Counting square-free values of random polynomials

Technical Summary: Counting Square-Free Values of Random Polynomials

Problem Statement
The paper addresses the problem of counting square-free values taken by integer polynomials. Let $P \in \mathbb{Z}[t]$ be a polynomial of degree $d$. The quantity of interest is $E_P(x)$, defined as the error term between the actual count of square-free values $P(n)$ for $1 \le n \le x$ and the expected main term $S_P x$, where $S_P$ is a singular series product over primes. While it is a folklore conjecture that $E_P(x) = o(x)$ for any irreducible polynomial, this remains unproven for degrees $d \ge 4$ (even the existence of infinitely many square-free values is only known for $d \le 3$).

The author investigates the behavior of $E_P(x)$ when averaged over a family of random polynomials. Specifically, the paper considers the set $P_d(H) = \{P \in \mathbb{Z}[t] : \deg(P)=d, |P| \le H\}$, where $|P|$ denotes the maximum modulus of the coefficients. The goal is to determine the magnitude of the second moment of the error term:
$$(2H)^{-d-1} \sum_{P \in P_d(H)} |E_P(x)|^2$$
Previous works by Filaseta, Shparlinski, Browning, and others have established bounds for this moment, but the precise order of magnitude of the error term had not been fully characterized in the context of random polynomials.

Methodology
The proof proceeds through several analytical stages, combining probabilistic number theory with complex analytic techniques:

1. Truncation and Model Transition: The author replaces the indicator function for square-free numbers, $\mu^2(P(n))$, with a truncated model $SP(z)$ involving a sum over $k \le z$. This allows the separation of the diagonal terms ($n=m$) from the off-diagonal terms ($n \neq m$) in the second moment expansion.
2. Combinatorial Counting: Lemmas 2.1 through 2.3 provide bounds on the number of roots of polynomials modulo $k^2$ and the distribution of polynomials divisible by squares. These bounds are crucial for estimating the error introduced by the truncation.
3. Cesàro Summation and Perron Integrals: A key innovation identified in the paper is the recognition that the second moment structure inherently incorporates Cesàro summation. The off-diagonal sum $\sum_{n \neq m} c(m,n)$ (where $c(m,n)$ depends on $m-n$) is transformed using a version of Perron's formula. This allows the use of complex integration with a kernel $1/(s(s+1))$, which provides extra convergence factors.
4. Dirichlet Series Analysis: The arithmetic function $b(k) = c(k, 0)$ is analyzed. Its Dirichlet series is shown to be $\zeta(s)\zeta(2+2s)G(s)/\zeta(2)^2$, where $G(s)$ is a holomorphic function for $\Re(s) > -3/4$.
5. Contour Shifting: The integral is shifted to the left, crossing poles at $s=1$, $s=0$, and $s=-1/2$. The author shifts the line of integration to $\Re(s) = -1/2 - \delta$. This is possible due to the convergence provided by the Cesàro kernel and the specific bounds on the Riemann zeta function in the critical strip. The paper explicitly notes that the Lindelöf hypothesis or subconvexity bounds do not offer an advantage in this specific setup.

Key Contributions and Results
The primary contribution is the determination of the asymptotic behavior of the second moment of the error term. The paper proves that the error term is typically of size $x^{1/4}$ (since the second moment is $x^{1/2}$).

Theorem 1.1 establishes the following asymptotics for the second moment, where $\gamma_0 \approx 0.192$ is a specific constant derived from the residue at $s=-1/2$:

1. For $x \le H^{1/(d+3)}$:
   $$(2H)^{-d-1} \sum_{P \in P_d(H)} |E_P(x)|^2 = \gamma_0 x^{1/2} + O(x^{1/2-\delta})$$
   This holds for any $\alpha < 1/(d+3)$ with $x \le H^\alpha$.

2. For $x \le H^{1/(d+28/9)}$:
   $$(2H)^{-d-1} \sum_{P \in P_d(H)} |E_P(x)|^2 = \gamma_0 x^{1/2} + O(x^{4/9+\varepsilon})$$
   This holds for any $\varepsilon > 0$ and $\beta < 1/(d+28/9)$.

The constant $\gamma_0$ arises from the residue of the Dirichlet series at $s = -1/2$, specifically involving $\zeta(-1/2)$ and the Euler product of $G(s)$. The term $x^{1/2}$ in the second moment implies that the typical error $E_P(x)$ is of order $x^{1/4}$.

Significance and Claims
The paper claims that the order of magnitude $x^{1/2}$ for the second moment (and thus $x^{1/4}$ for the error) is not immediately apparent, as the diagonal terms alone contribute a term of order $x H^{d+1}$. The novelty lies in the precise cancellation of the diagonal contribution by the off-diagonal terms, a phenomenon facilitated by the built-in Cesàro summation in the second moment of arithmetic functions over random polynomials.

The author emphasizes that this result provides a full asymptotic expansion for the contribution of the terms $n \neq m$. The work extends the understanding of square-free values beyond the first moment (averaged results) to the second moment, offering a sharper picture of the distribution of errors. The paper notes that while Mirsky's work suggests asymptotics for all moments of $\mu^2$ are possible, the computations for higher moments involving Cesàro summation appear "forbidding."

The results are conditional on the degree $d$ and the relationship between $x$ and $H$, but they represent a significant step in quantifying the "typical" behavior of the error term in the square-free value problem for random polynomials.