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This note presents corrections and additions to my paper (J. Number Theory 41 (1992) 322–358). [Copyright &y& Elsevier]

Text By Zeckendorf's theorem each positive integer is uniquely written as a sum of distinct non-adjacent terms of the Fibonacci sequence. This representability remains true for so called the Nth order Fibonacci sequence , and for a further generalization to linear recurrences with positive coefficients. In this paper we consider sequences { G n } that have the same linear recurrence relations as the N th order Fibonacci sequence but has different initial values, and investigate the number of positive integers up to X that are written as a sum of distinct terms of G n . We also introduce a converse of Zeckendorf's theorem that does not require the increasing condition. Our method extends to general linear recurrences, and a generalization is introduced in this paper. Video For a video summary of this paper, please visit https://youtu.be/vSwSJ_sppns . [ABSTRACT FROM AUTHOR]

Text Let m be an integer with m ≥ 2 . For A ⊆ Z m and n ∈ Z m , let R 1 ( A , n ) , R 2 ( A , n ) , R 3 ( A , n ) denote the number of solutions of the equation a + a ′ = n with ordered pairs ( a , a ′ ) ∈ A × A , unordered pairs ( a , a ′ ) ∈ A × A ( a ≠ a ′ ) and unordered pairs ( a , a ′ ) ∈ A × A , respectively. In this paper, for i ∈ { 1 , 2 , 3 } , we determine all sets A , B ⊆ Z m such that R i ( A , n ) = R i ( B , n ) for all n ∈ Z m when the cardinality of the symmetric difference of A and B is small or large. These extend some previous results. We also pose some problems for further research. Video For a video summary of this paper, please visit https://youtu.be/stBa9Uy5U0I . [ABSTRACT FROM AUTHOR]

Text Let K be a complete, algebraically closed, non-Archimedean valued field, and let ϕ ∈ K ( z ) with deg ⁡ ( ϕ ) ≥ 2 . In this paper we consider the family of functions ord Res ϕ n ( x ) , which measure the resultant of ϕ n at points x in P K 1 , the Berkovich projective line, and show that they converge locally uniformly to the diagonal values of the Arakelov–Green's function g μ ϕ ( x , x ) attached to the canonical measure of ϕ . Following this, we are able to prove an equidistribution result for Rumely's crucial measures ν ϕ n , each of which is a probability measure supported at finitely many points whose weights are determined by dynamical properties of ϕ . Video For a video summary of this paper, please visit https://youtu.be/YCCZD1iwe00 . [ABSTRACT FROM AUTHOR]

Text Two infinite sequences A and B of nonnegative integers are called additive complements , if their sum contains all sufficiently large integers. We also say that B is an additive complement of A if A and B are additive complements. In this paper, we consider a problem of Ben Green on additive complements of the squares: S = { 1 2 , 2 2 , … } . The following result is proved: if B = { b n } n = 1 ∞ with b n ≥ π 2 16 n 2 − 0.57 n 1 2 log ⁡ n − β n 1 2 for all positive integers n and any given constant β , then B is not an additive complement of S . In particular, B = { ⌊ π 2 16 n 2 ⌋ | n = 1 , 2 , … } is not an additive complement of S . Video For a video summary of this paper, please visit https://youtu.be/cVXWCP4Igp8 . [ABSTRACT FROM AUTHOR]

Text The aim of this paper is to present a method to calculate the rate of growth of divergent series. In the case of the harmonic series, we study an alternate of other classical methods presented by L. Comtet, S.M. Zeyman, W. DeTemple and S.-H. Wang. Moreover, the method presented in this paper is suitable for obtaining similar results on other divergent series, as we will show in the final part of our investigation. Video For a video summary of this paper, please visit http://youtu.be/SFR7zY4pvG4 . [ABSTRACT FROM AUTHOR]

