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The primary purpose of this paper is to generalize the classical Riemann zeta function to the setting of Krull monoids with torsion class groups. We provide a first study of the same generalization by extending Euler's classical product formula to the more general scenario of Krull monoids with torsion class groups. In doing so, the Decay Theorem is fundamental as it allows us to use strong atoms instead of primes to obtain a weaker version of the Fundamental Theorem of Arithmetic in the more general setting of Krull monoids with torsion class groups. Several related examples are exhibited throughout the paper, in particular, algebraic number fields for which the generalized Riemann zeta function specializes to the Dedekind zeta function. [ABSTRACT FROM AUTHOR]

A new algorithm for computing Hecke operators for SL n was introduced in [14]. The algorithm uses tempered perfect lattices , which are certain pairs of lattices together with a quadratic form. These generalize the perfect lattices of Voronoi [17]. The present paper is the first step in characterizing tempered perfect lattices. We obtain a complete classification in the plane, where the Hecke operators are for SL 2 (Z) and its arithmetic subgroups. The results depend on the class field theory of orders in imaginary quadratic number fields. [ABSTRACT FROM AUTHOR]

A group G is called residually finite if for every non-trivial element g ∈ G , there exists a finite quotient Q of G such that the element g is non-trivial in the quotient as well. Instead of just investigating whether a group satisfies this property, a new perspective is to quantify residual finiteness by studying the minimal size of the finite quotient Q depending on the complexity of the element g , for example by using the word norm ‖ g ‖ G if the group G is assumed to be finitely generated. The residual finiteness growth RF G : N → N is then defined as the smallest function such that if ‖ g ‖ G ≤ r , there exists a morphism φ : G → Q to a finite group Q with | Q | ≤ RF G (r) and φ (g) ≠ e Q. Although upper bounds have been established for several classes of groups, exact asymptotics for the function RF G are only known for very few groups such as abelian groups, the Grigorchuk group and certain arithmetic groups. In this paper, we show that the residual finiteness growth of virtually abelian groups equals log k for some k ∈ N , where the value k is given by an explicit expression. As an application, we show that for every m ≥ 1 and every 1 ≤ k ≤ m , there exists a group G containing a normal abelian subgroup of rank m and with RF G ≈ log k. [ABSTRACT FROM AUTHOR]

The solving degree of a system of multivariate polynomial equations provides an upper bound for the complexity of computing the solutions of the system via Gröbner basis methods. In this paper, we consider polynomial systems that are obtained via Weil restriction of scalars. The latter is an arithmetic construction which, given a finite Galois field extension k ↪ K , associates to a system F defined over K a system Weil (F) defined over k , in such a way that the solutions of F over K and those of Weil (F) over k are in natural bijection. In this paper, we find upper bounds for the complexity of solving a polynomial system Weil (F) obtained via Weil restriction in terms of algebraic invariants of the system F. [ABSTRACT FROM AUTHOR]

The power classes of a field are well-known for their ability to parameterize elementary p -abelian Galois extensions. These classical objects have recently been reexamined through the lens of their Galois module structure. Module decompositions have been computed in several cases, providing deep new insight into absolute Galois groups. The surprising result in each case is that there are far fewer isomorphism types of indecomposables than one would expect generically, with summands predominately free over associated quotient rings. Though non-free summands are the exception both in their form and prevalence, they play the critical role in controlling arithmetic conditions in the field which allow the rest of the decomposition to be so simple. Suppose m , n ∈ N and p is prime. In a recent paper, a surprising and elegant decomposition for p m th power classes has been computed when the underlying Galois group is a cyclic group of order p n. As with previous module decompositions, at most one non-free summand appears. Outside of a particular special case when p = 2 , the structure of this exceptional summand was determined by a vector a ∈ { − ∞ , 0 , ... , n } m and a natural number d. In this paper we give field-theoretic interpretations for a and d , showing they are related to the solvability of a family of Galois embedding problems and the cyclotomic character associated to K / F. [ABSTRACT FROM AUTHOR]

We develop a uniform method to derive Chudnovsky-Ramanujan type formulae for triangle groups based on a generalization of a method of Chudnovsky and Chudnovsky; in particular, we carry out the method systematically for non-compact arithmetic triangle groups and one non-Fuchsian covering. As a result, we derive all rational Ramanujan type series given by Chan-Cooper for levels 1-4, as well as two additional rational series of a similar form prescribed by Chan-Cooper for these levels, but not found in the paper of Chan-Cooper. These two additional series were first found by Z.-W. Sun in a slightly different form. We also derive additional rational series of a similar form, but not found in the papers of Chan-Cooper nor Z.-W. Sun. As an ingredient in the method, we give an algorithm to rigorously confirm the singular values of normalized Eisenstein series of weight 2 and give examples in higher class number, which may be of independent interest. We also prove some partial completeness results for Clausen type single summation series. [ABSTRACT FROM AUTHOR]

