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The additive monoid R + (x) is defined as the set of all nonnegative integer linear combinations of binomial polynomials ( x n ) for n ∈ Z +. This paper is concerned with the inquiry into the structure of R + (α) for complex numbers α. Particularly interesting is the case of algebraic α which are not non-negative integers. This question is motivated by the study of functors between Deligne categories Rep (S t) (and also Rep (GL t)) for t ∈ C ﹨ Z +. We prove that this object is a ring if and only if α is an algebraic number that is not a nonnegative integer. Furthermore, we show that all algebraic integers generated by α , i.e. all elements of O Q (α) , are also contained in this ring. We also give two explicit representations of R + (α) for both algebraic integers and general algebraic numbers α. One is in terms of inequalities for the valuations with respect to certain prime ideals and the other is in terms of explicitly constructed generators. We show how these results work in the context of the study of symmetric monoidal functors between Deligne categories in positive characteristic. Moreover, this leads to a particularly simple description of R + (α) for both quadratic algebraic numbers and roots of unity. [ABSTRACT FROM AUTHOR]

In 1975, Don Zagier obtained a new version of the Kronecker limit formula for a real quadratic field which involved an interesting function F (x) which is now known as the Herglotz function. As demonstrated by Zagier, and very recently by Radchenko and Zagier, F (x) satisfies beautiful properties which are of interest in both algebraic number theory as well as in analytic number theory. In this paper, we study F k , N (x) , an extension of the Herglotz function which also subsumes higher Herglotz function of Vlasenko and Zagier. We call it the extended higher Herglotz function. It is intimately connected with a certain generalized Lambert series as well as with a generalized cotangent Dirichlet series inspired from Krätzel's work. We derive two different kinds of functional equations satisfied by F k , N (x). Radchenko and Zagier gave a beautiful relation between the integral ∫ 0 1 log ⁡ (1 + t x) 1 + t d t and F (x) and used it to evaluate this integral at various rational as well as irrational arguments. We obtain a relation between F k , N (x) and a generalization of the above integral involving polylogarithm. The asymptotic expansions of F k , N (x) and some generalized Lambert series are also obtained along with other supplementary results. [ABSTRACT FROM AUTHOR]

We call an abelian variety over a finite field F q super-isolated if its (F q -rational) isogeny class contains a single isomorphism class. In this paper, we use the Honda-Tate theorem to characterize super-isolated ordinary simple abelian varieties by certain algebraic integers. Our main result is that for a fixed dimension g ≥ 3 , there are finitely many such varieties over all finite fields. [ABSTRACT FROM AUTHOR]

Abstract In this paper we describe how to use the algorithmic methods provided by Hunter and Pohst in order to give a complete classification of number fields of degree 8 and signature (2 , 3) with absolute discriminant less than a certain bound. The choice of this bound comes from the local corrections given by prime ideals to the lower estimates for discriminants obtained with the Odlyzko–Poitou–Serre method. [ABSTRACT FROM AUTHOR]

Abstract: This paper gives the answer to a question of R.M. Wilson regarding the existence of group divisible designs of large order. Let k and u be positive integers such that . Then there exists an integer such that there exists a group divisible design of group type with block size k and index one for any integer satisfying the necessary arithmetic conditions [1.] , [2.] . This paper also presents a large-index asymptotic existence theorem for group divisible t-designs with a fixed number of groups, fixed group size and fixed block size. [Copyright &y& Elsevier]

Abstract: Text: For each positive integer m, let , , where and A is a subset of . Recently Chen proved that for each positive integer m, there exists a set such that and for any . In this paper, the following results are proved: (i) for each positive integer m, there exists a set such that and for all with at most 3 exceptions; (ii) for each positive integer m, there exists a set with and such that and for all with at most 3 exceptions. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=GfZ07Fg5qXE. [Copyright &y& Elsevier]

