Let $S(x) = \sum\_{n=0}^{N-1} a\_n e^{2 \pi i n x}$ be a trigonometric polynomial of length $N$. The analytic/harmonic large sieve inequality in its sharpest form states that


$$ \sum\_{r=1}^R |S(x\_r)|^2 \leq (N + \delta^{-1}-1) \int\_{0}^{1} |S(x)|^2 dx$$


where $x\_1,x\_2,\ldots,x\_R$ are $\delta$ separated points. This can be thought of as a discretization of Parseval's identity on the circle.


In many of the applications of this inequality to number theory one takes $\delta = 1/N$ in which case the right hand side of the inequality has a factor $2N$. 


The factor of $2$ here leads to some very unfortunate inefficiencies in sieve theoretic applications, such as the $2$ in the Brun-Titchmarsh inequality. It seems that improving the factor of $2$ in these applications is related to both the parity problem in sieve theory and the Siegel zero problem. See, for instance, [this paper](https://arxiv.org/pdf/1201.1777.pdf) of Maynard.


However it is easy to see in some cases, such as when $x\_1,x\_2, \ldots, x\_R$ are equally spaced (using Fourier analysis on the group of residues mod $N$), that the inequality indeed holds with a constant $1$ in place of the constant $2$.



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Examples of the large sieve inequality where a constant larger than 1 is needed
Some comments:


1) Montgomery's survey paper
<https://projecteuclid.org/euclid.bams/1183540922>


page 559, Theorem 3 and equation (19). 
It essentially says that Selberg gave an example,
with the comment that this is the only situation where this is sharp.


See also page 563 for lower bounds on $\Delta$, and also upper bounds, when one is sieving modulo
some subset of the integers $q\leq Q$.


This paper gives links to the rich literature of that time.


2) Similarly, also some comments after Theorem 4.7 of
Tenenbaum's Introduction to analytic and probabilistic number theory, (3rd edition).


Two more recent items, maybe not so well known:


3)
Vandermonde Matrices with Nodes in the Unit Disk
and the Large Sieve,
C\'eline Aubel and Helmut Bölcskei
<https://arxiv.org/abs/1701.02538>


4) Refinements of Selberg's sieve, 
Sara Elizabeth Blight, PhD Thesis, Rutgers, 2010.


Especially chapter 4
<https://rucore.libraries.rutgers.edu/rutgers-lib/27420/>