Suppose that O L is the ring of integers of a number field L , and suppose that f (z) = ∑ n = 1 ∞ a f (n) q n ∈ S k (Γ 0 (N) +) ∩ O L [ [ q ] ] is a normalized Hecke eigenform for Γ 0 (N) +. We say that f is non-ordinary at p if there is a prime ideal p ⊂ O L above p for which a f (p) ≡ 0 (mod p). In the authors' previous paper with Ken Ono [10] it was proved that there are infinitely many Hecke eigenforms for SL 2 (Z) such that are non-ordinary at any given finite set of primes. In this paper, we extend this result to some genus 0 subgroups of SL 2 (R) , namely, the normalizers Γ 0 (N) + of the congruence subgroups Γ 0 (N). Our result also generalizes some of Choi and Kim's result in [2]. [ABSTRACT FROM AUTHOR]

It has been recently observed that fundamental aspects of the classical theory of factorization can be greatly generalized by combining the languages of monoids and preorders. This has led to various theorems on the existence of certain factorizations, herein called ⪯-factorizations, for the ⪯-non-units of a (multiplicatively written) monoid H endowed with a preorder ⪯, where an element u ∈ H is a ⪯-unit if u ⪯ 1 H ⪯ u and a ⪯-non-unit otherwise. The "building blocks" of these factorizations are the ⪯-irreducibles of H (i.e., the ⪯-non-units a ∈ H that cannot be written as a product of two ⪯-non-units each of which is strictly ⪯-smaller than a); and it is interesting to look for sufficient conditions for the ⪯-factorizations of a ⪯-non-unit to be bounded in length or finite in number (if measured or counted in a suitable way). This is precisely the kind of questions addressed in the present work, whose main novelty is the study of the interaction between minimal ⪯-factorizations (i.e., a refinement of ⪯-factorizations used to counter the "blow-up phenomena" that are inherent to factorization in non-commutative or non-cancellative monoids) and some finiteness conditions describing the "local behavior" of the pair (H , ⪯). Besides a number of examples and remarks, the paper includes many arithmetic results, a part of which are new already in the basic case where ⪯ is the divisibility preorder on H (and hence in the setup of the classical theory). [ABSTRACT FROM AUTHOR]

In this paper, we present some partial results for the geometrically m -step solvable Grothendieck conjecture in anabelian geometry. Among other things, we prove the geometrically 3-step solvable Grothendieck conjecture for genus 0 curves over fields finitely generated over the prime field of arbitrary characteristic. [ABSTRACT FROM AUTHOR]

In 2007, Andrews and Paule introduced the broken k -diamond partition function Δ k (n). Many researches on the arithmetic properties for Δ k (n) have been done. In this paper, we prove that D 3 log ⁡ Δ 1 (n − 1) > 0 for n ≥ 5 and D 3 log ⁡ Δ 2 (n − 1) > 0 for n ≥ 7 , where D is the difference operator with respect to n. We also conjecture that for any k ≥ 1 and r ≥ 1 , there exists a positive integer n k (r) such that for n ≥ n k (r) , (− 1) r D r log ⁡ Δ k (n) > 0. This is analogous to the positivity of finite differences of the logarithm of the partition function, which has been proved by Chen, Wang, and Xie. Furthermore, we obtain that both { Δ 1 (n) } n ≥ 0 and { Δ 2 (n) } n ≥ 0 satisfy the higher order Turán inequalities for n ≥ 6. [ABSTRACT FROM AUTHOR]

Triple correlation sums problem concerns the non-trivial power-saving bounds for the correlation of three objects. It is conjectured that these sums are non-trivial in any fixed but arbitrarily given ranges. In this paper, the uniform non-trivial bounds for the triple correlation sums ∑ m ≥ 1 , n ≥ 1 λ π (1 , m) λ ⋆ (n) λ f (m + p n) U (m / X) V (n / H) in the level aspect are derived, where π is any G L 3 -Maaß cuspidal form, f ∈ B k ⁎ (p) , any Hecke newform of prime level p and weight k ∈ N + , and λ ⋆ (n) , n ≥ 1 , are certain coefficients of arithmetic interest. As a result, we show that sums of this type follow the 1/3- significance level. We study the strength of the result by specifying ⋆ being the cusp forms on G L 3 and G L 2 , respectively, and further obtain the more significant cancellations in these sums. [ABSTRACT FROM AUTHOR]

In this paper we analyze the computational costs of various operations and algorithms in algebraic number fields using exact arithmetic. Let K be an algebraic number field. In the first half of the paper, we calculate the running time and the size of the output of many operations in K in terms of the size of the input and the parameters of K. We include some earlier results about these, but we go further than them, e.g. we also analyze some R -specific operations in K like less-than comparison. In the second half of the paper, we analyze two algorithms: the Bareiss algorithm, which is an integer-preserving version of the Gaussian elimination, and the LLL algorithm, which is for lattice basis reduction. In both cases, we extend the algorithm from Z n to K n , and give a polynomial upper bound on the running time when the computations in K are performed exactly (as opposed to floating-point approximations). [ABSTRACT FROM AUTHOR]

