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Square-free values of polynomials had been studied by various authors, including Estermann, Heath-Brown and Hooley. For 1 ≤ x , y ≤ H , Tolev proved that the number of the square-free values attained by the polynomial x 2 + y 2 + 1 has the asymptotic formula c 1 H 2 + O ( H 4 / 3 + e ) , where is c 1 is an absolute constant and e is an arbitrary small positive number. The key ingredient of his proof which leads to the elaborate error term is the estimate for the Kloosterman sum. In this paper, by using Tolev's method and some estimate for the Salie sum, we show that for any fixed integer k, there is an absolute constant c 2 such that the number of square-free values of the polynomial x 2 + y 2 + z 2 + k with 1 ≤ x , y , z ≤ H is c 2 H 3 + O ( H 7 / 3 + e ) .

In this paper it is shown that there is an absolute constant κ > 0 with the following property. For any prime p and nonempty subsets A , B of Z / p Z such that 1 | A | p 2 , B ∩ − B = ∅ and | B | κ | A | ln ⁡ ( | A | ) , we have that max b ∈ B ⁡ | ( A + b ) ∖ A | ≥ | B | . In 2011, V. Lev proved the former statement assuming also that | B | ≤ p 8 ; in the same paper, Lev suggested that this technical condition could be eliminated or weakened. In this paper his conjecture is confirmed.

Let χ be a Dirichlet character modulo a prime p . We give explicit upper bounds on q 1 q 2 … q n , the n smallest prime nonresidues of χ . More precisely, given n 0 and p 0 there exists an absolute constant C = C ( n 0 , p 0 ) > 0 such that q n ≤ C p 1 4 ( log ⁡ p ) n + 1 2 whenever n ≤ n 0 and p ≥ p 0 .

Let F be a self-dual Hecke-Maass cusp form for S L ( 3 , Z ) and let a F ( 1 , n ) denote the n-th coefficient of the Godement-Jacquet L-function L ( s , F ) . Then we show that there exists an absolute constant c 0 > 0 such that ∑ n ⩽ X a F ( 1 , n ) μ ( n ) e ( n α ) ≪ X exp ⁡ ( − c 0 log ⁡ X ) . Here the implied constant depends only on the form F and the bound is uniform in α ∈ R . Moreover, we notice that the aforementioned result generalises to self-dual automorphic cuspidal representations of G L 3 ( A Q ) , with unitary central character.

We prove that there exists an absolute constant c > 0 such that if an arithmetic progression P modulo a prime number p does not contain zero and has the cardinality less than cp, then it cannot be represented as a product of two subsets of cardinality greater than 1, unless P = − P or P = { − 2 r , r , 4 r } for some residue r modulo p.

We prove that there is an absolute constant $c>0$ with the following property: if $Z/pZ$ denotes the group of prime order $p$, and a subset $A\subset Z/pZ$ satisfies $1

We prove a formula for the Barban–Davenport–Halberstam average sum S ( Q , x ) = ∑ q ⩽ Q ∑ ( a , q ) = 1 1 ⩽ a ⩽ q ( ∑ n ≡ a ( mod q ) n ⩽ x Λ ( n ) − x φ ( q ) ) 2 , where x is sufficiently large, Λ ( n ) is the von Mangoldt function, and (α) x e − c ( log x ) 1 2 ⩽ Q ⩽ x , c > 0 being an absolute constant. The formula, which involves the exceptional zero of L-functions, comes from the intention of investigating the asymptotic behaviour of S ( Q , x ) via the circle method and the zero-density method for Q in the range (α) (presently unknown without assuming GRH). The formula not only implies a weaker version of the known asymptotic formula for S ( Q , x ) due to Montgomery and Hooley whenever x ( log x ) − A ⩽ Q ⩽ x for any constant A > 0 , but also improves a lower bound for S ( Q , x ) obtained by Hooley recently for Q satisfying (α) .

