In this paper, we study how close the terms of a finite arithmetic progression can get to a perfect square. The answer depends on the initial term, the common difference and the number of terms in the arithmetic progression.

In this paper, we prove some one level density results for the low-lying zeros of families of L-functions. More specifically, the families under consideration are that of L-functions of holomorphic Hecke eigenforms of level 1 and weight k twisted with quadratic Dirichlet characters and that of cubic and quartic Dirichlet L-functions.

For two real characters ψ,ψ′ of conductor dividing 8 define where and the subscript 2 denotes the fact that the Euler factor at 2 has been removed. These double Dirichlet series can be extended to possessing a group of functional equations isomorphic to D12. The convexity bound for Z(s,w;ψ,ψ′) is |sw(s+w)|1/4+ε for ℜs=ℜw=1/2. It is proved that Moreover, the following mean square Lindelöf-type bound holds: for any Y1,Y2≥1.

Let u(n)=f(gn), where g > 1 is integer and f(X) ∈ ℤ[X] is non-constant and has no multiple roots. We use the theory of -unit equations as well as bounds for character sums to obtain a lower bound on the number of distinct fields among for n ∈ . Fields of this type include the Shanks fields and their generalizations.

Here we derive a recursive formula for even-power moments of Kloosterman sums or equivalently for power moments of two-dimensional Kloosterman sums. This is done by using the Pless power-moment identity and an explicit expression of the Gauss sum for Sp(4,q).

We study the Gowers norm for periodic binary sequences and relate it to correlation measures for such sequences. The case of periodic binary sequences derived from inversive pseudorandom numbers is considered in detail.

The main goal of this paper is to provide asymptotic expansions for the numbers #{p≤x:p prime,sq(p)=k} for k close to ((q−1)/2)log qx, where sq(n) denotes the q-ary sum-of-digits function. The proof is based on a thorough analysis of exponential sums of the form (where the sum is restricted to p prime), for which we have to extend a recent result by the second two authors.

A two-bridge knot (or link) can be characterized by the so-called Schubert normal form Kp, q where p and q are positive coprime integers. Associated to Kp, q there are the Conway polynomial ▽kp, q(z) and the normalized Alexander polynomial Δkp, q(t). However, it has been open problem how ▽kp, q(z) and Δkp, q(t) are expressed in terms of p and q. In this note, we will give explicit formulae for the Conway polynomials and the normalized Alexander polynomials in the case of two-bridge knots and links. This is done using elementary number theoretical functions in p and q.

In this paper we study the pseudorandom properties of the signs of Kloosterman sums.

A diophantine system is studied from which is deduced an analogue of van der Corput's A5-process in order to bound analytic exponential sums of the form . The saving has now to be taken to the exponent 1/20 instead of 1/32. Our main application is a “ninth derivative test” for exponential sums which is essential for giving new exponent pairs in [3].

In this paper we derive a relation between character sums and partial sums of Dirichlet series.

Let ϑ be an integer of multiplicative order t≥1 modulo a prime p. Sums of the form

are introduced and estimated, with a sequence such that kz1, …, kzT is a permutation of z1, …, zT, both sequences taken modulo t, for sufficiently many distinct modulo t values of k. Such sequences include

xn for x = 1 ,…,t with an integer n≥1;

xn for x = 1 ,…,t and gcd (x, t) = 1 with an integer n≥1;

ex for x = 1 ,…,T with an integer e, where T is the period of the sequence ex modulo t.

Some of the results can be extended to composite moduli and to sums of multiplicative characters as well. Character sums with the above sequences have some cryptographic motivation and applications and have been considered in several papers by J. B. Friedlander, D. Lieman and I. E. Shparlinski. In particular several previous bounds are generalized and improved.

In the paper [5] bounds are found for the sum,

for a suitable Dirichlet character χ mod r, and real functionf(x). The proofs in that paper use Bombieri and Iwaniec's method [1], one formulation of which has as part of its first step the estimation of S in terms of a sum of many shorter sums of the form,

where e(x) = exp (2πix), mi∈ [M, 2M], and each mi, lies in its own interval, of length N ≥ M/4, that is disjoint from those of the others. This paper addresses a problem springing from above: to bound the numbers of ‘similar’ pairs, Si+, Si+, satisfying both

and

where ‖x‖ = min{|x − n|: n ∈ ℤ}. Lemma 5.2.1 of [3] (partial summation) shows that each sum in a similar pair is a bounded multiple of the other.

L'objet de cet article est d'étudier un procédé de summation associé á certaines séries. Notant P(n) le plus grand facteur premier d'un entier générique n, nous rappellons les définitions de P-convergence et de P-régularité d'une série, introduites dans [7].

The exponential sum S(x) = Σe(f(m + x)) has mean square size O(M), when m runs through M consecutive integers, f(x) satisfies bounds on the second and third derivatives, and x runs from 0 to 1.

In this short paper, we shall give a new estimate for the exponential sum S(H, M, N), where

e( ξ,) = exp (2πiξ;) for a real number ξ, am and bn are complex numbers with |am| ≤ 1 and |bn| ≤ l, H, M, N ≤1, , x is a large number, ε is a sufficiently small positive number, and Y ≤ x(½)−ε (h ∼ H means 1≤h/H < 2 and so on). In making application of the Rosser-Iwaniec linear sieve of Iwaniec [6] to find almost primes in short intervals of the type (x − y, x], Halberstam, Heath-Brown and Richert [4] first considered an estimate for S(H, M, N) to the effect that

with MN as large as possible. Later, better estimates were given in Iwaniec and Laborde [7], and Fouvry and Iwaniec [3]. Of course, the most interesting case would be finding P2 numbers in a short interval (x − y, x]. The related estimate of [7] implies that (1) holds provided that

Let r, k, h be positive integers. For any Dirichlet character χ modulo k write

In this article we establish an estimate for a sum over primes that is the analogue of an estimate for a sum over consecutive integers which has proved to be very useful in applications of exponential sums to problems in number theory.