This paper continues a research program on constructive investigations of non-commutative Ore localizations, initiated in our previous papers, and particularly touches the constructiveness of arithmetics within such localizations. Earlier we have introduced monoidal, geometric and rational types of localizations of domains as objects of our studies. Here we extend this classification to rings with zero divisors and consider Ore sets of the mentioned types which are commutative enough: such a set either belongs to a commutative algebra or it is central or its elements commute pairwise. By using the systematic approach we have developed before, we prove that arithmetic within the localization of a commutative polynomial algebra is constructive and give the necessary algorithms. We also address the important question of computing the local closure of ideals which is also known as the desingularization, and present an algorithm for the computation of the symbolic power of a given ideal in a commutative ring. We also provide algorithms to compute local closures for certain non-commutative rings with respect to Ore sets with enough commutativity. [ABSTRACT FROM AUTHOR]

The goal of the present paper is to push forward the frontiers of computations on mod ℓ Farrell–Tate cohomology for arithmetic groups. We deal with ℓ -rank 1 cases different from PSL 2. The conjugacy classification of cyclic subgroups of order ℓ is reduced to the classification of modules of C ℓ -group rings over suitable rings of integers which are principal ideal domains, generalizing an old result of Reiner. As an example of the number-theoretic input required for the Farrell–Tate cohomology computations, we discuss in detail the homological torsion in PGL 3 over principal ideal rings of quadratic integers, accompanied by machine computations in the imaginary quadratic case. [ABSTRACT FROM AUTHOR]

In this paper we study couples of finite separable extensions of the function field F q (T) which are arithmetically equivalent, i.e. such that prime ideals of F q [ T ] decompose with the same inertia degrees in the two fields, up to finitely many exceptions. In the first part of this work, we extend previous results by Cornelissen, Kontogeorgis and Van der Zalm to the case of non-geometric extensions of F q (T) , which are fields such that their field of constants may be bigger than F q. In the second part, we explicitly produce examples of non-geometric extensions of F 2 (T) which are equivalent and non-isomorphic over F 2 (T) and non-equivalent over F 4 (T) , solving a particular Inverse Galois Problem. [ABSTRACT FROM AUTHOR]

Let K be an algebraically closed field of characteristic p > 0. A pressing problem in the theory of algebraic curves is the determination of the p -rank of a (nonsingular, projective, irreducible) curve X over K. This birational invariant affects arithmetic and geometric properties of X , and its fundamental role in the study of the automorphism group Aut (X) has been noted by many authors in the past few decades. In this paper, we provide an extensive study of the p -rank of curves of Fermat type y m = x n + 1 over K = F ¯ p. We determine a combinatorial formula for this invariant in the general case and show how this leads to explicit formulas of the p -rank of several such curves. By way of illustration, we present explicit formulas for more than twenty subfamilies of such curves, where m and n are generally given in terms of p. We also show how the approach can be used to compute the p -rank of other types of curves. [ABSTRACT FROM AUTHOR]

This paper is a survey on sheaves of arithmetic differential operators with congruence level on formal schemes. We present first results about these sheaves and discuss some examples. [ABSTRACT FROM AUTHOR]

A basic problem in transcendental number theory is to determine the arithmetic properties of values of special functions. Many special functions, such as Bessel functions and certain hypergeometric functions, are E -functions which are a natural generalization of the exponential function and satisfy certain linear differential equations. In this case, there exists an algorithm which determines if f (α) is transcendental or algebraic if f (z) is an E -function and α ∈ Q ‾ ⁎ is a non-zero algebraic number. In this paper, we consider the analogous question when f (z) satisfies an integral equation, in particular, a Fredholm integral equation of the first or second kind where the kernel and forcing term satisfy strong arithmetic properties. We show that in both periodic and non-periodic cases, there exists no algorithm to determine if f (0) ∈ Q is rational. Our results are an application of the undecidability of the Generalized Collatz Problem due to Conway [6]. [ABSTRACT FROM AUTHOR]

A Diophantine m -tuple with elements in the field K is a set of m non-zero (distinct) elements of K with the property that the product of any two distinct elements is one less than a square in K. Let X : (x 2 − 1) (y 2 − 1) (z 2 − 1) = k 2 , be an affine variety over K. Its K -rational points parametrize Diophantine triples over K such that the product of the elements of the triple that corresponds to the point (x , y , z , k) ∈ X (K) is equal to k. We denote by X ‾ the projective closure of X and for a fixed k by X k a variety defined by the same equation as X. In this paper, we try to understand what can the geometry of varieties X k , X and X ‾ tell us about the arithmetic of Diophantine triples. First, we prove that the variety X ‾ is birational to P 3 which leads us to a new rational parametrization of the set of Diophantine triples. Next, specializing to finite fields, we find a correspondence between a K3 surface X k for a given k ∈ F p × in the prime field F p of odd characteristic and an abelian surface which is a product of two elliptic curves E k × E k where E k : y 2 = x (k 2 (1 + k 2) 3 + 2 (1 + k 2) 2 x + x 2). We derive an explicit formula for N (p , k) , the number of Diophantine triples over F p with the product of elements equal to k. Moreover, we show that the variety X ‾ admits a fibration by rational elliptic surfaces and from it we derive the formula for the number of points on X ‾ over an arbitrary finite field F q. Using it we reprove the formula for the number of Diophantine triples over F q from [DK21]. Curiously, from the interplay of the two (K3 and rational) fibrations of X ‾ , we derive the formula for the second moment of the elliptic surface E k (and thus confirming Steven J. Miller's Bias conjecture in this particular case) which we describe in terms of Fourier coefficients of a rational newform generating S 4 (Γ 0 (8)). Finally, in the Appendix, Luka Lasić defines circular Diophantine m -tuples, and describes the parametrization of these sets. For m = 3 this method provides an elegant parametrization of Diophantine triples. [ABSTRACT FROM AUTHOR]

