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We study extreme values of Dirichlet polynomials with multiplicative coefficients, namely D N (t) : = D f , N (t) = 1 N ∑ n ⩽ N f (n) n i t , where f is a completely multiplicative function with | f (n) | = 1 for all n ∈ N. We use Soundararajan's resonance method to produce large values of | D N (t) | uniformly for all such f. In particular, we improve a recent result of Benatar and Nishry, where they establish weaker lower bounds and only for almost all such f. [ABSTRACT FROM AUTHOR]

We consider finite two-player normal form games with random payoffs. Player A's payoffs are i.i.d. from a uniform distribution. Given p ∈ [ 0 , 1 ] , for any action profile, player B's payoff coincides with player A's payoff with probability p and is i.i.d. from the same uniform distribution with probability 1 − p. This model interpolates the model of i.i.d. random payoff used in most of the literature and the model of random potential games. First we study the number of pure Nash equilibria in the above class of games. Then we show that, for any positive p , asymptotically in the number of available actions, best response dynamics reaches a pure Nash equilibrium with high probability. [ABSTRACT FROM AUTHOR]

Llibre et al. in [ J. Nonlinear Sci. , 2019] studied the number of limit cycles of piecewise linear Liénard system with one anti-saddle at the central zone in some parameter regions. However, the exact number and hyperbolicity of limit cycles were still unclear as in these cases of parameter. The goal of this paper is to give the exact number of limit cycles and the hyperbolicity of limit cycles in these parameter cases, where the maximum number of limit cycles is two. Moreover, all corresponding global phase portraits in the Poincaré disc are obtained. [ABSTRACT FROM AUTHOR]

Every integer greater than two can be expressed as the sum of a prime and a square-free number. Expanding on recent work, we provide explicit and asymptotic results when divisibility conditions are imposed on the square-free number. For example, we show for odd k ≤ 10 5 and even k ≤ 2 ⋅ 10 5 that any even integer n ≥ 40 can be expressed as the sum of a prime and a square-free number coprime to k. We also discuss applications to other Goldbach-like problems. [ABSTRACT FROM AUTHOR]

In this paper, we mainly answer a Lampe's question [3] about the solutions of a Diophantine equation, that is, we give a criterion to determine which solutions of the Diophantine equation are in the orbit of the initial solution (ϵ , ϵ , ϵ , ϵ , ϵ) under the actions of the group G which is defined by mutations of a cluster algebra. In order to do this, using a rational map φ , we transform the Diophantine equation to a related equation whose all positive integral solutions form the orbit of an initial solution φ (ϵ , ϵ , ϵ , ϵ , ϵ) = (3 , 4 , 4) under the actions of the group G ˜ , and the set S (3 , 4 , 4) is shown to be the orbit of (ϵ , ϵ , ϵ , ϵ , ϵ) under the actions of a subgroup of G. Then the criterion is proved as the main conclusion. [ABSTRACT FROM AUTHOR]

Given two non-CM newforms with integral Fourier coefficients, in this paper we study the number of distinct prime divisors of their Fourier coefficients in a probability way. Based on a multivariate version of the Erdős-Kac theorem, using the Galois representations attached to newforms and the effective Chebotarev density theorem, and assuming the generalized Riemann hypothesis, we show that the distribution of the number of distinct primes dividing the Fourier coefficients behaves like the standard multivariate normal distribution if these newforms are not twists of each other. As a consequence, we prove a multiplicity one result for modular forms under the generalized Riemann hypothesis. [ABSTRACT FROM AUTHOR]

The main theorem of the paper [2] requires a restriction on the prime p. This statement is being modified based on a certain conjecture on the p -adic zeta functions. [ABSTRACT FROM AUTHOR]

In this work we develop a framework that enables the use of Shimura curve parametrizations of elliptic curves to approach the abc conjecture, leading to a number of new unconditional applications over Q and, more generally, totally real number fields. Several results of independent interest are obtained along the way, such as bounds for the Manin constant, a study of the congruence number, extensions of the Ribet-Takahashi formula, and lower bounds for the L 2 -norm of integral quaternionic modular forms. The methods require a number of tools from Arakelov geometry, analytic number theory, Galois representations, complex-analytic estimates on Shimura curves, automorphic forms, known cases of the Colmez conjecture, and results on generalized Fermat equations. [ABSTRACT FROM AUTHOR]

