关于大筛法不等式（Large Sieve Inequality）的表述、基础应用及d(p-1)求和估计的技术问询

Hey folks,

I've been digging into the Large Sieve Inequality recently, working through Chapter 27 of Davenport's Multiplicative Number Theory. Now that I've wrapped up the chapter, I'm eager to build a more practical, intuitive understanding of how this inequality is applied—both to get more comfortable with it and satisfy my curiosity.

I'd be really grateful if someone could share key statements of the Large Sieve Inequality along with trusted references (no detailed proofs needed, just clear formulations and where to find more depth). Additionally, I'm looking for simple, concrete applications to solidify my grasp, like:

• Counting perfect squares in $[1, N]$
• Counting primes in the interval $[M+1, M+N]$
• Counting twin primes in $[1, N]$

On top of that, I've heard the Large Sieve Inequality can be used to estimate the sum $\sum_{p \leq x} d(p-1)$ (where $d(n)$ is the divisor function, summing over all primes $p \leq x$). I've searched around diligently over the past few days but haven't found a clear reference for this specific application. Any guidance or pointers to relevant sources would be a huge help.

补充整理（基于常见资料） #

(Note: I've compiled some standard info to frame the discussion, but I'm looking for more targeted insights)

1. Core Statements of the Large Sieve Inequality

• Arithmetic Form: Let $A$ be a set of integers with $X \leq n < 2X$, $|A| = N$. For any collection of moduli $q_1, q_2, ..., q_r$, where each $q_j$ has $t_j$ distinct residue classes $S_j$, and all residue classes across moduli are pairwise non-overlapping (i.e., no two classes from different moduli are congruent), we have:
  $$\sum_{j=1}^r t_j^{-1} \sum_{a \in S_j} \left| \sum_{\substack{n \in A \ n \equiv a \mod q_j}} a_n \right|^2 \leq (X + Q^2) \sum_{n \in A} |a_n|^2$$
  where $Q = \max{q_1, ..., q_r}$, and $a_n$ are arbitrary complex numbers.
• Analytic Form: For any complex-valued function $f(n)$ and real numbers $\alpha_1, ..., \alpha_r$ with pairwise distances $\geq \delta > 0$, we have:
  $$\sum_{k=1}^r \left| \sum_{n \leq N} f(n) e^{2\pi i \alpha_k n} \right|^2 \leq \left(N + \delta^{-1}\right) \sum_{n \leq N} |f(n)|^2$$

2. Key References

• Davenport, H. Multiplicative Number Theory (Chapter 27, foundational treatment)
• Montgomery, H.L. Multiplicative Number Theory I: Classical Theory (Chapter 10, detailed deep dive into the large sieve)
• Iwaniec, H. & Kowalski, E. Analytic Number Theory (Chapter 7, modern formulations and extensions)

3. Basic Applications

• Perfect Squares in $[1, N]$: While the exact count is $\lfloor \sqrt{N} \rfloor$, the large sieve can be used to prove non-trivial upper bounds by restricting squares to their congruence classes modulo small integers, verifying that their distribution aligns with expected sparse behavior.
• Primes in $[M+1, M+N]$: The large sieve gives an upper bound for the number of primes in short intervals, such as:
  $$\pi(M+N) - \pi(M) \leq \frac{2N}{\log N} + O\left(N^{1/2} \log N\right)$$
  This relies on bounding the concentration of primes in any single residue class modulo small moduli.
• Twin Primes in $[1, N]$: For the count $T(N)$ of twin primes $\leq N$, the large sieve yields an upper bound:
  $$T(N) \leq C \frac{N}{(\log N)^2}$$
  where $C$ is a constant, by limiting the number of prime pairs $(p, p+2)$ that fall into forbidden congruence classes modulo small integers.

Again, I'm particularly stuck on finding references for the $\sum_{p \leq x} d(p-1)$ sum estimation using the large sieve. Any leads would be greatly appreciated!

备注：内容来源于stack exchange，提问作者zero2infinity