Abstract: This paper explores a simple yet powerful relationship between the problem of counting lattice points and the computation of Dedekind sums. We begin by constructing and proving a sharp upper estimate for the number of lattice points in tetrahedra with some irrational coordinates for the vertices. Besides providing a sharper estimate, this upper bound (Theorem 1.1) becomes an equality (i.e. gives the exact number of lattice points) in a tetrahedron where the lengths of the edges divide each other. This equality condition can then be applied to the explicit computation of the classical Dedekind sums, a topic that is the central focus in the second half of our paper. In this half of the paper, we come up with a number of interesting results related to Dedekind sums, based on our upper estimate (Theorem 1.1). Among these findings, Theorem 1.9 and Theorem 1.10 deserve special attention, for they successfully generalize two of Apostol''s formulas in [T.M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, New York, 1997], and also directly imply the famous Reciprocity Law of Dedekind sums. [Copyright &y& Elsevier]

Abstract: We give an alternative computation of the Galois group of the maximal 2-ramified and complexified pro-2-extension of any 2-rational number field (Theorem 2), a particular case of results of Movahhedi–Nguyen Quang Do. This short Note is motivated by the paper [J. Jossey, Galois 2-extensions unramified outside 2, J. Number Theory 124 (2007) 42–76] and, at this occasion, we bring into focus some classical technics of abelian ℓ-ramification which, unfortunately, are often ignored, especially those developed by J.-F. Jaulent with the ℓ-adic class field theory, and by G. Gras in his book on class field theory, and which considerably simplify the study of such subjects; for instance, our proof of Theorem 2 generalizes the purpose of Jossey''s paper in such a way using a result of Herfort–Zalesskii. This Note is mainly an attempt of clarification about ℓ-rationality. [Copyright &y& Elsevier]

Abstract: In the previous paper [O.J. Brison, J.E. Nogueira, Linear recurring sequence subgroups in finite fields, Finite Fields Appl. 9 (2003) 413–422] the authors investigated when, and how, the elements of a multiplicative subgroup of a finite field can be written, without repetition, as a cyclically-closed second order linear recurring sequence. Here, the earlier results are extended; in particular a general method is suggested for attempting to prove a given subgroup to be “standard,” in the terminology of the previous paper, while a lifting theorem for “nonstandard” subgroups is proved. [Copyright &y& Elsevier]

Abstract: This paper provides a method for identifying “sufficiently rich” simple groups of finite Morley rank with simple algebraic groups over algebraically closed fields. Special attention is given to the even type case, and the paper contains a number of structural results about simple groups of finite Morley rank and even type. [Copyright &y& Elsevier]

Abstract: Let be the set of all nonnegative integers, be a finite set, and 2A be the set of all numbers of the form for all a and b in A. In [G.A. Freiman, Foundations of a Structural Theory of Set Addition, Transl. Math. Monogr., vol. 37, American Mathematical Society, Providence, RI, 1973 (translated from the Russian)] the arithmetic structure of A was optimally characterized when . 2 In [G.A. Freiman, Foundations of a Structural Theory of Set Addition, Transl. Math. Monogr., vol. 37, American Mathematical Society, Providence, RI, 1973 (translated from the Russian)] the structure of A was also characterized without proof when . Since then the efforts have been made to generalize these results, see [V. Lev, P.Y. Smeliansky, On addition of two distinct sets of integers, Acta Arith. 70 (1) (1995) 85–91; V. Lev, On the structure of sets of integers with small doubling property, unpublished manuscripts, 1995; Y.O. Hamidoune, A. Plagne, A generalization of Freiman''s theorem, Acta Arith. 103 (2) (2002) 147–156] for example. However, no optimal characterization of the structure of A has been obtained without imposing extra conditions, until now, when . In this paper we optimally characterize, with the help of nonstandard analysis, the arithmetic structure of A when , where b is positive but not too large. Precisely, we prove that there is a real number and there is such that if and for , then A is either a subset of an arithmetic progression of length at most or a subset of a bi-arithmetic progression 3 of length at most . An application of this result to the inverse problem for upper asymptotic density is presented near the end of the paper. In the application we improve the most important part of the main theorem in [R. Jin, Solution to the inverse problem for upper asymptotic density, J. Reine Angew. Math. (Crelle''s J.) 595 (2006) 121–166]. [2] One can easily prove that is always true and implies that A is an arithmetic progression. [3] The definition of bi-arithmetic progression can be found in the beginning of Introduction. [Copyright &y& Elsevier]

