On general divisor functions over Piatetski-Shapiro sequences

1. Problem Statement

The paper investigates the asymptotic behavior of general divisor functions evaluated over Piatetski-Shapiro (PS) sequences.

• Piatetski-Shapiro Sequences: These are sequences of the form $[n^c]$, where $n$ is a positive integer and $c > 1$ is a real parameter. The floor function $[x]$ denotes the greatest integer less than or equal to $x$.
• The Core Problem: While the distribution of primes in PS sequences (the Piatetski-Shapiro Prime Number Theorem) and specific divisor functions (like $\tau_k([n^c])$) have been studied extensively, there is a lack of results for generalized divisor functions $f(n)$ over these sequences.
• The Function Class: The author considers arithmetic functions $f(n)$ defined by the convolution:
  $$f(n) = \sum_{n_1 n_2 = n} \tau(n_1)g(n_2)$$
  where $\tau(n)$ is the standard divisor function (number of divisors), and $g(n)$ is a weight function satisfying specific growth conditions ($\sum |g(n)| \ll x^{\theta + \epsilon}$).
• Goal: To establish an asymptotic formula for the sum $\sum_{n \le x} f([n^c])$ and extend these results to arithmetic progressions.

2. Methodology

The author employs analytic number theory techniques, specifically focusing on exponential sums and Vaughan's identity type decompositions adapted for PS sequences.

• Reduction to Exponential Sums:
  The sum over $[n^c]$ is transformed using the identity $[n^c] = m \iff -(m+1)\gamma < -n \le -m\gamma$ (where $\gamma = 1/c$). This allows the sum to be split into a main term (integral approximation) and an error term involving the fractional part function $\psi(t) = t - [t] - 1/2$.
• Exponential Sum Approximation:
  The fractional part function $\psi(t)$ is approximated using a trigonometric polynomial (Lemma 2.1), reducing the error term to exponential sums of the form:
  $$S = \sum_{m} f(m) e(h m^\gamma)$$
• Handling General Divisor Functions:
  The author utilizes the specific structure $f(n) = \sum \tau(n_1)g(n_2)$. By expanding $\tau(n_1)$ and applying partial summation, the problem reduces to estimating three-dimensional exponential sums.
• Key Technical Tools:
  • Van der Corput's Method & Exponential Sum Bounds: Lemma 2.3 (based on Jutila's work) is used to bound sums of the form $\sum e(X m_1^{\alpha_1} m_2^{\alpha_2} m_3^{\alpha_3})$.
  • Binary Additive Divisor Problem: To handle the arithmetic progression case, the paper relies on deep results regarding the sum $D(x, q) = \sum \tau(n)\tau(n+q)$ (Lemma 2.7, Jutila) and its error term properties.
  • Hecke-Maass Forms: For the arithmetic progression case involving Fourier coefficients $\lambda_\nu(n)$, the paper utilizes bounds related to the Ramanujan-Petersson conjecture (specifically the exponent $\alpha$ such that $\lambda_\nu(n) \ll n^\alpha$).

3. Key Contributions and Results

A. Main Theorem for General Divisor Functions (Theorem 1.1)

The paper establishes an asymptotic formula for $f([n^c])$ under the condition that the weight function $g(n)$ satisfies $\sum_{n \le x} |g(n)| \ll x^{5/8+\epsilon}$.

• Result: For fixed $c \in (1, 6/5)$,
  $$\sum_{n \le x} f([n^c]) = \int_1^{x^c} \gamma u^{\gamma-1} d\left( \sum_{n \le u} f(n) \right) + O(x^{1-\epsilon})$$
• Significance: This extends the range of admissible $c$ to $(1, 6/5)$ for a broad class of divisor functions, a range previously difficult to achieve using only three-dimensional exponential sums without deep results like Jutila's.

B. Specific Corollaries

The main theorem is applied to derive asymptotic formulas for several specific arithmetic functions:

1. Divisors of specific types: For $\tau_{(1,1,2,2)}(n)$ (related to the number of direct factors in finite Abelian groups), the paper provides an asymptotic formula with a main term involving $x \log x$ and $x$.
2. $k$-free Divisor Functions: For $\tau^{(k)}(n)$ (counting divisors that are $k$-free), an asymptotic formula is derived involving the Riemann zeta function $\zeta(s)$.
3. Standard Divisor Function: The result recovers and improves upon previous results for $\tau([n^c])$, matching the result obtained by using Jutila's deep theorem but derived here via the general framework.

C. Arithmetic Progressions (Theorem 1.5)

The paper generalizes the results to sums over arithmetic progressions $[n^c] \equiv a \pmod d$.

• Condition: The admissible range for $c$ depends on the exponent $\alpha$ in the bound for Hecke-Maass coefficients ($\lambda_\nu(n) \ll n^\alpha$). The range is $c \in (1, \frac{8+4\alpha}{7+4\alpha})$.
• Result:
  $$\sum_{\substack{n \le x \\ [n^c] \equiv a \pmod d}} f([n^c]) = \int_1^{x^c} \gamma u^{\gamma-1} d\left( \sum_{\substack{n \le u \\ n \equiv a \pmod d}} f(n) \right) + O(x^{1-\epsilon})$$
• Application: For the standard divisor function $\tau(n)$, using the best known bound $\alpha = 7/64$ (from the Kim-Sarnak bound), the admissible range is $c \in (1, 135/119)$. This provides an analogue of the Selberg-Hooley theorem for PS sequences.

4. Significance and Impact

• Unification: The paper provides a unified framework for studying various divisor functions over PS sequences, rather than treating them as isolated cases.
• Range Extension: It successfully extends the admissible range of the exponent $c$ to $(1, 6/5)$ for general divisor functions, which is a significant improvement over previous methods that relied on specific properties of $\tau_k$.
• Methodological Advancement: By combining the structure of general divisor functions with exponential sum estimates and results on the binary additive divisor problem, the author avoids the need for the most restrictive deep theorems in certain contexts, offering a more flexible approach.
• Arithmetic Progressions: The extension to arithmetic progressions connects PS sequences with the theory of Hecke-Maass forms, bridging gaps between prime distribution, divisor sums, and automorphic forms.

In summary, Wei Zhang's work significantly advances the understanding of the distribution of arithmetic functions over Piatetski-Shapiro sequences, providing robust asymptotic formulas that generalize previous results on primes and specific divisor functions.