A possible error in Elliot's book "Probabilistic Number Theory"

Question

In Elliot's book "Probabilistic Number Theory", there seems to be an inaccuracy. The author defines, for any sequence $a_n$, the quantity

$$V(p)=\sum_{r=0}^{p-1}\left|\sum_{\substack{n=1 \\n \equiv r\mathrm{mod}(p)}}^N a_n-p^{-1}\sum_{n=1}^N a_n \right|^2$$

He then asserts that, if $a_n$ assumes only the values 0,1, then

$$\sum_{p\leq Q}pV(p)\leq c_1 Q^2 \log(Q)\sum_{n=1}^N|a_n|^2$$

where $c_1$ is some absolute constant. The issue is, this would imply that

$$\limsup_{N\to\infty}\sum_{p\leq Q}p\frac{V(p)}{N^2}\leq c_1 Q^2 \log(Q)\limsup_{N\to\infty}\frac{1}{N^2}\sum_{n=1}^N|a_n|^2=0$$

which isn't always true. An easy counter-example is $a_n$ defined as $0$ when $n$ is even and $1$ when $n$ is odd, and $Q=2$. Namely, we have that

\begin{align*} \sum_{p\leq Q}p\frac{V(p)}{N^2} &= 2\frac{V(2)}{N^2}\\ &=\frac{2}{N^2}\sum_{r=0}^{1}\left|\sum_{\substack{n=1 \\n \equiv r\mathrm{mod}(2)}}^N a_n-\frac{1}{2}\sum_{n=1}^N a_n \right|^2\\ &=2\sum_{r=0}^{1}\left|\frac{1}{N}\sum_{\substack{n=1 \\n \equiv r\mathrm{mod}(2)}}^N a_n-\frac{1}{2N}\sum_{n=1}^N a_n\right|^2\\ &=2\left|\frac{1}{N}\sum_{\substack{n=1 \\n \equiv 0\mathrm{mod}(2)}}^N a_n-\frac{1}{2N}\sum_{n=1}^N a_n\right|^2+2\left|\frac{1}{N}\sum_{\substack{n=1 \\n \equiv 1\mathrm{mod}(2)}}^N a_n-\frac{1}{2N}\sum_{n=1}^N a_n \right|^2\\ \end{align*}

Since

$$\lim_{N\to\infty}\frac{1}{2N}\sum_{n=1}^Na_n=\frac{1}{4}$$ $$\lim_{N\to\infty}\frac{1}{N}\sum_{\substack{n=1 \\n \equiv 1\mathrm{mod}(2)}}^N a_n=\frac{1}{2}$$ $$\lim_{N\to\infty}\frac{1}{N}\sum_{\substack{n=1 \\n \equiv 0\mathrm{mod}(2)}}^N a_n=0$$

We see that

$$\lim_{N\to\infty}\sum_{p\leq Q}\frac{V(p)}{N^2}=2\left(\frac{1}{4}\right)^2+2\left(\frac{1}{4}\right)^2=\frac{1}{8}$$

Numerical computations show that $\frac{1}{8} \neq 0$ and thus this is a contradiction. The paper cited for this result is locked behind a paywall so I cannot access it and see what the true theorem is. Does anyone know what the actual result should have been? Where is the typo?

The paper cited with the result is

Roth, Klaus F., On the large sieves of Linnik and Renyi, Mathematika, Lond. 12, 1-9 (1965). ZBL0137.25904.

SIDE QUESTION:

In the book, there are many inequalities given for the sum $\sum_{p<Q}pV(p)$, but if you think about $V(p)$ as being on the order of $\frac{N^2}{p^2}$ for large $N,p$ (which is the worst-case scenario), then the sum $\sum_{p<Q}p^2V(p)$ feels much more natural to study, and $\sum_{p<Q}pV(p)$ feels like a logarithmically weighted version. Does anyone know of any inequalities for $\sum_{p<Q}p^2V(p)$?

Answer 1

You are right that the second display is false in general (Elliott might impose some conditions). The following version is well-known, and a consequence of Selberg's optimized large sieve inequality: $$\sum_{p\leq Q}pV(p)\leq (N+Q^2-1)\sum_{n=1}^N|a_n|^2.$$ This holds for any complex numbers $a_n$. See (23) in Montgomery: The analytic principle of the large sieve. Well, Montgomery has $N+Q^2$ instead of $N+Q^2-1$, but the latter is also valid in the light of Theorem 3 in Montgomery's survey.