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Let f be a normalized Hecke eigenform of weight k for the full modular group S L 2 (ℤ). In this paper, we obtain the asymptotic of higher moments of general divisor functions associated to the Fourier coefficients of Rankin–Selberg L-functions R (s , f × f) , supported at the integers represented by primitive integral positive-definite binary quadratic forms (reduced forms) of a fixed discriminant D < 0. We improve previous results in the case when the reduced form is given by (x 1 , x 2) = x 1 2 + x 2 2 . [ABSTRACT FROM AUTHOR]

Let f ∈ S k (Γ 0 (N)) be a normalized Hecke eigenforms of integral weight k and level N ≥ 1 . In the article, we establish the asymptotics of power moment associated to the sequences { λ f ⊗ f ⊗ f (Q (x ̲)) } Q ∈ S D , x ̲ ∈ Z 2 and { λ f ⊗ sym 2 f (Q (x ̲)) } Q ∈ S D , x ̲ ∈ Z 2 where S D denotes the set of inequivalent primitive integral positive-definite binary quadratic forms (reduced forms) of fixed discriminant D < 0. As a consequence, we prove results concerning the behaviour of sign changes associated to these sequences. [ABSTRACT FROM AUTHOR]

In this article, we establish an average behaviour of the Fourier coefficients of the Hecke eigenforms supported at the integers represented by any primitive integral positive definite binary quadratic form of fixed discriminant D < 0 when the class number h (D) = 1 . We also obtain a quantitative result for the number of sign changes of the sequence of Fourier coefficients a(n) of the Hecke eigenforms, where n is represented by any primitive integral positive definite binary quadratic form of fixed discriminant D < 0 when the class number h (D) = 1 in the interval (x, 2x], for sufficiently large x. [ABSTRACT FROM AUTHOR]

Let f and g be two distinct primitive holomorphic cusp forms of even integral weights k 1 and k 2 for the full modular group Γ = S L (2 , Z) , respectively. Denote by λ f (n) and λ g (n) the nth normalized Fourier coefficients of f and g, respectively. And set Q (x) a primitive integral positive-definite binary quadratic form of fixed discriminant D < 0 with the class number h (D) = 1 . In this paper, we establish a lower bound for the analytic density of the set { p : p = Q (x) for some x ∈ Z 2 , λ f (p i) λ f (p j) < λ g (p i) λ g (p j) } , where j ⩾ 1 , 0 ⩽ i ⩽ j are any fixed integers. Furthermore, we also consider the similar problem concerning the linear combinations of λ f (p j) and λ g (p j) in a given interval supported on the binary quadratic form Q (x) . Similar problems for triple product L-functions are also studied. [ABSTRACT FROM AUTHOR]

In this article, we prove a result concerning the infinitude of square-free integers represented by a class of polynomials in two variables. More precisely, we prove that infinitely many square-free positive integers are represented by a primitive integral positive-definite binary quadratic form of a given discriminant D. We obtain our result by deriving an asymptotic formula for the summatory function associated to it using some known L-functions. [ABSTRACT FROM AUTHOR]

Let λ f (n) and λ g (n) be the normalized n th -Fourier coefficient of distinct Hecke eigenforms f and g of integral weight for the full modular group, respectively, and Q (x ̲) is a primitive integral positive-definite binary quadratic form of fixed discriminant D < 0 with the class number h (D) = 1. We establish a lower bound for the analytic density of the set { p : p = Q (x ̲) for some x ̲ ∈ ℤ 2 and λ f (p m) < λ g (p m) } for each fixed m ∈ ℕ. With the above notation, we also obtain a lower bound for the analytic density of the set { p : p = Q (x ̲) for some x ̲ ∈ ℤ 2 and λ f 2 (p m) < λ g 2 (p m) } for each fixed m ∈ ℕ. As a consequence, we obtain a deterministic criterion for the equality of two Hecke eigenforms. [ABSTRACT FROM AUTHOR]