We point out an error in [2]. [ABSTRACT FROM AUTHOR]

Cavicchioli, O'Brien, and Spaggiari studied the number of isomorphism classes of irreducible groups of Fibonacci type as a function σ (n) of the number of generators n. In the case n = p l , where p is prime and l ≥ 1 , n ≠ 2 , 4 , they conjectured a function C (p l) , that is polynomial in p , for the value of σ (p l). We prove that C (p l) is an upper bound for σ (p l). We introduce a function τ (n) for the number of abelianised groups and conjecture a function D (p l) , that is polynomial in p , for the value of τ (p l) , when p l ≠ 2 , 4 , 5 , 7 , 8 , 13 , 23. We prove that D (p l) is an upper bound for τ (p l). We pose three questions that ask if particular pairs of groups with common abelianisations are non-isomorphic. We prove that if τ (p l) = D (p l) and each of these questions has a positive answer then σ (p l) = C (p l). [ABSTRACT FROM AUTHOR]

Clemm and Trebat-Leder (2014) proved that the number of quadratic number fields with absolute discriminant bounded by x over which there exist elliptic curves with good reduction everywhere and rational j -invariant is ≫ x log − 1 2 ⁡ (x). In this paper, we assume the abc -conjecture to show the sharp asymptotic ∼ c x log − 1 2 ⁡ (x) for this number, obtaining formulae for c in both the real and imaginary cases. Our method has three ingredients: (1) We make progress towards a conjecture of Granville: Given a fixed elliptic curve E / Q with short Weierstrass equation y 2 = f (x) for reducible f ∈ Z [ x ] , we show that the number of integers d , | d | ≤ D , for which the quadratic twist d y 2 = f (x) has an integral non-2-torsion point is at most D 2 3 + o (1) , assuming the abc -conjecture. (2) We apply the Selberg–Delange method to obtain a Tauberian theorem which allows us to count integers satisfying certain congruences while also being divisible only by certain primes. (3) We show that for a polynomially sparse subset of the natural numbers, the number of pairs of elements with least common multiple at most x is O (x 1 − ϵ) for some ϵ > 0. We also exhibit a matching lower bound. If instead of the abc -conjecture we assume a particular tail bound, we can prove all the aforementioned results and that the coefficient c above is greater in the real quadratic case than in the imaginary quadratic case, in agreement with an experimentally observed bias. [ABSTRACT FROM AUTHOR]

In this paper we initiate a systematic study of the Turán problem for edge-ordered graphs. A simple graph is called edge-ordered if its edges are linearly ordered. This notion allows us to study graphs (and in particular their maximum number of edges) when a subgraph is forbidden with a specific edge-order but the same underlying graph may appear with a different edge-order. We prove an Erdős-Stone-Simonovits-type theorem for edge-ordered graphs—we identify the relevant parameter for the Turán number of an edge-ordered graph and call it the order chromatic number. We establish several important properties of this parameter. We also study Turán numbers of edge-ordered paths, star forests and the cycle of length four. We make strong connections to Davenport-Schinzel theory, the theory of forbidden submatrices, and show an application in discrete geometry. [ABSTRACT FROM AUTHOR]

The property of Sheats compositions stated in Corollary 4.7 is false. As it enters into the proof of Theorem 5.7, one of the main results, we show that this proof is in fact independent of Corollary 4.7 and its derivatives Corollaries 4.9 and 4.10. Some other less important inaccuracies and typos are corrected. [ABSTRACT FROM AUTHOR]

Let p be a prime, and let f ∈ Z [ x ] be a non-constant polynomial whose leading coefficient is prime to p and that f has no repeated root modulo p. Suppose the mod- p irreducible factors of f all have the same degree. Then there are infinitely many Abelian extensions of number fields L / K such that O L / p O L ≃ (Z / p) [ x ] / f , where O L denotes the ring of integers of L and p ⊂ O K is a maximal ideal with O K / p O K ≃ Z / p. Moreover, under the generalized Riemann hypothesis for the Dedekind zeta functions of number fields, for fixed deg ⁡ (f) we can construct Abelian lifts in polynomial time in log ⁡ p. This confirms in a stronger form a question raised by Mihăilescu and Vuletescu concerning polynomial cyclic algebras. The proofs make use of a weak form of Artin's conjecture on primitive roots and polynomial time polynomial factoring algorithms over number fields. [ABSTRACT FROM AUTHOR]