A powerful new perspective in the analysis of absolute Galois groups has recently emerged from the study of Galois modules related to classical parameterizing spaces of certain Galois extensions. The recurring trend in these decompositions is their stunning simplicity: almost all summands are free over some quotient ring. The non-free summands which appear are exceptional not only because they are different in form, but because they play the key role in controlling arithmetic conditions that allow the remaining summands to be easily described. In this way, these exceptional summands are the lynchpin for a bevy of new properties of absolute Galois groups that have been gleaned from these surprising decompositions. In one such recent decomposition, a remarkable new exceptional summand was discovered which exhibited interesting properties that have not been seen before. The exceptional summand is drawn from a particular finite family that has not yet been investigated. The main goal of this paper is to introduce this family of modules and verify their indecomposability. We believe this module will be of interest to people working in Galois theory, representation theory, combinatorics, and general algebra. The analysis of these modules includes some interesting new tools, including analogs of p -adic expansions. [ABSTRACT FROM AUTHOR]

In 1999, Frank Schmidt noted that the number of partitions of integers with distinct parts in which the first, third, fifth, etc., summands add to n is equal to p (n) , the number of partitions of n. The object of this paper is to provide a context for this result which leads directly to many other theorems of this nature and which can be viewed as a continuation of our work on elongated partition diamonds. Again generating functions are infinite products built by the Dedekind eta function which, in turn, lead to interesting arithmetic theorems and conjectures for the related partition functions. [ABSTRACT FROM AUTHOR]

At the turn of the 19th century, arithmetic underwent a reform. Lagrange and Delambre developed diverse reflections on this discipline and contributed to the revival of the study of the ancient Greeks' mathematics. Through analyzing Lagrange's lectures on arithmetic and the popular arithmetic treatises, I reestablish how Lagrange's innovative ideas were adopted in the later books. Moreover, this paper clarifies the role that "complex numbers" played in these treatises and investigates Delambre's memoir on the ancient Greek arithmetic, so as to reveal the way in which Delambre read the ancients' texts and reconstructed their arithmetic using "complex numbers". [ABSTRACT FROM AUTHOR]

In this paper we establish accuracy bounds of Prony's method (PM) for recovery of sparse measures from incomplete and noisy frequency measurements, or the so-called problem of super-resolution, when the minimal separation between the points in the support of the measure may be much smaller than the Rayleigh limit. In particular, we show that PM is optimal with respect to the previously established min-max bound for the problem, in the setting when the measurement bandwidth is constant, with the minimal separation going to zero. Our main technical contribution is an accurate analysis of the inter-relations between the different errors in each step of PM, resulting in previously unnoticed cancellations. We also prove that PM is numerically stable in finite-precision arithmetic. We believe our analysis will pave the way to providing accurate analysis of known algorithms for the super-resolution problem in full generality. [ABSTRACT FROM AUTHOR]

The Shimura correspondence is a fundamental tool in the study of half-integral weight modular forms. In this paper, we prove a Shimura-type correspondence for spaces of half-integral weight cusp forms which transform with a power of the Dedekind eta multiplier twisted by a Dirichlet character. We prove that the lift of a cusp form of weight λ + 1 / 2 and level N has weight 2 λ and level 6 N , and is new at the primes 2 and 3 with specified Atkin-Lehner eigenvalues. This precise information leads to arithmetic applications. For a wide family of spaces of half-integral weight modular forms we prove the existence of infinitely many primes ℓ which give rise to quadratic congruences modulo arbitrary powers of ℓ. [ABSTRACT FROM AUTHOR]

This paper presents a model reduction technique to determine the optimal simulation model for high-dimensional systems within the confines of finite word-length computing. Such an optimal model is characterized by having minimal output error covariances in computational simulations when compared with the outputs of the physics model in reality. The round-off noise models for both floating- and fixed-point arithmetic are introduced first. Then, round-off error signals are incorporated into the dynamics models, representing the effects of finite precision. Analytical solutions for the simulation error covariance, perceived as a union of dynamics and round-off errors, are provided. Based on these insights, a general algorithm is developed to identify where the total simulation error is minimized, indicating the optimal model size. Two tensegrity structures, a two-dimensional morphing airfoil and a three-dimensional deployable tensegrity Levy cable dome, are analyzed to demonstrate this method. Our results show that large-scale models can be affected by increasing round-off errors in computational simulations, which might result in less accurate outcomes. However, by opting for a reduced-order simulation model, computational simulation performance can be improved considerably. Besides tensegrrity, the approach introduced in this research can be adapted to other high-dimensional dynamical system simulations. [ABSTRACT FROM AUTHOR]