Abstract: In this paper, we study the zeta function, named non-abelian zeta function, defined by Lin Weng. We can represent Weng''s rank r zeta function of an algebraic number field F as the integration of the Eisenstein series over the moduli space of the semi-stable -lattices with rank r. For , in the case of , Weng proved that it can be written by the Riemann zeta function, and Lagarias and Suzuki proved that it satisfies the Riemann hypothesis. These results were generalized by the author to imaginary quadratic fields and by Lin Weng to general number fields. This paper presents proofs of both these results. It derives a formula (first found by Weng) for Weng''s rank 2 zeta functions for general number fields, and then proves the Riemann hypothesis holds for such zeta functions. [Copyright &y& Elsevier]

Abstract: Let be a prime, and . In this paper we obtain a general criterion for to be a quartic residue in terms of appropriate binary quadratic forms. Let be a squarefree integer such that , where is the Legendre symbol, and let be the fundamental unit of the quadratic field . Since 1942 many mathematicians tried to characterize those primes so that is a quadratic or quartic residue . In this paper we will completely solve these open problems by determining the value of , where is an odd prime, and . As an application we also obtain a general criterion for , where is the Lucas sequence defined by and . [Copyright &y& Elsevier]

Abstract: This paper is motivated by Ito''s paper ((Hokkaido) Math. J. 22 (1993) 373), in which he constructed -relative difference sets in . Chen and Li have shown that , the alternating group of degree 5, has a (30,2,29,14)-relative difference set (see Discrete Math. 256 (2002) 301). These are the only examples of nontrivial relative difference sets in nonsolvable groups known to the author. In this paper, we study relative difference sets in and show that there exist exactly two nontrivial relative difference sets in up to equivalence with the aid of a computer. [Copyright &y& Elsevier]

The notion of a ∗-ordering on a skew-field with involution was introduced by S. Holland in [J. Algebra 101 (1986) 16–46] as an analogue to the notion of a total ordering on a skew-field and developed further in a series of papers of T. Craven, I. Idris, M. Marshall and T. Smith. While it is well known that every free skew-field has a total ordering (see [J. Lewin, Trans. Amer. Math. Soc. 192 (1974) 339–346]), it has not been known so far whether every free skew-field with some natural involution has a ∗-ordering. The aim of this paper is to give an explicit construction of a class of ∗-orderings on free associative algebras and to prove that ∗-orderings from this class extend uniquely to the corresponding free skew fields. The problem was posed by T. Craven and T. Smith in [J. Algebra 238 (2001) 314–327]. [Copyright &y& Elsevier]

Define D n ( x ) = ∑ k = 0 n ( n k ) 2 x k ( x + 1 ) n − k for n = 0 , 1 , 2 , … and s n ( x ) = ∑ k = 1 n 1 n ( n k ) ( n k − 1 ) x k − 1 ( x + 1 ) n − k for n = 1 , 2 , 3 , … . Then D n ( 1 ) is the n -th central Delannoy number D n , and s n ( 1 ) is the n -th little Schröder number s n . In this paper we obtain some surprising arithmetic properties of D n ( x ) and s n ( x ) . We show that 1 n ∑ k = 0 n − 1 D k ( x ) s k + 1 ( x ) ∈ Z [ x ( x + 1 ) ] for all n = 1 , 2 , 3 , … . Moreover, for any odd prime p and p -adic integer x ≢ 0 , − 1 ( mod p ) , we establish the supercongruence ∑ k = 0 p − 1 D k ( x ) s k + 1 ( x ) ≡ 0 ( mod p 2 ) . As an application we confirm Conjecture 5.5 in [S14a] , in particular we prove that 1 n ∑ k = 0 n − 1 T k M k ( − 3 ) n − 1 − k ∈ Z for all n = 1 , 2 , 3 , … , where T k is the k -th central trinomial coefficient and M k is the k -th Motzkin number. [ABSTRACT FROM AUTHOR]

