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Let p be a prime number, q = p r with r ≥ 2 and P ∈ F q [ X ] . In this paper, we first estimate the number of x ∈ F q such that P ( x ) has prescribed digits (in the sense of Dartyge and Sárközy). In particular, for a given proportion <0.5 of prescribed digits, we show that this number is asymptotically as expected. Then, we obtain similar results when x is allowed to run only in the set of generators (primitive elements) of F q ⁎ . In the case of special interest where P is a monomial of degree 2, our estimate for the number of x ∈ F q such that P ( x ) has prescribed digits is sharper than the estimate following from the Weil bound. We will need to study exponential sums of independent interest such as multiplicative character sums over affine subspaces and additive character sums with generator arguments. [ABSTRACT FROM AUTHOR]

Abstract: Let m be a positive integer and be a polynomial of the form . We call a polynomial a Rabinowitsch polynomial if for and consecutive integers , is either 1 or prime. In this paper, we show that there are exactly 14 Rabinowitsch polynomials . [Copyright &y& Elsevier]

Abstract: Let denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper it is proved that any sufficiently large integer N satisfying the congruence condition can be represented as the sum of twelve fourth powers of primes and the fourth power of one . This result constitutes an improvement upon that of Ren and Tsang. [Copyright &y& Elsevier]

Abstract: In this paper we study generalised prime systems for which the integer counting function is asymptotically well behaved, in the sense that , where ρ is a positive constant and . For such systems, the associated zeta function is holomorphic for . We prove that for , for any , and also for for all such σ except possibly one value. The Dirichlet divisor problem for generalised integers concerns the size of the error term in , which is for some . Letting denote the infimum of such θ, we show that . [Copyright &y& Elsevier]

Abstract: Let p be an odd prime and be the smallest positive integer s such that every integer is a sum of s kth powers . We establish and provided that k is not divisible by . Next, let , and q be any positive integer. We show that if then for some constant . These results generalize results known for the case of prime moduli. Video abstract: For a video summary of this paper, please visit http://www.youtube.com/watch?v=zpHYhwL1kD0. [Copyright &y& Elsevier]

Abstract: We call a quadratic extension of a cyclotomic field a quasi-cyclotomic field if it is non-abelian Galois over the rational number field. In this paper, we study the arithmetic of any quasi-cyclotomic field, including to determine the ring of integers of it, the decomposition nature of prime numbers in it, and the structure of the Galois group of it over the rational number field. We also describe explicitly all real quasi-cyclotomic fields, namely, the maximal real subfields of quasi-cyclotomic fields which are Galois over the rational number field. It gives a series of totally real fields and CM fields which are non-abelian Galois over the rational number field. [Copyright &y& Elsevier]