In this paper, we prove that there does not exist a set with more than 26 polynomials with integer coe#cients, such that the product of any two of them plus a linear polynomial is a square of a polynomial with integer coe#cients. 1991 Mathematics Subject Classification: 11D09. 1.

Abstract. In this paper, we develop a large sieve type inequality with characters to square moduli. One expects that the result should be weaker than the classical inequality, but, conjecturally at least, not by much. The method is generalizable to higher power moduli. 1. Introduction and

Abstract. In this paper, we develop a large sieve type inequality with quadratic amplitude. We use the double large sieve to establish non-trivial bounds. 1. Introduction and Statements of the Results The large sieve was an idea originated by Yu. V. Linnik [11] in 1941. He also made application to distributions of quadratic non-residues. Since then, the idea has been refined and perfected by many. We denote ‖x ‖ = min k∈Z |x − k | for x ∈ R. A set of real numbers {xk} is said to be δ-spaced modulo 1 if

Abstract. We verify the Hardy-Littlewood conjecture on primes in quadratic progressions on average. The

Abstract. In this paper, we establish a theorem on the distribution of primes in quadratic progressions on average.

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Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after the pioneering work of Viggo Brun (see [Bru16],[Bru19], [Bru22]). Using his formulation of the sieve Brun proved, that the sum

of logarithms of Dirichlet L-functions

On the number of prime factors of integers

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