There is a large literature on the asymptotic distribution of numbers free of large prime factors, so-called smooth or friable numbers. But there is very little known about this distribution that is numerically explicit. In this paper we follow the general plan for the saddle point argument of Hildebrand and Tenenbaum, giving explicit and fairly tight intervals in which the true count lies. We give two numerical examples of our method, and with the larger one, our interval is so tight we can exclude the famous Dickman–de Bruijn asymptotic estimate as too small and the Hildebrand–Tenenbaum main term as too large. [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve defined over a number field K . Then its Mordell–Weil group E ( K ) is finitely generated: E ( K ) ≅ E ( K ) t o r × Z r . In this paper, we discuss the cyclic torsion subgroup of elliptic curves over cubic number fields. For N = 169 , 143 , 91 , 65 , 77 or 55, we show that Z / N Z is not a subgroup of E ( K ) t o r for any elliptic curve E over a cubic number field K . [ABSTRACT FROM AUTHOR]

Let F be a totally real algebraic number field, with O F the ring of algebraic integers in F , let L be an O F -lattice on a d -dimensional ( d ≥ 2 ) positive definite quadratic space V over F and let u 0 be a primitive vector in L . The main objective of this paper is to study when a is a fixed (non-unit) algebraic integer in O F and n is a positive rational integer, how the class numbers of lattices translation L + u 0 a n grow as n tends to infinity. [ABSTRACT FROM AUTHOR]

Let a , b , c be fixed coprime positive integers with min ⁡ { a , b , c } > 1 . In this paper, combining the Gel'fond–Baker method with an elementary approach, we prove that if max ⁡ { a , b , c } > 5 × 10 27 , then the equation a x + b y = c z has at most three positive integer solutions ( x , y , z ) . [ABSTRACT FROM AUTHOR]

Let χ range over the non-trivial primitive characters associated with the abelian extensions L / K of a given number field K , i.e. over the non-trivial primitive characters on ray class groups of K . Let f χ be the norm of the finite part of the conductor of such a character. It is known that | L ( 1 , χ ) | ≤ 1 2 Res s = 1 ( ζ K ( s ) ) log ⁡ f χ + O ( 1 ) , where the implied constants in this O ( 1 ) are effective and depend on K only. The proof of this result suggests that one can expect better upper bounds by taking into account prime ideals of K dividing the conductor of χ , i.e. ramified prime ideals. This has already been done only in the case that K = Q . This paper is devoted to giving for the first time such improvements for any K . As a non-trivial example, we give fully explicit bounds when K is an imaginary quadratic number field. [ABSTRACT FROM AUTHOR]

In this paper we first of all determine all possible genera of (odd and even) definite unimodular lattices over an imaginary-quadratic field. The main questions are whether the partial class numbers of lattices with given Steinitz class within one genus are equal for all occurring Steinitz classes and whether the partial masses of those partial genera are equal. We show that the answer to the first question in general is “no” by giving a counter example, while the answer to the second question is “yes” by proving a mass formula for partial masses. Finally, we determine a list of all single-class partial genera and show that a partial genus consists of only one class if and only if the whole genus consists of only one class. [ABSTRACT FROM AUTHOR]