For a prime p, a subset S of Z p is a sumset if S = A + A for some A ⊂ Z p . Let f ( p ) denote the maximum integer so that every subset S ⊂ Z p of size at least p − f ( p ) is a sumset. The question of determining or estimating f ( p ) was raised by Green. He showed that for all sufficiently large p, f ( p ) ⩾ 1 9 log 2 p and proved, with Gowers, that f ( p ) c p 2 / 3 log 1 / 3 p for some absolute constant c. Here we improve these estimates, showing that there are two absolute positive constants c 1 , c 2 so that for all sufficiently large p, c 1 p log p ⩽ f ( p ) c 2 p 2 / 3 log 1 / 3 p . The proofs combine probabilistic arguments with spectral techniques.

We obtain an asymptotic formula for the number of square-free values among p−1, for primes p⩽x, and we apply it to derive the following asymptotic formula for L(x), the number of square-free values of the Carmichael function λ(n) for 1⩽n⩽x,L(x)=(κ+o(1))xln1−αx,where α=0.37395… is the Artin constant, and κ=0.80328… is another absolute constant.

Let 0⩽α = def {eiθ: θ∈[α, β]}. We show that for generalized (non–negative) polynomials P of degree r and p>0, we have ∑ j =1 m |P(a j )| p | a j − e iα | | a j − e iβ |+ β−α pr +1 2 1/2 ⩽cτ(pr+1) ∫ β α |P(e iθ )| p dθ, where a1, a2, …, am∈Δ, c is an absolute constant (and, thus, it is independent of α, β, p, m, r, P, {aj}) and τ is an explicitly determined constant which measures the number of points {aj} in a small interval. This implies large sieve inequalities for generalized (non–negative) trigonometric polynomials of degree r on subintervals of [0, 2π]. The essential feature is the uniformity of the estimate in α and β.

In this paper, we will show that there is a close connection between the vertical distribution of the zeros of the Riemann Zeta function ζ( s ) and that of the zeros of its derivative ζ′( s ). To be precise, we will prove, assuming the Riemann Hypothesis, the following theorem: THEOREM. Let T 0 be a large fixed positive real number and T ≥ T 0 . Let ρ′ 0 = β′ 0 + i γ′ 0 be a zero of ζ′( s ) with 1/2 0 g ( T ) where g ( T ) → 0 when T → ∞, and T ≤ γ′ 0 ≤ 2 T . Suppose there exists another zero ρ′ 1 of ζ′( s ) with |ρ′ 1 − ρ′ 0 | ≤ A (β′ 0 − 1/2) for some absolute constant A > 0. Then there exists a positive real number B depending only on A and a zero ρ = β + i γ of ζ( s ) with |γ′ 0 − γ| ≤ B (β′ 0 − 1/2) ( Note that here ρ′ 1 may be equal to ρ′ 0 , i.e ., ζ′( s ) may have a double zero at s = ρ′ 0 .) We will also give a slight generalization of this result.

A sightly improved classical transcendence measure for e will be given, by showing that the absolute constant in Mahler′s measure can be taken to be 1. We also give an improved linear independence measure for the system 1, e ,..., e n .

Let n > 2 be an integer, and for each integer 0 < x < n with (n, x) = 1, define x by the congruence xx ≡ 1 (mod n) and 0 < x < n. Let M(n, k) = ∑′n−1a=1 (a − a)2k. The main purpose of this paper is to study the asymptotic behaviour of M(n, k), and prove for any positive integer k that we have [formula] where φ(n) is the Euler function, d(n) is the divisor function, and the O(·) provides an absolute constant.

If α1, α2, α3 are algebraic numbers satisfying (i) the height of α1, α2, α3 do not exceed H (ii) the degree of the field generated by α1, α2, α3 over the field of rational numbers do not exceed D, then a positive lower bound for ∑k=13|2πk−αk| is determined explicitly (except for an absolute constant) in terms of D and H.

It is shown that for every positive integer s there is an absolute constant c(s) > 0, such that for every s-tupel of reals α1, …, α5 and every positive integer N there is a box B with sides not necessarily parallel to the coordinate axes in the s-dimensional unit-cube Is with volume greater than c(s) N −1 s which contains none of the points ({kα1}, …, {kαs}); k = 1, …, N.