Averages of diagonal Elliott-Halberstam problem twisted by Möbius function with Sobolev and Hölder-Zygmund weights

Technical Summary: Averages of Diagonal Elliott-Halberstam Problem Twisted by Möbius Function with Sobolev and Hölder-Zygmund Weights

Problem Statement
The paper investigates weighted average variants of the Elliott-Halberstam (EH) conjecture twisted by the Möbius function, denoted as $EH_\mu(N^\theta)$. The classical $EH_\mu$ conjecture posits that the distribution of primes weighted by the Möbius function in arithmetic progressions exhibits strong cancellation, a condition crucial for overcoming the "parity problem" in sieve methods. Specifically, the conjecture asserts that for $0 < \theta < 1$, the sum of discrepancies between the actual distribution of $\Lambda(n)\mu(N-n)$ in residue classes and their expected average is bounded by $N/\log(N)^A$.

The author notes that proving $EH_\mu(N^\theta)$ for $\theta > 1/2$, in conjunction with the classical Elliott-Halberstam conjecture, implies the binary Goldbach conjecture for sufficiently large even integers. However, the original formulation involves a maximum over the range $y \le N$, which complicates analysis. The paper focuses on a "diagonal" version of this problem, denoted $dEH_{\mu, \log}(N^\theta)$, where the summation is restricted to the diagonal case $n \equiv N \pmod q$ (effectively setting $y=N$ and removing the maximum over $y$). The primary objective is to establish upper bounds for weighted averages of this diagonal twisted sum, where the weights $f$ belong to specific function spaces, thereby demonstrating consistency with the conjectured bounds.

Methodology
The core methodological innovation is the application of a two-dimensional Abel summation formula to "decouple" the arithmetic functions involved in the weighted average. The paper utilizes two distinct approaches depending on the regularity of the weight function $f$:

1. Continuous Case (Sobolev Spaces): For weights $f$ in the Sobolev space $W^{2,1}$, the author employs a continuous two-dimensional Abel summation formula (Theorem 8). This identity relates the weighted average of the convolution of two arithmetic functions to the Laplace convolution of their unweighted explicit formulas. This allows the author to substitute the explicit formulas for the von Mangoldt function $\Lambda(n)$ and the Mertens function $M(n)$ (involving sums over non-trivial zeros of $L$-functions) directly into the integral representation.

2. Discrete Case (Hölder-Zygmund Spaces): For weights with lower regularity, specifically those in the Hölder-Zygmund spaces $C_\delta$ for $1 \le \delta < 2$, the author employs a discrete version of the Abel summation formula (Theorem 9). This approach replaces the second derivative of the weight with second-order forward differences ($\Delta^2_{1/N} f$). This shift allows the analysis to rely on the properties of Hölder-Zygmund spaces, which control the growth of these finite differences.

Key Assumptions
The results are conditional on several standard and conjectural hypotheses in analytic number theory:

• Generalized Riemann Hypothesis (GRH): Assumed for the relevant $L$-functions.
• Gonek-Hejhal Conjecture (Weak Version): Specifically, Conjecture 19, which asserts $\sum_{0<\gamma \le T} |\zeta'(\rho)|^{-2} \ll T$. This is used to ensure the absolute convergence of double series involving the non-trivial zeros of the Riemann zeta function and Dirichlet $L$-functions.
• Simplicity of Zeros: The analysis assumes the non-trivial zeros of $\zeta(s)$ are simple.

Key Contributions and Results

1. Reduction to Diagonal Conjecture: The paper first demonstrates (Section 2) that the diagonal conjecture $dEH_{\mu, \log}$ is sufficient to prove the binary Goldbach conjecture, justifying the study of this specific variant.

2. Explicit Formulas for Weighted Averages:

   • Sobolev Weights ($W^{2,1}$): Theorem 21 provides a truncated explicit formula for the weighted average involving a weight $f \in W^{2,1}$. The main term is expressed as a double sum over the non-trivial zeros of $\zeta(s)$ and $L(s, \chi)$, weighted by Gamma functions and an integral of the second derivative of $f$.
   • Hölder-Zygmund Weights ($C_\delta$): Theorem 23 provides a similar explicit formula for weights in $C_\delta$, where the integral involving $f''$ is replaced by a sum involving the second forward difference of $f$.

3. Upper Bounds for Weighted Averages:

   • Sobolev Case: Under GRH and the weak Gonek-Hejhal conjecture, Theorem 22 establishes that for weights in $W^{2,1}$, the weighted average of the diagonal twisted EH problem satisfies the bound $\ll_\varepsilon N^{2-\varepsilon} E(f'')$ for the full range $0 < \theta < 1$ (specifically $\theta = 1-2\varepsilon$).
   • Hölder-Zygmund Case: Theorem 24 extends these results to weights in $C_\delta$. The achievable range of $\theta$ depends on the regularity parameter $\delta$:
     • For $\delta \in [3/2, 2)$, the bound holds for $\theta < \delta - 1 - 2\varepsilon$.
     • For $\delta \in [1, 3/2)$, the bound holds for $\theta < 1/2 - 2\varepsilon$.
   • Logarithmic Weights: Sections 5 extends these results to the case where the Möbius function is multiplied by a logarithm, $\mu(n)\log(n)$, showing analogous bounds (Theorems 26 and 28).

4. Examples: Section 6 applies these general theorems to specific weight functions, including classical Cesàro-Riesz weights (which fall into the Sobolev class for $k>1$) and a specific Zygmund-type weight $f_{Zyg}(x) = (1-x)\sin(\log(1-x))$, demonstrating the applicability of the Hölder-Zygmund framework.

Significance and Claims
The paper claims to provide a consistent upper bound for the weighted average of the diagonal twisted Elliott-Halberstam problem. The primary significance lies in showing that these bounds are consistent with the "diagonal versions" of the conjecture (where $y=N$) under the assumption of GRH and a weak form of the Gonek-Hejhal conjecture.

The author emphasizes that the method successfully handles weights with varying degrees of regularity. By utilizing the discrete Abel summation formula, the paper extends the analysis beyond smooth Sobolev weights to the broader class of Hölder-Zygmund functions. The results suggest that the "diagonal" formulation, while weaker in terms of the maximum over $y$, retains the necessary strength to yield the expected cancellation in the weighted averages, provided the weights are sufficiently regular (or the regularity threshold $\theta$ is adjusted accordingly). The work does not claim to prove the full Elliott-Halberstam conjecture or the Goldbach conjecture unconditionally but rather establishes the validity of the weighted average variants under standard analytic hypotheses.