In this paper, we study multizeta values over function fields in characteristic p . For each d ≥ 2 , we show that when the constant field has cardinality >2, the field generated by all multizeta values of depth d is of infinite transcendence degree over the field generated by all single zeta values. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the 3-rank of the ideal class group of the imaginary quadratic field Q ( 4 − 3 18 n + 3 ) is at least 3 for every positive integer n . [ABSTRACT FROM AUTHOR]

Possible transcendental nature of Euler's constant γ has been the focus of study for sometime now. One possible approach is to consider γ not in isolation, but as an element of the infinite family of generalized Euler–Briggs constants. In a recent work [6] , it is shown that the infinite list of generalized Euler–Briggs constants can have at most one algebraic number. In this paper, we study the dimension of spaces generated by these generalized Euler–Briggs constants over number fields. More precisely, we obtain non-trivial lower bounds (see Theorem 5 and Theorem 6 ) on the dimension of these spaces and consequently establish the infinite dimensionality of the space spanned. Further, we study linear and algebraic independence of these constants over the field of all algebraic numbers. [ABSTRACT FROM AUTHOR]

Text In 2014, Herzog, Kaplan and Lev introduced the critical numbers cr ( r , n ) of natural intervals [ r , n ] and determined the values of cr ( 1 , n ) and cr ( 2 , n ) for all n , and for r ≥ 3 , they determined the values of cr ( r , n ) for n ≤ ( 3 r 2 + 3 r ) / 2 and n ≥ ( 5 r 2 + r ) / 2 . In this note, we determine the remaining values of cr ( r , n ) for r ≥ 3 . Video For a video summary of this paper, please visit https://youtu.be/hZSI-Ul3K94 . [ABSTRACT FROM AUTHOR]

Let A K denote the ring of algebraic integers of an algebraic number field K = Q ( θ ) where the algebraic integer θ has minimal polynomial F ( x ) = x n + a x m + b over the field Q of rational numbers with n = m t + u , t ∈ N , 0 ≤ u ≤ m − 1 . In this paper, we characterize those primes which divide the discriminant of F ( x ) but do not divide [ A K : Z [ θ ] ] when u = 0 or u divides m ; such primes p are important for explicitly determining the decomposition of p A K into a product of prime ideals of A K in view of the well known Dedekind theorem. As a consequence, we obtain some necessary and sufficient conditions involving only a , b , m , n for A K to be equal to Z [ θ ] . [ABSTRACT FROM AUTHOR]

In this paper we deal with Diophantine equations involving products of consecutive integers, inspired by a question of Erdős and Graham. [ABSTRACT FROM AUTHOR]

The goal of this paper is to generalize and refine the classical ramification theory of complete discrete valuation rings to more general valuation rings, in the case of Artin–Schreier extensions. We define refined versions of invariants of ramification in the classical ramification theory and compare them. Furthermore, we can treat the defect case. [ABSTRACT FROM AUTHOR]

In this paper, we sharpen and simplify our earlier results based on Thue's Fundamentaltheorem and use it to obtain effective irrationality measures for certain roots of particular polynomials of the form ( x − t ) n + ( x + t ) n , where n ≥ 4 is a positive integer and t is a negative integer. For n = 4 and n = 5 , we find infinitely many such algebraic numbers. [ABSTRACT FROM AUTHOR]

