---
title: 18.156 problem set 04
description: Linnik developed the large sieve to prove an estimate related to the distribution of quadratic residues.
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In this problem set, we digest the large sieve and its applications. First we recall the large sieve from class (Lecture 5). Recall that $[N] = \{1, ..., N\}$. If $f: [N] \rightarrow \mathbb{C}$, and $q \in \mathbb{N}$, we define $\pi_q f: \mathbb{Z}_q \rightarrow \mathbb{C}$ by
$$
\pi_q f(a) = \sum_{n \in a} f(n).
$$
We let $P_M:= \{ p \text{ prime } | p \sim M \}$.
> \[!theorem|latex\] Theorem 1.
>
> (Linnik large sieve) If $f: [N] \rightarrow \mathbb{C}$, and $M \le N^{1/2}$, then
>
> $$
> \sum_{p \in P_M} \| (\pi_p f)_h \|_{L^2}^2 \le C \frac{N}{M} \| f \|_{L^2}^2.
> $$
^theorem-1
**problem 1.** In order to digest the proof of Theorem [[courses/18-156-projection-theory-spring-2025/notes/05#^theorem-5-1|1]], sketch a proof for the following variation when $M > N^{1/2}$. This variation comes up in applications, like in the Bombieri-Vinogradov theorem.
^problem-1
> \[!theorem|latex\] Theorem 2.
>
> (Linnik large sieve) If $f: [N] \rightarrow \mathbb{C}$, and $M > N^{1/2}$, then
>
> $$
> \sum_{p \in P_M} \| (\pi_p f)_h \|_{L^2}^2 \le C M \| f \|_{L^2}^2.
> $$
^theorem-2
In your proof sketch, describe the Fourier analysis technical details in the level of detail that you find most helpful. Your proof sketch should illustrate why we have a factor $M$ on the right hand side in Theorem [[courses/18-156-projection-theory-spring-2025/notes/pset-04#^theorem-2|2]] as opposed to a factor of $N/M$ in Theorem [[courses/18-156-projection-theory-spring-2025/notes/05#^theorem-5-1|1]].
Linnik developed the large sieve to prove an estimate related to the distribution of quadratic residues. Our second goal for the problem set is to prove Linnik’s result.
**Background on quadratic residues.** Suppose that $p$ is a prime. Recall that a number $a \in \mathbb{Z}_p$ is called a quadratic residue if there is a solution to the equation $b^2 = a$ in $\mathbb{Z}_p$. (Here we count 0 as a quadratic residue, although sometimes one restricts attention to non-zero quadratic residues.) There are $\frac{p+1}{2}$ quadratic residues in $\mathbb{Z}_p$ and $\frac{p-1}{2}$ quadratic non-residues. Linnik wanted to understand how the quadratic residues are distributed in $\mathbb{Z}_p$. It appears that quadratic residues are distributed fairly randomly.
Suppose we took a large prime $p$ and colored the quadratic residues in $\mathbb{Z}_p$. Then for comparison suppose we randomly colored $\mathbb{Z}_p$ in two colors. These two colorings would look pretty similar. In the random coloring, the longest string of consecutive numbers that are the same color would have length around $\log p$. Something like this appears to be true for quadratic residues as well.
Define $q(p)$ to be the smallest $a$ so that $a$ is a quadratic non-residue in $\mathbb{Z}_p$. (The smallest quadratic residue is always 0.) If $q(p)$ is large, then it means that there are many quadratic non-residues in a row. Experiments suggest that $q(p)$ is always $\lesssim \log p$ or so. The general Riemann hypothesis implies that $q(p) \lesssim (\log p)^2$. However, the best proven bound is much worse, roughly $q(p) \lesssim p^{0.15...}$.
Linnik proved that, while there may be a few primes with $q(p)$ very large, these primes are quite rare. Let us make a little notation. We define
$$
P_{N^{1/2},L} := \{ p \text{ prime} | p \sim N^{1/2}, q(p) > L \} \tag{1}
$$
Then we define
$$
X_{N,L} := \{ n \in \mathbb{N} | n \le N, \text{ each prime factor of $n$ is at most } L \} \tag{2}
$$
> \[!theorem|latex\] Theorem 3.
>
> (Linnik) There is a universal constant $C > 0$ so that
>
> $$
> |P_{N^{1/2},L}| \le C \frac{N}{X_{N,L}}
> $$
^theorem-3
Number theorists have a good sense of $|X_{N,L}|$. For instance, for any $\epsilon > 0$, as $N \rightarrow \infty$, $|X_{N, N^\epsilon}| \sim C(\epsilon) N$, for some $C(\epsilon) > 0$. So Linnik’s theorem implies that $|P_{N^{1/2}, N^\epsilon}| \le C(\epsilon)$ uniformly in $N$.
In the rest of the problem set, you will prove Theorem [[courses/18-156-projection-theory-spring-2025/notes/pset-04#^theorem-3|3]].
**problem 2.** Prove that if $p \in P_{N^{1/2}, L}$, then $| \pi_p(X_{N,L})| \le \frac{p+1}{2}$.
^problem-2
**problem 3.** Suppose that $X \subset [N]$. Suppose that for some $p$, $| \pi_p(X) | \le (0.99) p$. Prove that there is a constant $c > 0$ so that
^problem-3
$$
\| ( \pi_p 1_X)_h \|_{L^2}^2 \ge c \frac{|X|^2}{p}.
$$
**problem 4.** Prove Theorem [[courses/18-156-projection-theory-spring-2025/notes/pset-04#^theorem-3|3]] using the large sieve and problems 1,2.
^problem-4
**Optional exploration.** To pursue this direction, it would be helpful to have a little background in restriction theory in Fourier analysis, but anyone can understand the problem.
In class, we used the large sieve to prove the following estimate.
> \[!theorem|latex\] Theorem 4.
>
> If $X \subset [N]$ and $| \pi_p(X) | \le (0.99) p$ for every $p \in P_{N^{1/2}}$, then $|X| \lessapprox N^{1/2}$
^theorem-4
This theorem is essentially sharp when $X$ is the set of squares.
We could explore what happens if we know $| \pi_p(X) \le (0.99) p$ for every $p \in P_{N^\alpha}$ for some other exponent $\alpha$, such as $\alpha = 1/4$.
Applying the large sieve as in Theorem [[courses/18-156-projection-theory-spring-2025/notes/05#^corollary-5-3|4]] shows that, if $|\pi_p(X)| \le (0.99)p$ for every $p \in P_{N^{1/4}}$, then $|X| \lessapprox N^{3/4}$. I don’t know any example where this is roughly sharp, and I suspect there is no such example. Examining the proof of the large sieve, we see that if the bound is almost tight, then $\int_0^1 | \hat 1_X(\xi)|^2 d \xi$ must be dominated by $\xi$ very close to fractions of the form $\frac{a}{p}$, $p \in P_{N^{1/4}}$, $a \not= 0$. The region close to these fractions has a very small measure, around $N^{-1/2}$, so it is striking for this region to contribute a large fraction of the integral.
There is a vague principle in Fourier analysis called the Heisenberg uncertainty principle, which says that it is difficult for both $f$ and $\hat f$ to concentrate in a small region. (The uncertainty principle can be made precise in various ways.) Here, the set $X \subset [N]$ is a small fraction of $[N]$ (because we already know $|X| \lessapprox N^{3/4}$). And if the large sieve argument is near tight, then $\hat 1_X$ concentrates in a small region near to the fractions $a/p$, with $p \in P_{N^{1/4}}$. I suspect that there is no such set $X$, and one might be able to prove it using ideas related to the Heisenberg uncertainty principle or to restriction theory.