We study one-dimensional Cohen-Macaulay rings whose trace ideal of the canonical module is as small as possible. In this paper we call such rings far-flung Gorenstein rings. We investigate far-flung Gorenstein rings in relation with the endomorphism algebras of the maximal ideals and numerical semigroup rings. We show that the solution of the Rohrbach problem in additive number theory provides an upper bound for the multiplicity of far-flung Gorenstein numerical semigroup rings. Reflexive modules over far-flung Gorenstein rings are also studied. [ABSTRACT FROM AUTHOR]

For a matrix incidence algebra over a field, there is a full description of its generating subsets. We generalise this result to the case of matrix incidence rings over an arbitrary unital associative ring. Generating subsets of the following type are under consideration: if one adds all scalar matrices to the subset, then this greater set will generate the incidence ring. We obtain the full characterisation of such subsets. Moreover, their minimum cardinality is calculated precisely when the ring of coefficients is simple or semilocal. In order to obtain these results we introduce the graph of full differences of a ring and study its clique number. [ABSTRACT FROM AUTHOR]

In this paper, we compute the q -adic slopes of the L-functions of an important class of exponential sums arising from analytic number theory. Our main tools include Adolphson-Sperber's work on toric exponential sums and Wan's decomposition theorems. This paper is a generalization of our previous paper. [ABSTRACT FROM AUTHOR]

Given positive integers n ⩾ 2 k , the Kneser graph K G n , k is a graph whose vertex set is the collection of all k -element subsets of the set { 1 , ... , n } , with edges connecting pairs of disjoint sets. One of the classical results in combinatorics, conjectured by Kneser and proved by Lovász, states that the chromatic number of K G n , k is equal to n − 2 k + 2. In this paper, we study the chromatic number of the random Kneser graph K G n , k (p) , that is, the graph obtained from K G n , k by including each of the edges of K G n , k independently and with probability p. We prove that, for any fixed k ⩾ 3 , χ (K G n , k (1 / 2)) = n − Θ ( log 2 ⁡ n 2 k − 2) , as well as χ (K G n , 2 (1 / 2)) = n − Θ ( log 2 ⁡ n ⋅ log 2 ⁡ log 2 ⁡ n 2). We also prove that, for k ⩾ (1 + ε) log ⁡ log ⁡ n , we have χ (K G n , k (1 / 2)) ⩾ n − 2 k − 10. This significantly improves previous results on the subject, obtained by Kupavskii and by Alishahi and Hajiabolhassan. The bound on k in the second result is also tight up to a constant. We also discuss an interesting connection to an extremal problem on embeddability of complexes. [ABSTRACT FROM AUTHOR]

The Katz-Sarnak Density Conjecture states that on a large scale, central point vanishing of suitably indexed L -functions is well-modeled by eigenvalues of random matrices drawn from the classical compact groups. In particular, over suitably restricted eigenvalues, an average order vanishing can bounded using only a chosen test function ϕ. For each type of symmetry, we find the optimal ϕ , and we give very explicit descriptions for small support. We compute the optimal bound on average rank, an improvement on the bounds in the existing literature. In general, we show that, for a given support, the optimal test function lies in a finite-dimensional family. We prove this using a new method of solving certain integral equations on finite intervals. We differentiate under the integral sign and solve the resulting system of delay differential equations using a priori symmetries of the solution. This method may prove useful in a variety of applications. [ABSTRACT FROM AUTHOR]

We show that, on the Riemann hypothesis, lim sup X → ∞ I (X) / X 2 ⩽ 0.8603 , where I (X) = ∫ X 2 X (ψ (x) − x) 2 d x. This proves (and improves on) a claim by Pintz from 1982. We also show unconditionally that 1.86 ⋅ 10 − 4 ⩽ I (X) / X 2 for sufficiently large X , and that the I (X) / X 2 has no limit as X → ∞. [ABSTRACT FROM AUTHOR]

