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Let denote the set of all nonnegative integers and be a subset of . The set is called an asymptotic basis of order if every sufficiently large integer can be written as the sum of two elements of . Otherwise, is called an asymptotic nonbasis of order . Let denote the number of representations of in the form , where and . An asymptotic nonbasis of order is called a maximal asymptotic nonbasis of order if is an asymptotic basis of order for every . In this paper, a maximal asymptotic nonbasis is constructed satisfying for all and as , where is an increasing sequence of . [ABSTRACT FROM AUTHOR]

For and , , let denote the classical Dedekind sum. We show that Dedekind sums take this value infinitely many times in the following sense. There are pairs , , with tending to infinity as grows, such that for all . [ABSTRACT FROM AUTHOR]

In this paper, we prove a multiple Abel summation formula and give some applications of this formula. We use these evaluations in tandem with some recent results to obtain various asymptotic evaluations for multiple sums. [ABSTRACT FROM AUTHOR]

We introduce a new factorization algorithm based on the optical computation by multi-path interference of the periodicity of a "factoring" function given by exponential sums at continuous arguments. We demonstrate that this algorithm allows, in principle, the prime number decomposition of several large numbers by exploiting a remarking rescaling property of this periodic function. Such a function is recorded by measuring optical interferograms with a multi-path Michelson interferometer, a polychromatic light source and a spectrometer. The information about factors is encoded in the location of the inteferogram maxima. [ABSTRACT FROM AUTHOR]

We specialize a problem studied by Elliott, the behavior of arbitrary sequences ap of complex numbers on residue classes to prime moduli to the case ap = e(αp). For these special cases, we obtain under certain additional conditions improvements on Elliott's results. [ABSTRACT FROM AUTHOR]

In this paper, we study Kloosterman sums twisted by multiplicative characters modulo a prime power. We show, by an elementary calculation, that these sums become equidistributed on the real line with respect to a suitable measure. [ABSTRACT FROM AUTHOR]

Let q > 2 be an odd number. For each integer a with 1 ≤ a ≤ q and (a, q) = 1, there exists one and only one $\overline{a}$ with $1\leq \overline{a} \leq q$ and $(\overline{a},q)=1$ such that $a\overline{a}\equiv 1\ (\bmod\ q)$. A Lehmer number is defined to be any integer a with $2\nmid a+\overline{a}$. Let a = (a1, ..., ak+1), b = (b1, ..., bk+1) with 0 ≤ bi < ai for i = 1, ..., k + 1, (a1 ⋯ ak+1, q) = 1, t = (t1, ..., tk) satisfying 0 < t1, ..., tk ≤ 1. Define \[ N(\mathbf{a},\mathbf{b},\mathbf{t};q)=\mathop{\mathop{\mathop{\sum_{n_1\leq t_1q}}_{(n_1, q)=1}}_{n_1\equiv b_1 ({\rm mod}\ a_1)}\cdots \mathop{\mathop{\sum_{n_{k}\leq t_kq}}_{(n_k, q)=1}}_{n_k\equiv b_k ({\rm mod}\ a_k)}}_{\overline{n_1\cdots n_k} \equiv b_{k+1} ({\rm mod}\ a_{k+1})}1. \] The main purpose of this paper is to study the properties of N(a, b, t; q), and give a sharp asymptotic formula, by using the estimate for hyper-Kloosterman sums and properties of trigonometric sums. [ABSTRACT FROM AUTHOR]

We find new conditions on a polynomial over a finite field that guarantee that the exponential sum defined by the polynomial has only one nonvanishing p-adic cohomology group, hence the L-function associated to the exponential sum is a polynomial or the reciprocal of a polynomial. [ABSTRACT FROM AUTHOR]

Let n be a positive integer and φ(n) denotes the Euler phi function. It is well known that the power sum of n can be evaluated in closed form in terms of n. Also, the sum of all those φ(n) positive integers that are coprime to n and not exceeding n, is expressible in terms of n and φ(n). Although such results already exist in literature, but here we have presented some new analytical results in these connections. Some functional and integral relations are derived for the general power sums. [ABSTRACT FROM AUTHOR]

We give bounds on the number of integers 1 ≤ n ≤ N such that p | s(n), where p is a prime and s(n) is the sum of aliquot divisors function given by s(n) = σ(n) - n, where σ(n) is the sum of divisors function. Using this result, we obtain nontrivial bounds in certain ranges for rational exponential sums of the form \[ S_p(a,N) = \sum \limits_{n\le N}\exp(2\pi ias(n)/p), \quad \gcd(a,p) = 1. \] [ABSTRACT FROM AUTHOR]

We give a proof for the approximate functional equation for exponential sums related to holomorphic cusp forms and derive an upper bound for the error term. [ABSTRACT FROM AUTHOR]

The nonlinear congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. In this paper we present a new type of discrepancy bound for sequences of s-tuples of successive nonlinear congruential pseudorandom numbers over a ring of integers ℤM. [ABSTRACT FROM AUTHOR]

Niederreiter and Shparlinski obtained a nontrivial discrepancy bound for sequences of inversive congruential pseudorandom numbers with odd prime-power modulus. Because of technical difficulties they had to leave open the case of greatest practical interest, namely where the modulus is a power of 2. In the present paper we successfully treat this case by using recent advances in the theory of exponential sums. [ABSTRACT FROM AUTHOR]

In this paper we study the general exponential sum related to multiplicative functions f(n) with |f(n)| ≤ 1, namely we study the sum \[ F(x,\alpha) = \sum_{n \le x} f(n){\bf e}(n\alpha) \] and obtain a non-trivial upper bound when α is a certain type of rational number. [ABSTRACT FROM AUTHOR]