Missing digits and sums of two prime squares

Technical Summary: Missing Digits and Sums of Two Prime Squares

Problem Statement
This paper investigates the intersection of two distinct areas in analytic number theory: integers with missing digits in their base-$g$ expansion and the representation of integers as sums of two prime squares. Specifically, the author studies the set $A(X)$ of integers up to $X = g^k$ whose base-$g$ expansion omits a fixed digit $b$ ($0 \le b < g$). The primary objective is to determine the asymptotic behavior and cardinality of the subset of $A(X)$ that can be written as $n = p^2 + q^2$ for primes $p, q$.

The study focuses on two main quantities:

1. The weighted count of representations, defined by the sum of the von Mangoldt function over representations: $r_2(n) = \sum_{a^2+b^2=n} \Lambda(a)\Lambda(b)$.
2. The unweighted count of representations, defined by the number of prime pairs: $r^*(n) = \#\{(p, q) \in \mathbb{P}^2 : p^2 + q^2 = n\}$.

Methodology
The paper employs a combination of the Hardy–Littlewood circle method and sieve theory, adapting techniques previously used by Maynard [14] for primes with missing digits and by the author [17] for sums of two prime squares.

1. Circle Method for Weighted Sums (Theorem 1.1):
   The proof for the average of $r_2(n)$ over the missing digit set utilizes the circle method. The unit interval is split into major arcs (near rationals with small denominators) and minor arcs.

   • Major Arcs: The analysis involves evaluating the singular series, which requires understanding the local density of solutions to $u^2 + v^2 \equiv a \pmod q$ where $u, v$ are coprime to the modulus. A key novelty is the derivation of a local factor $S(b, g)$ that depends on the missing digit $b$ and the prime factorization of the base $g$.
   • Minor Arcs: The author leverages the strong Fourier decay properties of the indicator function of the missing digit set (established in Maynard [14]) combined with exponential sum estimates for primes (Lemma 2.4) to bound the contribution from minor arcs.

2. Second Moment and Sieve Methods (Theorem 1.4):
   To study the unweighted function $r^*(n)$, the paper analyzes the second moment $\sum (r^*(n)^2 - 2r^*(n))$, which counts off-diagonal solutions to $p_1^2 + q_1^2 = p_2^2 + q_2^2$.

   • Factorization in $\mathbb{Z}[i]$: The author adapts an idea from [17] by factoring the equation $p_1^2 + q_1^2 = p_2^2 + q_2^2$ in the Gaussian integers as $(a+ib)(c+id) = (a-ib)(c-id)$ (up to units). This transforms the problem into counting solutions where linear forms in $a, b, c, d$ are prime.
   • Sieve and Circle Method Hybrid: The primality condition is relaxed using an upper bound sieve (Lemma 2.11) to allow for "rough" integers. The resulting sum is then analyzed using the circle method. Crucially, the major arcs for this second moment problem are defined with a much thinner width ($\exp(O(\log X)^{1/2})$) compared to the first part, necessitating refined exponential sum estimates (Lemma 2.10) to handle the minor arcs.

Key Results

• Theorem 1.1 (Weighted Average): For a sufficiently large base $g$, the weighted sum of representations satisfies:
  $$\sum_{n \in A(X)} r_2(n) = S(b, g) \cdot \frac{\pi}{4} \#A(X) + O_A\left(\frac{\#A(X)}{(\log X)^A}\right)$$
  where the local factor $S(b, g) = \frac{g}{g-1}\left(1 - \frac{\rho(b; g)}{\phi^2(g)}\right)$ depends on the missing digit $b$. The paper notes an interesting bias: $S(b, g)$ is maximized when $b=0$ and $g$ has a prime factor $p \equiv 3 \pmod 4$, and minimized when $b=0$ and all prime factors of $g$ are $\equiv 1 \pmod 4$.

• Corollary 1.3: Using Theorem 1.1 and partial summation, the paper establishes a lower bound for the unweighted sum:
  $$\sum_{n \in A(X)} r^*(n) \asymp \frac{\#A(X)}{(\log X)^2}$$

• Theorem 1.4 (Second Moment Bound): The paper establishes an upper bound for the off-diagonal contribution:
  $$\sum_{n \in A(X)} (r^*(n)^2 - 2r^*(n)) \ll \frac{\#A(X)(\log \log X)^4}{\log X}$$
  The author explicitly attributes the loss of a $(\log X)^2$ factor in the error term to the inability to take the major arcs thin enough, a limitation stemming from the lack of $L^\infty$ bounds for the Fourier transform of the missing digit indicator function.

• Theorem 1.6 (Cardinality of Representable Integers): By combining the first moment (Corollary 1.3) and the second moment (Theorem 1.4) via Cauchy-Schwarz and Markov's inequalities, the paper derives a nontrivial lower bound for the number of missing-digit integers representable as a sum of two prime squares:
  $$\frac{\#A(X)}{(\log X)^3 (\log \log X)^4} \ll \#\{n \in A(X) : r^*(n) > 0\} \ll \frac{\#A(X)}{(\log X)^2}$$

Significance and Claims
The paper contributes to the growing body of literature on "missing digit" problems by extending the scope from single primes (Maynard [14]) and three primes (Maier–Rassias [12], Leng–Sawhney [10]) to additive problems involving squares of primes.

The author claims the following significance:

1. Bias Discovery: The derivation of the local factor $S(b, g)$ reveals a specific arithmetic bias in the distribution of such numbers depending on the missing digit and the base's prime factors.
2. Nontrivial Existence: Theorem 1.6 proves that there are infinitely many integers with missing digits that are sums of two prime squares, providing a lower bound that, while not asymptotic, is significantly better than trivial bounds.
3. Methodological Synthesis: The work demonstrates the viability of combining the circle method with sieve estimates to handle the second moment of representation functions over sparse sets defined by digit restrictions.

The paper remains modest regarding the sharpness of the lower bound in Theorem 1.6. The author explicitly states that obtaining the expected asymptotic $\sim \frac{\pi}{2} S(b, g) \frac{\#A(X)}{(\log X)^2}$ is currently obstructed by the loss of the $(\log X)^2$ factor in the second moment estimate, which is a direct consequence of the limitations in understanding the $L^\infty$ bounds of the Fourier transform of the missing digit set. The results are also noted to extend to cases where multiple digits are missing, provided the number of missing digits is sufficiently small relative to the base.