Let G be an additive finite abelian group and Γ ⊂ End (G) be a subset of the endomorphism group of G. A sequence S = g 1 · ... · g ℓ over G is a (Γ -)weighted zero-sum sequence if there are γ 1 , ... , γ ℓ ∈ Γ such that γ 1 (g 1) + ... + γ ℓ (g ℓ) = 0 . We construct transfer homomorphisms from norm monoids (of Galois algebraic number fields with Galois group Γ ) and from monoids of positive integers, represented by binary quadratic forms, to monoids of weighted zero-sum sequences. Then we study algebraic and arithmetic properties of monoids of weighted zero-sum sequences. [ABSTRACT FROM AUTHOR]

We systematically investigate the expressions and congruences for both a one-parameter family {Gn(x)} as well as a two-parameter family {Gn(r, m)} of sequences. [ABSTRACT FROM AUTHOR]

Let $ \{S_n\} $ be the Apéry-like sequence given by $ S_n = \sum_{k = 0}^n\binom nk\binom{2k}k\binom{2n-2k}{n-k} $. We show that for any odd prime $ p $, $ \sum_{n = 1}^{p-1}\frac {nS_n}{8^n}{\equiv} (1-(-1)^{\frac{p-1}2})p^2\ (\text{ mod}\ {p^3}) $. Let $ \{Q_n\} $ be the Apéry-like sequence given by $ Q_n = \sum_{k = 0}^n\binom nk(-8)^{n-k}\sum_{r = 0}^k\binom kr^3 $. We establish many congruences concerning $ Q_n $. For an odd prime $ p $, we also deduce congruences for $ \sum_{k = 0}^{p-1}\binom{2k}k^3\frac 1{64^k}\ (\text{ mod}\ {p^3}) $, $ \sum_{k = 0}^{p-1}\binom{2k}k^3\frac 1{64^k(k+1)^2}\ (\text{ mod}\ {p^2}) $ and $ \sum_{k = 0}^{p-1}\binom{2k}k^3\frac 1{64^k(2k-1)}\ (\text{ mod}\ p) $, and pose lots of conjectures on congruences involving binomial coefficients and Apéry-like numbers.

We show that the Liouville function \lambda changes sign infinitely often on n^2+ d for any non-zero integer d. [ABSTRACT FROM AUTHOR]

In this methodological paper, we first review the classic cubic Diophantine equation a³ + b³ + c³ = d³, and consider the specific class of solutions q1³ + q2² + q3³ = q4³ with each qi being a binary quadratic form. Next we turn our attention to the familiar sums of powers of the first n positive integers, Sk = 1k+2k+· · ·+nk, and express the squares Sk², Sm², and the product SkSm as a linear combination of power sums. These expressions, along with the above quadratic-form solution for the cubic equation, allows one to generate an infinite number of relations of the form Q1³ + Q2³ + Q3³ = Q4³, with each Qi being a linear combination of power sums. Also, we briefly consider the quadratic Diophantine equations a² +b² +c² = d² and a² +b² = c², and give a family of corresponding solutions Q1² + Q2² + Q3² = Q4² and Q1² + Q2² = Q3² in terms of sums of powers of integers. [ABSTRACT FROM AUTHOR]

We consider the quadratic form x²+mxy+ny², where |m²-4n| is a prime number. Under the assumption that a particular, small, finite set of integers is representable, we determine all integers representable by this quadratic form. [ABSTRACT FROM AUTHOR]

For let and . Then and are Apéry-like numbers. In this paper we obtain some congruences and pose several challenging conjectures for sums involving or . [ABSTRACT FROM AUTHOR]

Let a , b , c be any positive integers such that c | a b and d i ± is a square free positive integer of the form d i ± = a 2 k b 2 l ± i c m where k , l ≥ m and i = 1 , 2. The main focus of this paper is to find the fundamental solution of the equation x 2 − d i ± y 2 = 1 , with the help of the continued fraction of d i ± . We also obtain all the positive solutions of the equations x 2 − d i ± y 2 = ± 1 and x 2 − d i ± y 2 = ± 4 by means of the Fibonacci and Lucas sequences. Furthermore, in this work, we derive some algebraic relations on the Pell form F d i ± ( x , y ) = x 2 − d i ± y 2 including cycle, proper cycle, reduction and proper automorphism of it. We also determine the integer solutions of the Pell equation F Δ d i ± ( x , y ) = 1 in terms of d i ±. We extend all the results of the papers [3, 10, 27, 37]. [ABSTRACT FROM AUTHOR]