It is well-known that the central path of semi-definite optimization, unlike linear optimization, has no analytic extension to μ = 0 in the absence of the strict complementarity condition. In this paper, we consider a reparametrization μ ↦ μ ρ , with ρ being a positive integer, that recovers the analyticity of the central path at μ = 0. We investigate the complexity of computing ρ using algorithmic real algebraic geometry and the theory of complex algebraic curves. We prove that the optimal ρ is bounded by 2 O (m 2 + n 2 m + n 4) , where n is the matrix size and m is the number of affine constraints. Our approach leads to a symbolic algorithm, based on the Newton-Puiseux algorithm, which computes a feasible ρ using 2 O (m + n 2) arithmetic operations. [ABSTRACT FROM AUTHOR]

In his 1984 AMS memoir, Andrews introduced the family of k -colored generalized Frobenius partition functions. For any positive integer k , let c ϕ k (n) denote the number of k -colored generalized Frobenius partitions of n. Among many other things, Andrews proved that for any n ≥ 0 , c ϕ 2 (5 n + 3) ≡ 0 (mod 5). Since then, many scholars subsequently considered congruence properties of various k -colored generalized Frobenius partition functions, typically with a small number of colors. In 2019, Chan, Wang and Yang systematically studied arithmetic properties of C Φ k (q) with 2 ≤ k ≤ 17 by employing the theory of modular forms, where C Φ k (q) denotes the generating function of c ϕ k (n). We notice that many coefficients in the expressions of C Φ k (q) are not integers. In this paper, we first observe that C Φ k (q) is related to the constant term of a family of bivariable functions, then establish a general symmetric and recurrence relation on the coefficients of these bivariable functions. Based on this relation, we next derive many bivariable identities. By extracting and computing the constant terms of these bivariable identities, we establish the expressions of C Φ k (q) with integral coefficients. As an immediate consequence, we prove some infinite families of congruences satisfied by c ϕ k (n) , where k is allowed to grow arbitrary large. [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]

In this paper, we deal with the étale cohomology of a proper regular arithmetic scheme X with Z p (r) and Q p (r) -coefficients, where the coefficients are complexes of étale sheaves that the author introduced in [SH]. We will prove that the étale cohomology of X with Q p (r) -coefficients agrees with the Selmer group of Bloch-Kato for any r ≧ dim (X). Using this fundamental result, we further discuss an approach to the study of zeta values (or residue) at s = r , via the étale cohomology with Z p (r) -coefficients, relating Tamagawa number conjecture of Bloch-Kato with a zeta value formula. As a consequence, we will obtain an unconditional example of an arithmetic surface for which the residue of its zeta function at s = 2 is computed modulo rational numbers prime to p , for infinitely many p 's. [ABSTRACT FROM AUTHOR]

In this paper, I study Richard Dedekind and Heinrich Weber's 1882 Theorie der algebraischen Funktionen einer Veränderlichen , with a focus on the inherently arithmetical aspects of their work. I show that their paper provides an arithmetical rewriting of Riemannian function theory, i.e. a rewriting built on elementary arithmetical notions such as divisibility. I start with contextual elements concerning what is “arithmetical”, to put Dedekind and Weber's works into perspective from that viewpoint. Then, through a detailed analysis of the 1882 paper and using elements of their correspondence, I suggest that Dedekind and Weber deploy a strategy of rewriting parts of mathematics using arithmetic, and that this strategy is essentially related to Dedekind's specific conception of numbers and arithmetic as intrinsically linked to the human mind. [ABSTRACT FROM AUTHOR]

This paper presents new and fast differential addition (i.e., the addition of two points with the known difference) and doubling formulas, as the core step in Montgomery scalar multiplication, for various forms of elliptic curves over binary fields. The formulas are provided for binary Edwards , binary Hessian and binary Huff elliptic curves with cost of 5 M + 4 S + 1 D when the given difference point is in affine form. Here, M , S , D denote the costs of a field multiplication, a field squaring and a field multiplication by a constant, respectively. In this paper, we show that every ordinary elliptic curve E over binary field F 2 n has Lopez and Dahab differential addition and doubling formulas with suitable function. This paper also presents, new complete differential addition formulas for binary Edwards curves with cost of 5 M + 4 S + 2 D. Also, new complete Montgomery ladder and scalar multiplication algorithms are presented using the complete differential addition formulas. • Differential addition and doubling formulas. • Efficient arithmetic on elliptic curves. • Scalar multiplication algorithms using Montgomery ladder. • Isogeny based Lopez and Dahab-like formulas. [ABSTRACT FROM AUTHOR]