Abstract The geometry of integer continued fractions and in particular, simple continued fractions has been recorded by exploring the underlying relationship | a d − b c | = 1 for a , b , c , d integers as it arises in the Farey tessellation of the hyperbolic plane H 2 and the array of Ford circles in the upper-half of the real plane R 2. Simple continued fractions may also be represented as a path on a graph whose vertices are reduced rationals and on a dual graph with vertices that are the Farey triangles in the tessellation of H 2 under the modular group. This paper produces an analogue of the above results for Gaussian integer continued fractions by examining the condition | α γ − β δ | = 1 for α , β , γ , δ Gaussian integers. Through this exploration it is possible to extend the concept of Farey neighbors to Gaussian rationals, introduce Farey sum sets, and establish the Farey tessellation of H 3 by Farey octahedrons under the action of the Picard groups without reference to the fundamental domains of the groups. A geodesic algorithm to extract a Gaussian integer continued fraction for complex numbers is introduced that is a geometrical analogue of the simple continued fraction for real numbers. [ABSTRACT FROM AUTHOR]

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: Given a set let denote the number of ordered pairs such that . The celebrated Erdős–Turán conjecture states that if such that for all sufficiently large n, then the representation function must be unbounded. For each positive integer m, let be the least positive integer r such that there exists a set with and . Ruzsa''s method in [I.Z. Ruzsa, A just basis, Monatsh. Math. 109 (1990) 145–151] implies that must be bounded. It is pleasure to call a Ruzsa''s number. In this paper we prove that all Ruzsa''s numbers . This improves the previous bound . Several related open problems are proposed. Video abstract: For a video summary of this paper, please visit http://www.youtube.com/watch?v=hgDwkwg_LzY. [Copyright &y& Elsevier]

Abstract: A splitting field of a central simple algebra is said to be absolute Galois if it is Galois over some fixed subfield of the centre of the algebra. The paper proves an existence theorem for such fields over global fields with enough roots of unity. As an application, all twisted function fields and all twisted Laurent series rings over symbol algebras (or p-algebras) over global fields are crossed products. An analogous statement holds for division algebras over Henselian valued fields with global residue field. The existence of absolute Galois splitting fields in central simple algebras over global fields is equivalent to a suitable generalization of the weak Grunwald–Wang theorem, which is proved to hold if enough roots of unity are present. In general, it does not hold and counter examples have been used in noncrossed product constructions. This paper shows in particular that a certain computational difficulty involved in the construction of explicit examples of noncrossed product twisted Laurent series rings cannot be avoided by starting the construction with a symbol algebra. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to characterize QM-abelian surfaces which has a model of -type over . The “special” involutions on a corresponding indefinite quaternion algebra (which is defined in Section 2) play an essential role in this paper. [Copyright &y& Elsevier]

Abstract: The aim of the paper is to determine all free separable quadratic algebras over the rings of integers of quadratic fields in terms of the properties of the fundamental unit in the real case. The paper corrects some earlier published results on the subject. [Copyright &y& Elsevier]

In this paper, we continue the investigation of the zeta function of divisors, as introduced by the first author in Wan (in: D. Jungnickel, H. Niederreiter (Eds.), Finite Fields and Applications, Springer, Berlin, 2001, pp. 437–461; Manuscripta Math. 74 (1992) 413), for a projective variety over a finite field. Assuming that the set of effective divisors in the divisor class group forms a finitely generated monoid, then there are four conjectures about this zeta function: p-adic meromorphic continuation, rank and pole relation, p-adic Riemann hypothesis, and simplicity of zeros and poles. This paper proves all four conjectures when the Chow the group of divisors is of rank one. Also, an example with higher rank is provided where all four conjectures hold. [Copyright &y& Elsevier]

In this paper, we prove two results. The first theorem uses a paper of Kim (J. Number Theory 74 (1999) 307) to show that for fixed primes p1,…,pk, and for fixed integers m1,…,mk, with pi∤mi, the numbers (ep1(n),…,epk(n)) are uniformly distributed modulo (m1,…,mk), where ep(n) is the order of the prime p in the factorization of n!. That implies one of Sander''s conjectures from Sander (J. Number Theory 90 (2001) 316) for any set of odd primes. Berend (J. Number Theory 64 (1997) 13) asks to find the fastest growing function f(x) so that for large x and any given finite sequence ϵi∈{0,1}, i⩽f(x), there exists n such that the congruences epi(n)≡ϵi (mod 2) hold for all i⩽f(x). Here, pi is the ith prime number. In our second result, we are able to show that f(x) can be taken to be at least c1(log x/(log log x)6)1/9, with some absolute constant c1, provided that only the first odd prime numbers are involved. [Copyright &y& Elsevier]