The canonical structure of DNA has four bases – Thymine (T), Adenine (A), Cytosine (C), and Guanine (G) – and DNA codes are regarded as words over the alphabet set Σ = { A , C , G , T } , satisfying certain combinatorial conditions. Good DNA codes are desirable for DNA computation, DNA microarray technologies and molecular barcodes, etc. One of the main tasks in DNA code designing is to build more codewords and better GC -content for given fixed word length n . Existing heuristic methods work well for small n . In this paper, we present a systematic method for constructing good DNA codes for large n by using irreducible cyclic codes. Being different from traditional DNA constructions, our method is based on algebraic number theory rather than classical heuristic algorithms and the conventional coding theory. Furthermore, comparing with the traditional DNA codes, our codes have larger number of codewords and better GC -content. As far as we know, it is the very first time to utilize irreducible cyclic codes for constructing a type of DNA codes. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate rotated versions of the densest known lattices in dimensions 2, 3, 4, 5, 6, 7, 8, 12 and 24 constructed via ideals and free Z -modules that are not ideals in subfields of cyclotomic fields. The focus is on totally real number fields and the associated full diversity lattices which may be suitable for signal transmission over both Gaussian and Rayleigh fading channels. We also discuss on the existence of a number field K such that it is possible to obtain the lattices A 2 , E 6 and E 7 via a twisted embedding applied to a fractional ideal of O K . [ABSTRACT FROM AUTHOR]

Let K be an algebraic number field and B ≥ 1 . For an endomorphism φ : P n → P n defined over K of degree d let R φ ⊂ O K denote its minimal resultant ideal. For a fixed height function h M d n on the moduli space of dynamical systems this paper shows that all such morphisms φ of bounded resultant N K / Q ( R φ ) ≤ B and bounded height h M d n ( 〈 φ 〉 ) ≤ B are contained in finitely many PGL n + 1 ( K ) -equivalence classes. This answers a question of Silverman in the affirmative. [ABSTRACT FROM AUTHOR]

Abstract: Let K be a complete discrete valuation field of mixed characteristic with residue field such that . Let be the absolute Galois group of K and a p-adic representation. When is perfect, Shankar Sen described the Lie algebra of in terms of so-called Sen's operator Θ for ρ. When may not be perfect, Olivier Brinon defined operators for ρ, which reduce to Sen's operator Θ in the case of . In this paper, we describe the Lie algebra of in terms of Brinon's operators , which is a generalization of Sen's result. [Copyright &y& Elsevier]

Abstract: Text: Let K be an algebraic number field and the ring of integers in K. Let be the set of all elements which are sums of squares in and the minimal number of squares necessary to represent −1 in . Let be the smallest positive integer t such that every element in is a sum of t squares in . In this note, for , where are two distinct positive square-free integers, we show that and if , then . Video: For a video summary of this paper, please click here or visit http://youtu.be/F6UHHgzfdzA. [Copyright &y& Elsevier]

In this paper, LCD cyclic, self dual and isodual codes over finite chain rings are investigated. It was proven recently that a non-free LCD cyclic code does not exist over finite chain rings. Based on algebraic number theory, we introduce necessary and sufficient conditions for which all free cyclic codes over a finite chain ring are LCD. We have also obtained conditions on the existence of non trivial self dual cyclic codes of any length when the nilpotency index of the maximal ideal of a finite chain ring is even. Further, several constructions of isodual codes are given based on the factorization of the polynomial x n − 1 over a finite chain ring. [ABSTRACT FROM AUTHOR]

Abstract: In the paper, the author establishes several integral representations and properties of Stirling numbers of the first kind. [Copyright &y& Elsevier]

Abstract: Text: We prove two congruence results concerning Kloosterman sums over finite fields. The first result concerns the coefficients of the characteristic polynomial over of a Kloosterman sum, and the second result gives a characterisation of ternary Kloosterman sums modulo 27. We use methods from algebraic number theory such as Stickelbergerʼs theorem and the Gross–Koblitz formula, as well as Fourier analysis. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=VJcB6W_PQ0s. [Copyright &y& Elsevier]

Abstract: Let K be a perfect field of characteristic not equal to two, an algebraic closure of K and let be the Galois group of the extension . Let T be an n-point set in . The field of moduli of T is contained in each field of definition but it is not necessarily a field of definition. In this paper we show that point sets of odd cardinality in with field of moduli K are defined over their field of moduli. We, also, show that, except for the special case of the 4-point sets, this does not hold in general for point sets of even cardinality . Finally we prove that the following local-to-global principle holds for point sets of cardinality : if an n-point set T in is defined over for each prime p (including the prime at ∞), then it is defined over . From this result we derive an analogue local-to-global principle for hyperelliptic curves. [Copyright &y& Elsevier]