In this paper, we study arithmetic properties of weighted Catalan numbers. Previously, Postnikov and Sagan found conditions under which the 2-adic valuations of the weighted Catalan numbers are equal to the 2-adic valuations of the Catalan numbers. We obtain the same result under weaker conditions by considering a map from a class of functions to 2-adic integers. These methods are also extended to q -weighted Catalan numbers, strengthening a previous result by Konvalinka. Finally, we prove some results on the periodicity of weighted Catalan numbers modulo an integer and apply them to the specific case of the number of combinatorial types of Morse links. Many open questions are mentioned. [ABSTRACT FROM AUTHOR]

Multiplicative relations between the roots of a polynomial in Q [ x ] have drawn much attention in the field of arithmetic and algebra, while the problem of computing these relations is interesting to researchers in many other fields. In this paper, a sufficient condition is given for a polynomial f ∈ Q [ x ] to have only trivial multiplicative relations between its roots, which is a generalization of those sufficient conditions proposed in Smyth (1986) , Baron et al. (1995) and Dixon (1997). Based on the new condition, a subset E ⊂ Q [ x ] is defined and proved to be generic (i.e. , the set Q [ x ] \ E is very small). We develop an algorithm deciding whether a given polynomial f ∈ Q [ x ] is in E and returning a basis of the lattice consisting of the multiplicative relations between the roots of f whenever f ∈ E. The numerical experiments show that the new algorithm is very efficient for the polynomials in E. A large number of polynomials with much higher degrees, which were intractable before, can be handled successfully with the algorithm. [ABSTRACT FROM AUTHOR]

In their 2006 seminal paper in Distributed Computing, Angluin et al. present a construction that, given any Presburger predicate, outputs a leaderless population protocol that decides the predicate. The protocol for a predicate of size m runs in O (m ⋅ n 2 log ⁡ n) expected number of interactions, which is almost optimal in n , the number of interacting agents. However, the number of states is exponential in m. Blondin et al. presented at STACS 2020 another construction that produces protocols with a polynomial number of states, but exponential expected number of interactions. We present a construction that produces protocols with O (m) states that run in expected O (m 7 ⋅ n 2) interactions, optimal in n , for all inputs of size Ω (m). For this, we introduce population computers, a generalization of population protocols, and show that our computers for Presburger predicates can be translated into fast and succinct population protocols. [ABSTRACT FROM AUTHOR]

We consider the existence of parabolic invariant tori for a class of quasi-periodically forced analytic skew-product maps φ : R n × T d → R n × T d : φ ( z θ ) = ( z + ϕ (z) + h (z , θ) + ϵ f (z , θ) θ + ω ) , where ϕ : R n → R n is a homogeneous function of degree l with l ≥ 2 and h = O (| z | l + 1). We obtain the following results: (a) For n = 1 , l being odd and ϵ sufficiently small, parabolic invariant tori exist if ω satisfies the Brjuno-Rüssmann's non-resonant condition. (b) For n = 1 , and ϵ sufficiently small, parabolic invariant tori also exist if one of the following conditions holds: (i) First order average is non-zero, first order non-average part is small enough and the forcing frequency ω does not need any arithmetic condition. (ii) First order average is non-zero and ω satisfies the Brjuno-type weak non-resonant condition; (iii) l = 2 , first order average is zero, both first and second order non-average parts are small enough and ω satisfying Brjuno-type weak non-resonant condition; (iv) l > 2 , first order average is zero, the second order average is non-zero, both first and second order non-average parts are small enough and ω satisfies the Brjuno-type weak non-resonant condition. (c) In the case n > 1 , if first order average belongs to the interior of the range of ϕ , S p e c (D ϕ) ∩ i R = ∅ , first order non-average part is small enough and the forcing frequency ω does not need any arithmetic condition, then the quasi-periodically forced skew-product maps above admit parabolic invariant tori for ϵ sufficiently small. The main methods of this paper are KAM theory and fixed point theorem, which are finally shown that it can be directly applied to the existence problem of quasi-periodic response solutions of degenerate harmonic oscillators. [ABSTRACT FROM AUTHOR]

The reconstruction of high-dimensional sparse signals is a challenging task in a wide range of applications. In order to deal with high-dimensional problems, efficient sparse fast Fourier transform algorithms are essential tools. The second and third authors have recently proposed a dimension-incremental approach, which only scales almost linear in the number of required sampling values and almost quadratic in the arithmetic complexity with respect to the spatial dimension d. Using reconstructing rank-1 lattices as sampling scheme, the method showed reliable reconstruction results in numerical tests but suffers from relatively large numbers of samples and arithmetic operations. Combining the preferable properties of reconstructing rank-1 lattices with small sample and arithmetic complexities, the first author developed the concept of multiple rank-1 lattices. In this paper, both concepts — dimension-incremental reconstruction and multiple rank-1 lattices — are coupled, which yields a distinctly improved high-dimensional sparse fast Fourier transform. Moreover, the resulting algorithm is analyzed in detail with respect to success probability, number of required samples, and arithmetic complexity. In comparison to single rank-1 lattices, the utilization of multiple rank-1 lattices results in a reduction in the complexities by an almost linear factor with respect to the sparsity. Various numerical tests confirm the theoretical results, the high performance, and the reliability of the proposed method. [ABSTRACT FROM AUTHOR]