The investigation of the graph G p associated with the finite p -groups of maximal class was initiated by Blackburn (1958) and became a deep and interesting research topic since then. Leedham-Green & McKay (1976–1984) introduced skeletons of G p , described their importance for the structural investigation of G p and exhibited their relation to algebraic number theory. Here we go one step further: we partition the skeletons into so-called Galois trees and study their general shape. In the special case p ⩾ 7 and p ≡ 5 mod 6 , we show that they have a significant impact on the periodic patterns of G p conjectured by Eick, Leedham-Green, Newman & O'Brien (2013). In particular, we use Galois trees to prove a conjecture by Dietrich (2010) on these periodic patterns. [ABSTRACT FROM AUTHOR]

Modal parameter estimation of operational structures is often a challenging task when confronted with unwanted distortions (outliers) in field measurements. Atypical observations present a problem to operational modal analysis (OMA) algorithms, such as stochastic subspace identification (SSI), severely biasing parameter estimates and resulting in misidentification of the system. Despite this predicament, no simple mechanism currently exists capable of dealing with such anomalies in SSI. Addressing this problem, this paper first introduces a novel probabilistic formulation of stochastic subspace identification (Prob-SSI), realised using probabilistic projections. Mathematically, the equivalence between this model and the classic algorithm is demonstrated. This fresh perspective, viewing SSI as a problem in probabilistic inference, lays the necessary mathematical foundation to enable a collection of new OMA approaches. To this end, a statistically robust SSI algorithm (robust Prob-SSI) is developed, capable of providing a principled and automatic way of handling outlying or anomalous data in the measured timeseries, such as may occur in field recordings, e.g. intermittent sensor dropout. Robust Prob-SSI is shown to outperform conventional SSI when confronted with 'corrupted' data, exhibiting improved identification performance and higher levels of confidence in the found poles when viewing stabilisation diagrams. Similar benefits are also demonstrated on the Z24 Bridge benchmark dataset, highlighting enhanced performance on measured systems. [Display omitted] • Novel probabilistic formulation of stochastic subspace identification. • Novel robust formulation of stochastic subspace identification. • Statistically robust to outliers in the time series responses. • Improvements to identification performance over Cov-SSI in such cases. • Demonstrated on Z24 data series. [ABSTRACT FROM AUTHOR]

We study the existence and the number of k -neighborly reorientations of an oriented matroid. This leads to k -variants of McMullen's problem and Roudneff's conjecture, the case k = 1 being the original statements. Adding to results of Larman and García-Colín, we provide new bounds on k -McMullen's problem and prove the conjecture for several ranks and k by computer. Further, we show that k -Roudneff's conjecture for fixed rank and k reduces to a finite case analysis. As a consequence we prove the conjecture for odd rank r and k = r − 1 2 as well as for rank 6 and k = 2 with the aid of the computer. [ABSTRACT FROM AUTHOR]

Recently, Dixit et al. [24] established a very elegant generalization of Hardy's theorem concerning the infinitude of zeros that the Riemann zeta function possesses at its critical line. By introducing a general transformation formula for the theta function involving the Bessel and modified Bessel functions of the first kind, we extend their result to a class of Dirichlet series satisfying Hecke's functional equation. In the process, we also find new generalizations of classical identities in Analytic number theory. [ABSTRACT FROM AUTHOR]

We combine the language of monoids with the language of preorders so as to refine some fundamental aspects of the classical theory of factorization and prove an abstract factorization theorem with a variety of applications. In particular, we obtain a generalization, from cancellative to Dedekind-finite (commutative or non-commutative) monoids, of a classical theorem on "atomic factorizations" that traces back to the work of P.M. Cohn in the 1960s; recover a theorem of D.D. Anderson and S. Valdes-Leon on "irreducible factorizations" in commutative rings; improve on a theorem of A.A. Antoniou and the author that characterizes atomicity in certain "monoids of sets" naturally arising from additive number theory and arithmetic combinatorics; and give a monoid-theoretic proof that every module of finite uniform dimension over a (commutative or non-commutative) ring is a direct sum of finitely many indecomposable modules (this is in fact a special case of a more general decomposition theorem for the objects of certain categories with finite products). [ABSTRACT FROM AUTHOR]