In an earlier paper we considered the effects that finite submodules can have on μ-invariants of Selmer groups. In this paper we examine some of the consequences of that theory to elliptic curves and their symmetric powers. One main result is the construction of an isogeny class of elliptic curves, all of which have positive μ-invariants. A second result is a connection between the behaviors of μ-invariants associated with the symmetric powers of an elliptic curve over Q, and the behavior of the μ-invariant of that elliptic curve over different extensions of Q. [Copyright &y& Elsevier]

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]

In this paper, we prove some results on the sum-product problem over arbitrary fields which improve and generalize results given by Hegyvári and Hennecart [5] . More precisely, we prove that, for related pairs of two-variable functions f ( x , y ) and g ( x , y ) , if A and B are two sets in an arbitrary field F with | A | = | B | , then max ⁡ { | f ( A , B ) | , | g ( A , B ) | } ≫ | A | 1 + c , for some c > 0 . [ABSTRACT FROM AUTHOR]

There exists a lattice map from the set of pure uniformizable Anderson t-motives to the set of lattices. It is not known what is the image and the fibers of this map. We prove a local result that sheds the first light to this problem and suggests that maybe this map is close to 1–1. Namely, let M ( 0 ) be a t-motive of dimension n and rank r = 2 n — the n -th power of the Carlitz module of rank 2, and let M be a t-motive which is in some sense “close” to M ( 0 ) . We consider the lattice map M ↦ L ( M ) , where L ( M ) is a lattice in C ∞ n . We show that the lattice map is an isomorphism in a “neighborhood” of M ( 0 ) . Namely, we compare the action of monodromy groups: (a) from the set of equations defining t-motives to the set of t-motives themselves, and (b) from the set of Siegel matrices to the set of lattices. The result of the present paper gives that the size of a neighborhood, where we have an isomorphism, depends on an element of the monodromy group. We do not know whether there exists a universal neighborhood. Method of the proof: explicit solution of an equation describing an isomorphism between two t-motives by a method of successive approximations using a version of the Hensel lemma. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the expected number of points in F q 2 of multiplicity m , for 0 ≤ m ≤ q + 1 , with respect to a randomly chosen arrangement of q + 1 lines with different slopes, is ( 1 / ( m ! e ) ) q 2 + O ( q ) , as q → ∞ . We further state that the distance between the number of such points in a randomly chosen arrangement and ( 1 / ( m ! e ) ) q 2 is lower than q ln ⁡ q with probability close to 1 for large q . [ABSTRACT FROM AUTHOR]

Text We develop geometric methods to study the generating weights of free modules of vector-valued modular forms of half-integral weight, taking values in a complex representation of the metaplectic group. We then compute the generating weights for modular forms taking values in the Weil representation attached to cyclic quadratic modules of order 2 p r , where p ≥ 5 is a prime. We also show that the generating weights approach a simple limiting distribution as p grows, or as r grows and p remains fixed. Video For a video summary of this paper, please visit https://youtu.be/QNbPSXXKot4 . [ABSTRACT FROM AUTHOR]

We investigate level p Eisenstein congruences for GSp 4 , generalisations of level 1 congruences predicted by Harder. By studying the associated Galois and automorphic representations we see conditions that guarantee the existence of a paramodular form satisfying the congruence. This provides theoretical justification for computational evidence found in the author's previous paper. [ABSTRACT FROM AUTHOR]

In this paper, we study the polynomial Gauss sums over finite fields, and present an analogue of Davenport–Hasse's theorem for the polynomial Gauss sums, which is a generalization of the previous result obtained by Hayes. [ABSTRACT FROM AUTHOR]

Let k n ( x ) be the n -th partial quotient of the generalized continued fraction (GCF) expansion of x . This paper is concerned with the growth rate of k n ( x ) . When the parameter function satisfies − 1 < ϵ ( k ) ≤ 1 , we obtain the Hausdorff dimension of the sets E ϕ = { x ∈ ( 0 , 1 ) : lim n → ∞ ⁡ log ⁡ k n ( x ) ϕ ( n ) = 1 } for any nondecreasing ϕ with lim n → ∞ ⁡ ( ϕ ( n + 1 ) − ϕ ( n ) ) = ∞ and lim n → ∞ ⁡ ϕ ( n + 1 ) / ϕ ( n ) = 1 . Applications are given to several kinds of exceptional sets related to the GCF expansion. [ABSTRACT FROM AUTHOR]