This paper studies the Sanskrit and Malayalam versions of Citrabhānu's Twenty-One Problems: a discussion of quadratic and cubic problems from 16th-century Kerala. It reviews the differences in the approaches of the two versions, highlighting the distinction between the Sanskrit indeterminate integer-remainder arithmetical techniques and the Malayali fixed point iterations. The paper concludes with some speculations on the possible transmission of algebraic knowledge between Kerala and the west. [ABSTRACT FROM AUTHOR]

According to the real τ -conjecture, the number of real roots of a sum of products of sparse univariate polynomials should be polynomially bounded in the size of such an expression. It is known that this conjecture implies a superpolynomial lower bound on the arithmetic circuit complexity of the permanent. In this paper, we use the Wronksian determinant to give an upper bound on the number of real roots of sums of products of sparse polynomials of a special form. We focus on the case where the number of distinct sparse polynomials is small, but each polynomial may be repeated several times. We also give a deterministic polynomial identity testing algorithm for the same class of polynomials. Our proof techniques are quite versatile; they can in particular be applied to some sparse geometric problems that do not originate from arithmetic circuit complexity. The paper should therefore be of interest to researchers from these two communities (complexity theory and sparse polynomial systems). [ABSTRACT FROM AUTHOR]

We construct, for specific imaginary quadratic number fields with class number 1, an arithmetic site of Connes-Consani type. The main difficulty here is that their constructions and part of their results strongly rely on the natural order existing on real numbers which is compatible with basic arithmetic operations. We first define what we call an arithmetic site for such number fields, we then calculate the points of those arithmetic sites and we express them in terms of the adèle class space considered by Connes to give a spectral interpretation of zeroes of Hecke L functions of number fields. We get that the points of our arithmetic site are related not only to the zeroes of the Dedekind zeta function of the number field considered but also to other Hecke L functions involving non-trivial characters at the archimedean place. We then study the relation between the spectrum of the ring of integers of the number field and the arithmetic site. An appendix by Alain Connes shows that the compatibility with the Frobenius yields the correct notion of morphisms of semirings in this context. For a video summary of this paper, please visit https://www.youtube.com/watch?v=lWF9gbfvcYY&feature=youtu.be. [ABSTRACT FROM AUTHOR]

Basic Linear Algebra Subprograms (BLAS) are the building blocks for various numerical algorithms and are widely used in scientific computations. However, some linear algebra applications need more precision than the standard double precision available in most existing BLAS libraries. In this paper, we implement and evaluate multiple-precision scalar and vector BLAS functions on graphics processing units (GPUs). We use the residue number system (RNS) to represent arbitrary length floating-point numbers. The non-positional nature of RNS enables parallelism in multiple-precision arithmetic and makes RNS a good tool for high-performance computing applications. We first present new data-parallel algorithms for multiplying and adding RNS-based floating-point representations. Next, we suggest algorithms for multiple-precision vectors specially designed for parallel computations on GPUs. Using these algorithms, we develop and evaluate four GPU-accelerated multiple-precision BLAS functions, ASUM, DOT, SCAL, and AXPY. It is shown through experiments that in many cases, the implemented functions achieve significantly better performance compared to existing multiple-precision software for CPU and GPU. • Residue number system enables parallelism in arithmetic with multiple precision. • New algorithms for arbitrary length floating-point numbers are presented. • Multiple-precision ASUM, DOT, SCAL, and AXPY are implemented using CUDA and evaluated. • We show that multiple-precision BLAS can be accelerated using GPU. [ABSTRACT FROM AUTHOR]

We study the topos of sets equipped with an action of the monoid of regular 2 × 2 matrices over the integers. In particular, we show that the topos-theoretic points are given by the double quotient GL 2 (Z ˆ) \ M 2 (A f) / GL 2 (Q) , so they classify the groups Z 2 ⊆ A ⊆ Q 2 up to isomorphism. We determine the topos automorphisms and then point out the relation with Conway's big picture and the work of Connes and Consani on the Arithmetic Site. As an application to number theory, we show that classifying extensions of Q by Z up to isomorphism relates to Goormaghtigh conjecture. For a video summary of this paper, please visit https://youtu.be/5cA1MOG26ng. [ABSTRACT FROM AUTHOR]

