Lower bounds for low moments of character sums, I: Short sums with general multiplicative weights

Technical Summary: Lower Bounds for Low Moments of Character Sums, I

Problem Statement
This paper addresses the problem of establishing sharp lower bounds for the low moments ($0 \le q \le 1$) of Dirichlet character sums and continuous character sums (zeta sums). Specifically, it investigates the average size of the quantity:
$$\frac{1}{r-1} \sum_{\chi \pmod r} \left| \sum_{n \le x} \chi(n) \right|^{2q}$$
where $r$ is a large prime and $1 \le x \le r^{0.499}$. The author also considers weighted sums $\sum_{n \le x} h(n)\chi(n)$ with multiplicative weights $h(n)$ (where $|h(p)|=1$ and $|h(p^k)| \le 1$) and continuous sums $\sum_{n \le x} n^{it}$.

Prior work by the author [10] established upper bounds for these moments that exhibit "better than square-root cancellation," specifically of the order:
$$\left( \frac{x}{1 + (1-q)\sqrt{\log \log (10L)}} \right)^q$$
where $L = \min\{x, r/x\}$. While these upper bounds were conjectured to be sharp, existing lower bounds (e.g., by La Bretèche, Munsch, and Tenenbaum [1]) fell short by powers of logarithmic factors. This paper aims to close that gap and prove that the upper bounds are indeed sharp for the specified range of $x$.

Methodology
The proofs rely on a "proxy" method, a staple in the analysis of $L$-function moments, involving the construction of a specific auxiliary object $I(\chi)$ designed to mimic the behavior of the character sum $\sum_{n \le x} \chi(n)$ but be more tractable for analysis.

1. Construction of the Proxy $I(\chi)$:
   The proxy is defined via a "barrier-adjusted" Perron integral:
   $$I(\chi) = \frac{1}{2\pi i} \int_{1/2-iT}^{1/2+iT} \left( \sum_{x^{1-3\beta} < p \le x^{1-\beta}} \frac{\chi(p)}{p^s} \right) F^*_\chi(s) G_\chi(s) \frac{x^s}{s} ds$$

   • Truncation: The integral is truncated at height $T$.
   • Structured Polynomial: The inner sum is replaced by a product of Dirichlet polynomials over disjoint prime ranges: large primes $(x^{1-3\beta}, x^{1-\beta}]$, $P$-smooth numbers ($\le x^\beta$), and $P$-rough numbers ($> P$).
   • The Barrier: The crucial component is $G_\chi(s)$, a short Dirichlet polynomial approximating a "barrier" function. This barrier restricts the growth of partial Euler products $\prod_{p \le P} |1 - \chi(p)p^{-s}|^{-1}$ across multiple scales (defined by $P e^{-j}$). This barrier is inspired by results on random multiplicative functions and is designed to suppress "abnormal" large values that would otherwise cause the fourth moment to blow up.

2. Correlation and Moment Bounds:
   The proof strategy reduces the main theorem to two Key Propositions involving the proxy $I(\chi)$:

   • Correlation: A lower bound for the correlation between the character sum and the proxy: $E_{\text{char}} |\sum \chi(n)| |I(\chi)|$.
   • Moments: Upper bounds for the second and fourth moments of the proxy: $E_{\text{char}} |I(\chi)|^2$ and $E_{\text{char}} |I(\chi)|^4$.
     By applying Hölder's inequality, the lower bound for the target moment is derived from these quantities.

3. Transition to Random Multiplicative Functions:
   A critical step involves passing from character sums to Steinhaus random multiplicative functions $f(n)$. Using the orthogonality relation $E_{\text{char}} \chi(n)\overline{\chi(m)} = \delta_{n,m}$ (valid for $n,m < r$), the author shows that provided the lengths of the Dirichlet polynomials involved are sufficiently small relative to $r$ (specifically $x^{2+4\beta} \ll r$), the character averages match the expectations over random multiplicative functions. This allows the use of probabilistic tools (Khintchine's inequality, properties of random Euler products) to estimate the moments of $I(\chi)$.

4. Handling the Barrier:
   The core technical difficulty lies in estimating the fourth moment of the proxy. The barrier $G_\chi(s)$ is constructed to ensure that the "blow-up" factor in the fourth moment (arising from correlations of Euler products at nearby points $s$ and $s'$) remains negligible. The barrier restricts the partial sums of the logarithm of the Euler product to lie within specific bounds, effectively controlling the "multiplicative chaos" behavior.

Key Contributions and Results
The paper establishes the following sharp lower bounds, matching the previously known upper bounds:

• Theorem 1: For a large prime $r$, uniformly for $1 \le x \le r^{0.499}$ and $0 \le q \le 1$:
  $$\frac{1}{r-1} \sum_{\chi \pmod r} \left| \sum_{n \le x} \chi(n) \right|^{2q} \gg \left( \frac{x}{1 + (1-q)\sqrt{\log \log (10x)}} \right)^q$$
• Theorem 2: The same sharp lower bound holds for weighted sums $\sum_{n \le x} h(n)\chi(n)$ where $h(n)$ is a multiplicative function with $|h(p)|=1$ and $|h(p^k)| \le 1$.
• Theorem 3: The same sharp lower bound holds for continuous character sums (zeta sums) $\sum_{n \le x} n^{it}$ for $1 \le x \le T^{0.499}$.

Significance and Claims

• Sharpness: The results confirm the conjecture that the upper bounds obtained in [10] are sharp for the range $x \le r^{0.499}$. This provides a complete order-of-magnitude understanding of low moments in this regime.
• Methodological Innovation: The paper introduces a specific "barrier-adjusted" proxy object that successfully mimics the behavior of character sums while allowing for rigorous fourth-moment estimates. This approach serves as a substitute for the conditioning arguments used in the random multiplicative function setting, which are difficult to apply directly to character sums due to the constraints on the conductor $r$.
• Range Limitations: The author notes that the restriction $x \le r^{0.499}$ (specifically $x < r^{1/2}$) arises naturally from the length constraints of the proxy's Dirichlet polynomials required to invoke the orthogonality relation (1.3).
• Future Work: The paper mentions that a companion paper [14] extends these arguments to the full range $x \le 0.99r$ for unweighted sums, utilizing special properties of unweighted sums to overcome the length restrictions encountered here. This extension leads to non-vanishing results for Dirichlet theta functions.

The paper does not claim to resolve the behavior of moments for $1 < q < 2$ (the gap between the second and fourth moments), noting this remains an open challenge, though it is conjectured to follow a similar logarithmic pattern.