Abstract: For any language L, let pow(L) = {uj | j ? 0, u ? L} be the set of powers of elements of L. Given a rational language L (over a finite alphabet), we study the question, posed in [3], whether pow(L) is rational or not. While leaving open the problem in general, we provide an algorithmic solution for the case of one-letter alphabets. This case is still non trivial; our solution is based on Dirichlet's result that for two relatively prime numbers, their associated arithmetic progression contains infinitely many primes.

Abstract: Assuming the Generalized Riemann Hypothesis, we obtain a lower bound within a constant factor of the conjectured asymptotic result for the second moment for primes in an individual arithmetic progression in short intervals. Previous re- sults were averaged over all progression of a given modulus. The method uses a short divisor sum approximation for the von Mangoldt function, together with some new results for binary correlations of this divisor sum approximation in arithmetic progressions.

Abstract: It is shown that the set {1, 2,? , 2 n+ 3} ? {p } can be partitioned into differences 1, 3,? , 2 n+ 1 precisely whenn? 1, p is odd and (n, p) ?= (1, 3). All sets whose elements are in arithmetic progression and which can be partitioned into differences that are again in arithmetic progression are classified.

Abstract: In this paper, we give an explicit numerical upper bound for the moduli of arithmetic progressions, in which the ternary Goldbach problem is solvable. Our result implies a quantitative upper bound for the Linnik constant.

Abstract: Dirichlet's theorem asserts that every arithmetic progression m, m + n, m + 2n, . . .. with m and n relatively prime, contains infinitely many primes. The simplest proofs are analytic, using properties of Dirichlet L-series [1], although Atle Selberg gave a complicated elementary proof in 1949 [5]. Certain individual cases, such as 3, 3 + 4, 3 + 8, ... and 5, 5 + 6, 5 + 12, ..., are easy to prove. Other special cases, notably 1, 1 + 4, 1 + 8, . . ., can be proved using simple properties of quadratic residues II]. In this note, we use elementary arguments to cover an infinite number of cases. While these have been given other elementary proofs (see, for instance, Dickson [2, vol. 1, p. 418] or Ribenboim [4, p. 268]), the proof presented here is the simplest and shortest that the author knows. In his recent paper in this MAGAZINE [3], Lionel Levine proved the following interesting theorem:

Abstract: We study M(n,k,r), the number of orbits of {(a_1,...,a_k)\in Z_n^k | a_1+...+a_k = r (mod n)} under the action of S_k. Equivalently, M(n,k,r) sums the partition numbers of an arithmetic sequence: M(n,k,r) = sum_{t \geq 0} p(n-1,k,r+nt), where p(a,b,t) denotes the number of partitions of t into at most b parts, each of which is at most a. We derive closed formulas and various identities for such arithmetic partition sums. These results have already appeared in Elashvili/Jibladze/Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Alg. Combin. 10 (1999) 173-188, and the main result was also published by Von Sterneck in Sitzber. Akad. Wiss. Wien. Math. Naturw. Class. 111 (1902), 1567-1601 (see Lemma 2 and references in math.NT/9909121). Thanks to Don Zagier and Robin Chapman for bringing these references to our attention.

Abstract: Here we describe some recent advances that have been made regarding the arithmetic of the unrestricted partition function p(n). A partition of a non- negative integer n is any nonincreasing sequence of positive integers whose sum is n. As usual, we let p(n) denote the number of partitions of n. For example, it is easy to see that p(4) = 5 since the partitions of 4 are: $$ 4, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1. $$

Abstract: In 1837 Dirichlet proved by an ingenious analytic method that there are infinitely many primes in the arithmetic progression $$ a,\quad a + q,a + 2q,a + 3q.... $$ in which a and q have no common factor and q is prime. The general case, for arbitrary q, was completed only later by him, in 1840, when he had finished proving his celebrated class number formula. In fact, many are of the view that the subject of analytic number theory begins with these two papers. It is also accurate to say that character theory of finite groups also begins here.

Abstract: In this paper, we give an estimation of general linear exponential sums over primes in an arithmetic progression by means of Vaughan's identity.

Abstract: Elementary proofs of unique factorization in rings of arithmetic functions using a simple variant of Euclid's proof for the fundamental theorem of arithmetic.

Abstract: Let be a finite arithmetic sequence in C with elements the values of a Dirichlet character Ξ mod n. If Xis the circulant n × n matrix with elements Ξ(i) then the eigenvalues of X are the Gauss sums that correspond to Ξ. Moreover, if Ξ=Ξ1 is the principal Dirichlet character mod n, then the eigenvalues of X are the Ramanujan sums C n(K).

Abstract: We construct an asymptotic formula for a sum function for σ a (α), where σ a (α) is the sum of the ath powers of the norms of divisors of the Gaussian integer α on an arithmetic progression α ≡ α0 (mod γ) and in a narrow sector ϕ1 ≤ arg α < ϕ2. For this purpose, we use a representation of σ a (n) in the form of a series in the Ramanujan sums.

Abstract: This paper contains an elementary derivation of formulas for multiplicative functions of m which exactly yield the following numbers: the number of distinct arithmetic progressions of w reduced residues modulo m; the number of the same with first term n; the number of the same with mean n; the number of the same with common difference n. With m and odd w fixed, the values of the first two of the last three functions are fixed and equal for all n relatively prime to m; other similar relations exist among these three functions.

Abstract: has rational solutions if and only if n = r(F2m+i ? 1) and the solutions are F2m/(F2m+i ? 1) and ? F2m+2/(F2m+i ? 1) independent of r where Fk denotes the kth Fibonacci number. This and similar results can be found in [2] and eventually lead to a remarkable result concerning Pell's equation in [3]. In this paper, we consider a somewhat related question that has an equally surprising answer and that provides an interesting and informative study that students can readily pursue. In particular, what can be said about the solutions to linear systems of equations whose coefficients are in arithmetic progression?systems like

Abstract: In this paper we show that if A is a subset of the primes with positive lower relative density then A +A must have positive lower density at least C1= log log(1= ) in the natural numbers. Our argument uses the techniques developed by the author and I.Ruzsa in their work on additive properties of dense subsequences of suciently sifted sequences. The result is optimal and improves on recent work of K.Chipeniuk & M.Hamel. We continue by proving several similar results, by successively replacing the sequence of primes by the sequence of sums of two squares, by the sequence of those integers n that are such that n and n + 1 are both a sum of two squares and nally by the sequence of primes p that are such that p + 1 is a sum of two squares. The second part of this paper contains a heuristical argument that leads to several conjectures concerning the existence of k-term arithmetic progressions within these sequences. We conclude with some conjectures belonging to the Ramsey part of additive number theory.

Abstract: We extend a result of Ramare and Rumely, 1996, about the Chebyshev function θ in arithmetic progressions. We find a map e(x) such that | θ(x; k, l) - x/φ(k) | 0), whereas e(x) is a constant. Now we are able to show that, for x ≥ 1531, |θ(x; 3, l) - x/2 | x/2 ln x.