Dirichlet's version of Gauss's reduction theory for indefinite binary quadratic forms includes a map from Gauss-reduced forms to strings of natural numbers. It attaches to a form the minimal period of the continued fraction of a quadratic irrationality associated with the form. When Zagier developed his own reduction theory, parallel to Dirichlet's, he omitted an analogue of this map. We define a new map on Zagier-reduced forms that serves as this analogue. We also define a map from the set of Gauss-reduced forms into the set of Zagier-reduced forms that gives a near-embedding of the structure of Gauss's reduction theory into that of Zagier's. From this perspective, Zagier-reduction becomes a refinement of Gauss-reduction. [ABSTRACT FROM AUTHOR]

In this article we estimate the number of integers up to X which can be properly represented by a positive-definite, binary, integral quadratic form of small discriminant. This estimate follows from understanding the vector of signs that arises from computing the Legendre symbol of small integers n at multiple primes. [ABSTRACT FROM AUTHOR]

Let p be a prime number. In this paper we show that p can be expressed as p = a 2 ± a b − b 2 with integers a and b if and only if p is congruent to 0, 1 or −1 ( mod 5 ) and p can be expressed as p = a 2 ± a b + b 2 with integers a and b if and only if p is congruent to 0, 1 ( mod 3 ) . [ABSTRACT FROM AUTHOR]

We present an improved algorithm for tabulating class groups of imaginary quadratic fields of bounded discriminant. Our method uses classical class number formulas involving theta-series to compute the group orders unconditionally for all Δ ≢ 1 (mod 8). The group structure is resolved using the factorization of the group order. The 1 mod 8 case was handled using the methods of Jacobson, Ramachandran, and Williams including the batch verification method based on the Eichler-Selberg trace formula to remove dependence on the Extended Riemann Hypothesis. Our new method enabled us to extend the previous bound of |Δ| < 2·1011 to 240. Statistical data in support of a variety of conjectures is presented, along with new examples of class groups with exotic structures. [ABSTRACT FROM AUTHOR]

We present an improved algorithm for tabulating class groups of imaginary quadratic fields of bounded discriminant. Our method uses classical class number formulas involving theta-series to compute the group orders unconditionally for all Δ ≢ 1 (mod 8). The group structure is resolved using the factorization of the group order. The 1 mod 8 case was handled using the methods of Jacobson, Ramachandran, and Williams including the batch verification method based on the Eichler-Selberg trace formula to remove dependence on the Extended Riemann Hypothesis. Our new method enabled us to extend the previous bound of |Δ| < 2·1011 to 240. Statistical data in support of a variety of conjectures is presented, along with new examples of class groups with exotic structures. [ABSTRACT FROM AUTHOR]

We investigate the relationship between the numbers of representations of certain integers by a primitive integral binary quadratic form of discriminant and the order of the class of in the form class group of discriminant , in the case when this order is even. The explicit form of the solutions obtained is used to give a partial answer to a question regarding which multiples of can be parameterized in a particular way. [ABSTRACT FROM AUTHOR]

Let ℤ be the set of integers, and let p be a prime of the form 4k + 1 and so p = c2 + d2 with c, d ∈ ℤ. Let q be an integer of the form 4k + 3. Assume that 4n2p = x2 + qy2 with c, d, n, x, y ∈ ℤ and (q, n) = (x, y) = 1, where (a, b) is the greatest common divisor of integers a and b. In this paper, we establish congruences for (-q)[p/8] ( p) in terms of c, d, n, x and y, where [⋅] is the greatest integer function. In particular, we establish a reciprocity law and give an explicit criterion for (-11)[p/8] ( p). [ABSTRACT FROM AUTHOR]

Let ℤ be the set of integers, and let p be a prime of the form 4k + 1 and so p = c2 + d2 with c, d ∈ ℤ. Let q be an integer of the form 4k + 3. Assume that 4n2p = x2 + qy2 with c, d, n, x, y ∈ ℤ and (q, n) = (x, y) = 1, where (a, b) is the greatest common divisor of integers a and b. In this paper, we establish congruences for (-q)[p/8] ( p) in terms of c, d, n, x and y, where [⋅] is the greatest integer function. In particular, we establish a reciprocity law and give an explicit criterion for (-11)[p/8] ( p). [ABSTRACT FROM AUTHOR]

