We obtain asymptotic formulas for sums over arithmetic progressions of coefficients polynomials the form $$ \prod _{j=1}^n\prod _{k=1}^{p-1}(1-q^{pj-k})^s, where p is an odd prime and n, s are positive integers. Precisely, let $$a_i$$ denote coefficient $$q^i$$ in above polynomial suppose that b integer. prove \Big |\sum _{i\equiv b\ \mathrm{mod}\ 2pn}a_i-\frac{v(b)p^{sn}}{2pn}\Big |\le p^{sn/2...

The rings considered in this article are commutative with identity $1neq 0$. By a proper ideal of a ring $R$,  we mean an ideal $I$ of $R$ such that $Ineq R$.  We say that a proper ideal $I$ of a ring $R$ is a  maximal non-prime ideal if $I$ is not a prime ideal of $R$ but any proper ideal $A$ of $R$ with $ Isubseteq A$ and $Ineq A$ is a prime ideal. That is, among all the proper ideals of $R$,...

In this paper we extend the exponential sum results from [B-K] and [B-G-K] for prime moduli to composite moduli q involving a bounded number of prime factors. In particular, we obtain nontrivial bounds on the exponential sums associated to multiplicative subgroups H of size q, for any given δ > 0. The method consists in first establishing a ‘sum-product theorem’ for general subsets A of Z. If q...

A set X is prime bounding if for every complete atomic decidable (CAD) theory T there is a prime model A of T decidable in X. It is easy to see that X = 0′ is prime bounding. Denisov claimed that every X <T 0′ is not prime bounding, but we discovered this to be incorrect. Here we give the correct characterization that the prime bounding sets X ≤T 0′ are exactly the sets which are not low2. Reca...

In the following: x denotes any real number greater than 1 ; y, c, t denote real positive numbers; ¿ = log (x); d, n, m, v, A denote integers satisfying w^O, d>0, m>0, 0<v^k, (v,k) = l; Aim, v, k) denotes the set of integers miv + nk) for varying n; fim, v, A, x, c) denotes the number of integers in A im, v, k) less than or equal to x and prime to primes greater than x°; y¡/(x, k) denotes the n...

Let R be an integral domain. Note some obvious facts. Every ele­ ment a of R divides 0. Indeed 0 = 0 · a. On the other hand, 0 only divides 0. Indeed if a = q · 0, then a = 0 (obvious!). Finally every unit u divides any other element a. Indeed if v ∈ R is the inverse of u, so that uv = 1 then a = a · 1 = (av)u. Lemma 19.3. Let R be an integral domain and let p ∈ R. Then p is prime if and only i...

Let q be a prime with primitive root 2. We show that (a) if (pi) q−2 i=0 is a sequence of primes such that pi = 2pi−1 + 1 for all 1 ≤ i ≤ q − 2, then q divides p0 + 1 or p0 ∈ {2, 3, 5} and (b) if (pi) q−2 i=0 is a sequence of primes such that pi = 2pi−1 − 1 for all 1 ≤ i ≤ q − 2, then q divides p0 − 1 or p0 ∈ {2, 3}.

Two general constructions of linear codes with functions over finite fields have been extensively studied in the literature. The first one is given by C(f)={ Tr(af(x)+bx)x ∈ \mathbb Fqm*: a,b Fqm }, where q a prime power, Fqm* = \{0}, Tr trace function from to Fq, and f(x) f(0)=0. Almost bent functions, quadratic some monomials on F2m were used construction, many families binary few weights obt...