emphasizes that the restriction to rational exponents is essential for the algebraic construction (as it relies on the $b$-th root of an integer power), and the restriction $|z| \le 1$ ensures the short coefficients remain bounded by 1. The paper does not claim to extend these results to irrational exponents or exponents outside $[-1, 1]$ without significant modification.