Abstract: Text: For any integer , we show that there are infinitely many complex quadratic fields whose 2-class groups are cyclic of order . The proof combines the circle method with an algebraic criterion for a complex quadratic ideal class to be a square. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=H49W8Wyl47o. [Copyright &y& Elsevier]

Abstract: Text: The isomorphism number of an F-crystal over an algebraically closed field of positive characteristic is the smallest non-negative integer such that the -th level truncation of determines the isomorphism class of . When is isoclinic, namely it has a unique Newton slope λ, we provide an efficiently computable upper bound for in terms of λ and the Hodge slopes of . This is achieved by providing an upper bound for the level torsion of introduced by Vasiu. We also check that this upper bound is optimal for many families of isoclinic F-crystals that are of special interest (such as isoclinic F-crystals of K3 type). Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=gVObAZZ1DKE. [Copyright &y& Elsevier]

Abstract: In this paper we give two proofs of the mass formula for definite central division algebras over global function fields, due to Denert and Van Geel. The first proof is based on a calculation of Tamagawa measures. The second proof is based on analytic methods, in which we establish the relationship directly between the mass and the value of the associated zeta function at zero. [Copyright &y& Elsevier]

Abstract: Let p be an odd prime number. For the cyclotomic -extension of a finite algebraic number field F, we denote by the maximal unramified pro-p-extension of . In this paper, using Iwasawa theory and the theory of central class fields, we give necessary conditions for to be abelian, and give sufficient conditions in certain special cases. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce and study several norms which are constructed in order to satisfy an extremal property with respect to the Mahler measure. These norms are a natural generalization of the metric Mahler measure introduced by Dubickas and Smyth. We show that bounding these norms on a certain subspace implies Lehmerʼs conjecture and in at least one case that the converse is true as well. We evaluate these norms on a class of algebraic numbers that include Pisot and Salem numbers, and for surds. We prove that the infimum in the construction is achieved in a certain finite dimensional space for all algebraic numbers in one case, and for surds in general, a finiteness result analogous to that of Samuels and Jankauskas for the t-metric Mahler measures. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to introduce a new class of Noetherian rings of prime characteristic via perfect closure and study their basic properties. If the perfect closure of a Noetherian ring is coherent, we call it an F-coherent ring. Some applications are given to tight closure theory. In particular, we discuss some relationship between F-coherent rings and F-pure, F-regular, and F-injective rings. As a main tool, we use techniques from valuation theory. The final section discusses how the coherent property effects the behavior of tight closure of finitely generated ideals on general perfect rings. [Copyright &y& Elsevier]

Abstract: In this paper we show that, for a K3 surface within a certain class of surfaces and over a number field, the orbit of a point under the group of automorphisms is either finite or its exponent of growth is exactly the Hausdorff dimension of a fractal associated to the ample cone. In particular, the exponent depends on the geometry of the surface and not its arithmetic. For surfaces in this class, the exponent is . [ABSTRACT FROM AUTHOR]

Abstract: Let be a number field and an absolutely irreducible polynomial of such that the algebraic curve defined by the equation is rational. In this paper we present practical algorithms for the determination of all solutions of the Diophantine equation in algebraic integers of . [ABSTRACT FROM AUTHOR]

Abstract: In this paper we give irrationality results for numbers of the form where the numbers behave like a geometric progression for a while. The method is elementary, not using differentiation or integration. In particular, we derive elementary proofs of the irrationality of π and for Gaussian integers . [Copyright &y& Elsevier]

Abstract: In this paper, we determine the closure in the full topology over of the set , where is a nondegenerate binary recurrent sequence with integer coefficients whose characteristic roots are quadratic units. This generalizes the result for the case when was the nth Fibonacci number. [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]