Leibniz published his Euclidean construction of a catenary in Acta Eruditorum of June 1691, but he was silent about the methods used to discover it. He explained how he used his differential calculus only in a private letter to Rudolph Christian von Bodenhausen and specified a number that was key to his construction, 2.7182818, with no clue about how he calculated it. Apparently, the calculations were never divulged to anyone but were discovered later among his personal papers. They may be the earliest record of an accurate approximation of the number we label e and a demonstration of its role as the base of the natural logarithm and exponential function. This, at that time, was a remarkably precise estimate for e , accomplished more than 22 years before Roger Cotes published e to 12 significant digits, and some 57 years before Euler's treatment of the logarithm in his Introductio in Analysin Infinitorum. The Leibniz construction reveals a hyperbolic cosine built on an exponential curve based on his estimated value, which implies that he understood the number as the base of his logarithmic curve. The sheets of arithmetic used by Leibniz preserved at the Gottfried Wilhelm Leibniz Bibliothek (GWLB) in Hannover, confirm this. Those sheets show how Leibniz calculated e and applied it to his catenary construction. The data actually yield e to 12 significant figures: 2.71828182845, missed by Leibniz because of a misplaced decimal point. We summarize the construction and examine the worksheets. The unpublished methods seem entirely modern to us and could serve as enrichening examples in modern calculus texts. [ABSTRACT FROM AUTHOR]

In this paper, we study twisted arithmetic divisors on the modular curve X 0 (N) with N square-free. For each pair (Δ , r) , where Δ ≡ r 2 mod 4 N and Δ is a fundamental discriminant, we construct a twisted arithmetic theta function ϕ ˆ Δ , r (τ) which is a generating function of arithmetic twisted Heegner divisors. We prove that the arithmetic pairing 〈 ϕ ˆ Δ , r (τ) , ω ˆ N 〉 is equal to the special value, rather than the derivative, of some Eisenstein series, thanks to some cancellation, where ω ˆ N is a normalized metric Hodge line bundle. We also prove the modularity of ϕ ˆ Δ , r (τ). [ABSTRACT FROM AUTHOR]

In this article, we undertake the study of the function F (x ; u , v) , which we refer to as the Herglotz-Zagier-Novikov function. This function appears in Novikov's work on the Kronecker limit formula, which was motivated by Zagier's paper where he obtained the Kronecker limit formula in terms of the Herglotz function F (x). Two, three, and six-term functional equations satisfied by F (x ; u , v) are exhibited. These are cohomological relations coming from the action of an involution and SL 2 (Z) on C × D 1 2 (the unit circle D 1). We also provide the special values of F (x ; u , v) at rational arguments of x. Importantly, F (x ; u , v) serves as a unified generalization of three other interesting functions, namely F (x) , J (x) , and T (x) , which also appear in various Kronecker limit formulas and are previously studied by Cohen, Herglotz, Muzaffar and Williams, and Radchenko and Zagier. Consequently, our study not only reveals the numerous elegant properties of F (x ; u , v) but also helps us to further develop the theories of functions related to its special cases such as J (x) and T (x). [ABSTRACT FROM AUTHOR]

In this paper we study the moduli spaces of nodal sextic curves. We realize each irreducible component of the GIT space of sextic curves with given number of nodes as an open subspace of type IV arithmetic quotients. We then focus on the compactifications of the moduli spaces, one side is the geometric (GIT) compactifications, the other side is the Hodge theoretic compactifications such as Looijenga compactifications and Baily-Borel compactifications. The main result is the isomorphism between GIT and Looijenga compactifications. Some examples are closely related to del Pezzo surfaces. We also extend our results to moduli of nodal sextic curves with specified symmetry. [ABSTRACT FROM AUTHOR]

In this paper, we study some arithmetic properties of certain Toeplitz-type matrices over finite fields. For example, suppose { α 1 , ... , α n } are all the nonzero k -th powers in a finite field and for 1 ⩽ i , j ⩽ n , we set x i j = { 1 α i − α j if i ≠ j , 0 if i = j. We give an explicit formula for the characteristic polynomial of [ x i j ] 1 ⩽ i , j ⩽ n and confirm several conjectures posed by Zhi-Wei Sun. [ABSTRACT FROM AUTHOR]

Abstract: This paper is devoted to (discrete) p-adic dynamical systems, an important domain of algebraic and arithmetic dynamics. We consider the following open problem from theory of p-adic dynamical systems. Given continuous function . Let us represent it via special convergent series, namely van der Put series. How can one specify whether this function is measure-preserving or not for an arbitrary p? In this paper, for any prime p, we present a complete description of all compatible measure-preserving functions in the additive form representation. In addition we prove the criterion in terms of coefficients with respect to the van der Put basis determining whether a compatible function preserves the Haar measure. [Copyright &y& Elsevier]

Abstract: The notation means the finite sum of all numbers as ranges over the integers that divide including and . The purpose of this paper is to define a theory of summability suggested by this type of sum and an associated theory of convergence. This paper studies five sequence spaces that are related to the arithmetical summability thus defined. [Copyright &y& Elsevier]

Abstract: In the present paper we investigate the sum-of-digits function for canonical number systems. We are interested in its re-partition in arithmetic progressions and similar results for sum sets. The original problem goes back to Gelfond, who proved the independence of the distribution of the digits and their sum of digits. The present paper extends results by Thuswaldner and Mauduit and Sárközy to this kind of numeration systems. [Copyright &y& Elsevier]