Let B be a finite set of natural numbers or complex numbers. Product set corresponding to B is defined by B . B : = { a b : a , b ∈ B } . In this paper we give an upper bound for longest length of consecutive terms of a polynomial sequence present in a product set accurate up to a positive constant. We give a sharp bound on the maximum number of Fibonacci numbers present in a product set when B is a set of natural numbers and a bound which is accurate up to a positive constant when B is a set of complex numbers. [ABSTRACT FROM AUTHOR]

In this paper we focus on finding all the factorials expressible as a product of a fixed number of 2 k -nacci numbers with k ≥ 2 . We derive the 2-adic valuation of the 2 k -nacci sequence and use it to establish bounds on the solutions of the initial equation. In addition, we specify a more general family of sequences, for which we can perform a similar procedure. We also investigate a possible connection of these results with p -regular sequences. [ABSTRACT FROM AUTHOR]

The paper presents formulae for certain series involving the Riemann zeta function. These formulae are generalizations, in a natural way, of well known formulae, originating from Leonhard Euler. Formulae that existed only for initial values n = 0 , 1 are now found for every natural n . Relevant connections with various known results are also pointed out. [ABSTRACT FROM AUTHOR]

A new class of partitions, partitions with designated summands, was introduced by Andrews, Lewis and Lovejoy. Let PD ( n ) denote the number of partitions of n with designated summands. Andrews, Lewis and Lovejoy established many congruences modulo 3 and powers of 2 for PD ( n ) by using the theory of modular forms. In this paper, we prove several infinite families of congruences modulo 9 and 27 for PD ( n ) by employing the generating functions of PD ( 3 n ) and PD ( 3 n + 1 ) which were discovered by Chen, Ji, Jin and Shen. For example, we prove that for n ≥ 0 and k ≥ 1 , PD ( 2 18 k − 1 ( 12 n + 1 ) ) ≡ 0 ( mod 27 ) . Furthermore, using some results due to Newman, we find some strange congruences modulo 27 for PD ( n ) . For example, we prove that for k ≥ 0 , PD ( 13 9 k ( 75 p + 2 ) ) ≡ 0 ( mod 27 ) and PD ( 2 × 13 9 k + 8 ) ≡ 0 ( mod 27 ) , where p is a prime and p ≡ 1 ( mod 12 ) . [ABSTRACT FROM AUTHOR]

This corrigendum describes a number of calculation and typographical errors, primarily in the examples, in the 2017 JNT paper Some Real Quadratic Number Fields with their Hilbert 2-Class Field Having Cyclic 2-Class Group , by Elliot Benjamin. [ABSTRACT FROM AUTHOR]

A univariate polynomial f ( x ) is said to be nice if all of its coefficients as well as all of the roots of both f ( x ) and its derivative f ′ ( x ) are integers. The known examples of nice polynomials with distinct roots are limited to quadratic polynomials, cubic polynomials, symmetric quartic polynomials and, up to equivalence, only a finite number of nonsymmetric quartic polynomials and one quintic polynomial. In this paper we find parametrized families of nice nonsymmetric quartic polynomials with distinct roots, as well as infinitely many nice quintic and sextic polynomials with distinct roots. [ABSTRACT FROM AUTHOR]

Abstract: We say the sets of nonnegative integers and are additive complements if their sum contains all sufficiently large integers. In this paper we prove a conjecture of Chen and Fang about additive complement of a finite set by using analytic tools. [Copyright &y& Elsevier]

Abstract: Let be positive integers. In this paper, we prove that the equation has no solutions in integers and k with , and . [Copyright &y& Elsevier]

Abstract: Let G be a finite cyclic group. Every sequence S over G can be written in the form where and , and the index ind S of S is defined to be the minimum of over all possible such that . A conjecture says that if G is finite such that , then for every minimal zero-sum sequence S. In this paper, we prove that the conjecture holds if has two prime factors. [Copyright &y& Elsevier]

Abstract: In the present paper, we show the equality of the γ-factors defined by Jacquet, Piatetski-Shapiro and Shalika with those obtained via the Langlands–Shahidi method. Our results are new in the case of positive characteristic, where we establish a refined version of the local–global principle for which has independent interest. In characteristic zero, the results are due to Shahidi. The comparison of γ-factors is made via a uniqueness result for Rankin–Selberg γ-factors. [Copyright &y& Elsevier]

