Related papers: The square sieve and the large sieve with square m…

We show a new large sieve version of the Brun-Titchmarsh theorem.

In this article, we continue our recent investigations on bilinear sums and additive energies with modular square roots. Here we improve our recent results for the case when the ranges of variables are large. We use these results to make…

In this paper, we establish a general version of the large sieve with additive characters for restricted sets of moduli in arbitrary dimension for function fields. From this, we derive function field versions for the large sieve in high…

We extend bounds on additive energies of modular square roots by Dunn, Kerr, Shparlinski, Shkredov and Zaharescu and apply these results to obtain bounds on certain bilinear exponential sums with modular square roots. From here, we make…

We introduce a variant of the large sieve and give an example of its use in a sieving problem. Take the interval [N] = {1,...,N} and, for each odd prime p <= N^{1/2}, remove or ``sieve out'' by all n whose reduction mod p lies in some…

The purpose of this paper is twofold: 1) Applications of Gallagher's larger sieve modulo prime squares do not work. In some relevant cases we can transform the residue class information modulo $p^2$ to more suitable residue information…

We continue our investigations of bilinear sums with modular square roots and the large sieve for square moduli in our recent article "On bilinear sums with modular square roots and applications II", arXiv:2603.00768. In the present…

We improve the large sieve inequality with $k$th-power moduli, for all $k\ge 4$. Our method relates these inequalities to a restricted variant of Waring's problem. Firstly, we input a classical divisor bound on the number of representations…

In this paper, we develop a large sieve type inequality for some special characters whose moduli are squares of primes. Our result gives non-trivial estimate in certain ranges.

We improve on the spectral large sieve inequality for symmetric-squares. We also prove a lower bound showing that the most optimistic upper bound is not true for this family.

We prove an analogue of Heath-Brown's bound on the twelfth moment of the Riemann zeta function for Dirichlet L-functions with smooth moduli.

We establish large sieve inequalities for power moduli in imaginary quadratic number fields, extending earlier work of Baier and Bansal for the Gaussian field.

In this article, we establish a large sieve inequality for additive characters to moduli in the range of appropriate integer polynomials of degree two. As an application, we derive a weighted zero-density estimate for twists of…

We prove a conjecture of Heath-Brown on the number of rational points of bounded height for a large class of projective varieties.

We improve on the best available bounds for the square-free sieve and provide a general framework for its applicability. The failure of the local-to-global principle allows us to obtain results better than those reached by a classical…

We use the Burgess bound and Selberg sieve to obtain an upper bound on the second moment of sums over an interval $[u+1,u+h]$ of Legendre symbols modulo primes $p$ in a dyadic interval $[Q,2Q]$. The bound is nontrivial and gives a power…

We prove a power saving upper bound for the sum of Fourier coefficients $\rho_f(\cdot)$ of a fixed cubic metaplectic cusp form $f$ over primes. Our result is the cubic analogue of a celebrated 1990 Theorem of Duke and Iwaniec, and the…

Suppose that an infinite set $A$ occupies at most $\frac{1}{2}(p+1)$ residue classes modulo $p$, for every sufficiently large prime $p$. The squares, or more generally the integer values of any quadratic, are an example of such a set. By…

In this paper, we prove a large sieve inequality for quartic Dirichlet characters. The result is analogous to large sieve inequalities for the quadratic and cubic Dirichlet characters.

We prove an asymptotic formula for the number of representations of squares by nonsingular cubic forms in six or more variables. The main ingredients of the proof are Heath-Brown's form of the Circle Method and various exponential sum…