Let F be an algebraic number field and let PF(r) denote the number of nonassociated prime elements of absolute field norm less than or equal to r in the corresponding ring of integers. Using information about the absolute field norms of prime elements and Chebotarev's density theorem, we readily show that when F is a Galois extension of Q, it is the case that PF is asymptotic to 1/hπ, where π is the standard prime-counting function and h is the class number of F. Along the way, we pick up some well-known facts on the realizability of certain prime numbers in terms of those binary quadratic forms associated with the field norm over a ring of integers that is a unique factorization domain. [ABSTRACT FROM AUTHOR]

Let p be a prime greater than 3. In this paper, by using expansions and congruences for Lucas sequences and the theory of cubic residues and cubic congruences, we establish some congruences for and modulo p, where [x] is the greatest integer not exceeding x, and m is a rational p-adic integer with m ≢ 0 ( p). [ABSTRACT FROM AUTHOR]

Abstract: Let be a prime, and let m be an integer with . In the paper we solve some conjectures of Z.W. Sun concerning , and . In particular, we show that for . Let be the Legendre polynomials. In the paper we also show that , where t is a rational p-adic integer, is the greatest integer not exceeding x and is the Legendre symbol. As consequences we determine in the cases and confirm many conjectures of Z.W. Sun. [Copyright &y& Elsevier]

Abstract: Let be a prime, and let m be an integer with . In this paper, based on the work of Brillhart and Morton, by using the work of Ishii and Deuringʼs theorem for elliptic curves with complex multiplication we solve some conjectures of Zhi-Wei Sun concerning . [Copyright &y& Elsevier]

Abstract: Let a and b be positive integers and let p be an odd prime such that for some integers x and y. Let be given by . In this paper, using Jacobiʼs identity , we construct in terms of . For example, if and , then . We also give formulas for , and . [Copyright &y& Elsevier]

Let f( X) and g( Y) be nondegenerate quadratic forms of dimensions m and n, respectively, over K, char K ≠ 2. The problem of birational composition of f( X) and g( Y) is considered: When is the product f( X) · g( Y) birationally equivalent over K to a quadratic form h( Z) over K of dimension m + n? The solution of the birational composition problem for anisotropic quadratic forms over K in the case of m = n = 2 is given. The main result of the paper is the complete solution of the birational composition problem for forms f( X) and g( Y) over a local field P, char P ≠ 2. [ABSTRACT FROM AUTHOR]

Abstract: Let be a prime, , , and , where is the Legendre symbol. In the paper we mainly determine the value of by expressing p in terms of appropriate binary quadratic forms. As applications, for we obtain a general criterion for and a criterion for to be a cubic residue of p, where is the fundamental unit of the quadratic field . We also give a general criterion for , where is the Lucas sequence defined by , and (). Furthermore, we establish a general result to illustrate the connections between cubic congruences and binary quadratic forms. [Copyright &y& Elsevier]

In this paper, we solve Goldbach’s ternary problem involving primes expressible by given primitive positive definite binary quadratic forms whose discriminants coincide with the discriminants of imaginary quadratic fields in which quadratic forms split into linear multipliers. [ABSTRACT FROM AUTHOR]

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]

We study the arithmetic codings of hyperbolic automorphisms of the 2-torus, i.e., the continuous mappings acting from a certain symbolic space of sequences with a finite alphabet endowed with an appropriate structure of the additive group onto the torus which preserve this structure and turn the two-sided shift into a given automorphism of the torus. This group is uniquely defined by an automorphism, and such an arithmetic coding is a homomorphism of that group onto $$\mathbb{B}^2 $$ . The necessary and sufficient condition of the existence of a bijective arithmetic coding is obtained; it is formulated in terms of a certain binary quadratic form constructed by means of a given automorphism. Furthermore, we describe all bijective arithmetic codings in terms of the Dirichlet group of the corresponding quadratic field. The minimum of that quadratic form over the nonzero elements of the lattice coincides with the minimal possible order of the kernel of a homomorphism described above. [ABSTRACT FROM AUTHOR]