Abstract: In this paper, we determine all sets A of integers such that, for any integral-valued polynomial which has no fixed divisor, for all integers and n, there are infinitely many integers and a choice of such that . The earlier result shows that is such a set. [Copyright &y& Elsevier]

Abstract: In this paper we consider non-linear differential equations which are closely related to the generating functions of Frobenius–Euler polynomials. From our non-linear differential equations, we derive some new identities between the sums of products of Frobenius–Euler polynomials and Frobenius–Euler polynomials of higher order. [Copyright &y& Elsevier]

Abstract: In this paper, we show that there is a frame of norm k in the odd Leech lattice for every . [Copyright &y& Elsevier]

Abstract: In this paper, we construct for arbitrary reductive group a full eigenvariety, which parameterizes all p-adic overconvergent cohomological eigenforms of the group in the sense of Ash–Stevens and Urban. We also see that the eigenvariety constructed by Eric Urban (2011) is the union of those irreducible components of codimension zero in the eigenvariety we constructed, therefore we can formulate Urbanʼs conjectures. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the Pell sequence and the Perrin sequence and we derive some relationships between these sequences and permanents and determinants of one type of Hessenberg matrices. [Copyright &y& Elsevier]

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: In this paper, we consider mod ℓ Galois representations of . In particular, we develop an effective criterion to decide whether or not two mod ℓ Galois representations are isomorphic. The proof uses methods from Khare–Wintenbergerʼs recent theorem on Serreʼs conjecture along with theorems by Sturm and Kohnen. [Copyright &y& Elsevier]

Abstract: Let n be a fixed odd integer with . In this paper, using a recent result on the existence of primitive divisors of Lehmer numbers give by Y. Bilu, G. Hanrot and P.M. Voutier, we prove that the equation has no positive integer solution with . [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: In this paper, the convergence of the Euler product of the Hecke zeta-function is proved on the line with . A certain functional identity between and is found. An analogue of Tate''s adelic Poisson summation is obtained for the global Hankel transformation, which is constructed in Li (2010) . [Copyright &y& Elsevier]

Abstract: Let be an integer, and π an irreducible unitary cuspidal representation for , whose attached automorphic L-function is denoted by . Let be the sequence of coefficients in the Dirichlet series expression of in the half-plane . It is proved in this paper that, if π is such that the sequence is real, then there are infinitely many sign changes in the sequence , and the first sign change occurs at some , where is the conductor of π, and the implied constant depends only on m and ε. This generalizes the previous results for . A result of the same quality is also established for , the sequence of coefficients in the Dirichlet series expression of in the half-plane . [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: Text: Let denote a Leopoldt–Kubota p-adic L-function, where and χ is a nonprincipal even character of the first kind. The aim of this article is to study how the values assumed by this function depend on the Iwasawa λ-invariant associated to χ. Assuming that , it turns out that behaves, in some sense, like a polynomial of degree λ. The results lead to congruences of a new type for (generalized) Bernoulli numbers. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=5aaB1d6fZDs. [Copyright &y& Elsevier]

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: We call a set A of positive integers an asymptotic basis of order h if every sufficiently large integer n can be written as a sum of h elements of A. If no proper subset of A is an asymptotic basis of order h, then A is a minimal asymptotic basis of that order. Erdős and Nathanson showed that for every there exists a minimal asymptotic basis A of order h with , where denotes the density of A. Erdős and Nathanson asked whether it is possible to strengthen their result by deciding on the existence of a minimal asymptotic bases of order such that . Moreover, they asked if there exists a minimal asymptotic basis with . In this paper we answer these questions in the affirmative by constructing a minimal asymptotic basis A of order 2 fulfilling a very restrictive condition [Copyright &y& Elsevier]

Abstract: As part of his work to develop an explicit trace formula for Hecke operators on congruence subgroups of , Hijikata (1974) defines and characterizes the notion of a split order in , where k is a local field. In this paper, we generalize the notion of a split order to for and give a natural geometric characterization in terms of the affine building for . In particular, we show that there is a one-to-one correspondence between split orders in and a collection of convex polytopes in apartments of the building such that the split order is the intersection of all the maximal orders representing the vertices in the polytope. This generalizes the geometric interpretation in the case in which split orders correspond to geodesics in the tree for with the split order given as the intersection of the endpoints of the geodesic. [Copyright &y& Elsevier]