Let E be the infinite dimensional Grassmann algebra over a field F of characteristic zero. In this paper we investigate the structures of Z -gradings on E of full support. Using methods of elementary number theory, we describe the Z -graded polynomial identities for the so-called 2-induced Z -gradings on E of full support. As a consequence of this fact we provide examples of Z -gradings on E which are PI-equivalent but not Z -isomorphic. This is the first example of graded algebras with infinite support that are PI-equivalent, but non-isomorphic as graded algebras. We also present the notion of central Z -gradings on E and we show that their Z -graded polynomial identities are closely related to the Z 2 -graded polynomial identities of E. [ABSTRACT FROM AUTHOR]

• Integer lattice. • Discrepancy. • Annuli. • Gauss circle problem. Given a domain Ω ⊆ R 2 , let D (Ω , x , R) be the number of lattice points from Z 2 in R Ω − x , for R ≥ 1 and x : = (x 1 , x 2) ∈ T 2 , minus the area of R Ω: D (Ω , x , R) = # { (j , k) ∈ Z 2 : (j − x 1 , k − x 2) ∈ R Ω } − R 2 | Ω |. We call ∫ T 2 | D (Ω , x , R) | p d x the p -th moment of the discrepancy function D. In 2014, Huxley showed that for convex domains with sufficiently smooth boundary, the fourth moment of D is bounded by O (R 2 log ⁡ R) , and in 2019, Colzani, Gariboldi, and Gigante extended this result to higher dimensions. In this paper, our contribution is twofold: first, we present a simple direct proof of Huxley's 2014 result; second, we establish new estimates for the p -th moments of lattice point discrepancy of annuli of radius R , and any fixed thickness 0 < t < 1 for p ≥ 2. For a video summary of this paper, please visit https://youtu.be/YWIe1IBIi9Q. [ABSTRACT FROM AUTHOR]

We define a new metric between natural numbers induced by the ℓ ∞ norm of their unique prime signatures. In this space, we look at the natural analog of the number line and study the arithmetic function L ∞ (N) , which tabulates the cumulative sum of distances between consecutive natural numbers up to N in this new metric. Our main result is to identify the positive and finite limit of the sequence L ∞ (N) / N as the expectation of a certain random variable. The main technical contribution is to show with elementary probability that for K = 1 , 2 or 3 and ω 0 , ... , ω K ≥ 2 the following asymptotic density holds lim n → ∞ ⁡ | { M ≤ n : ‖ M − j ‖ ∞ < ω j for j = 0 , ... , K } | n = ∏ p : prime (1 − ∑ j = 0 K 1 p ω j ). This is a generalization of the formula for k -free numbers, i.e. when ω 0 = ... = ω K = k. The random variable is derived from the joint distribution when K = 1. As an application, we obtain a modified version of the prime number theorem. Our computations up to N = 10 12 have also revealed that prime gaps show a considerably richer structure than on the traditional number line. Moreover, we raise additional open problems, which could be of independent interest. [ABSTRACT FROM AUTHOR]

We study Tamagawa numbers of elliptic curves with torsion Z / 2 Z ⊕ Z / 14 Z over cubic fields and of elliptic curves with an n -isogeny over Q , for n ∈ { 6 , 8 , 10 , 12 , 14 , 16 , 17 , 18 , 19 , 37 , 43 , 67 , 163 }. Bruin and Najman [3] proved that every elliptic curve with torsion Z / 2 Z ⊕ Z / 14 Z over a cubic field is a base change of an elliptic curve defined over Q. We find that Tamagawa numbers of elliptic curves defined over Q with torsion Z / 2 Z ⊕ Z / 14 Z over a cubic field are always divisible by 142, with each factor 14 coming from a rational prime with split multiplicative reduction of type I 14 k , one of which is always p = 2. The only exception is the curve 1922.e2 , with c E = c 2 = 14. The same curves defined over cubic fields over which they have torsion subgroup Z / 2 Z ⊕ Z / 14 Z turn out to have the Tamagawa number divisible by 143. As for n -isogenies, Tamagawa numbers of elliptic curves with an 18-isogeny must be divisible by 4, while elliptic curves with an n -isogeny for the remaining n from the mentioned set must have Tamagawa numbers divisible by 2, except for finite sets of specified curves. [ABSTRACT FROM AUTHOR]

We correct an error found in Section 3.4.1 of Linear operators, the Hurwitz zeta function and Dirichlet L-functions published in JNT 217 (2020) 422–442. The error is related to the convergence of the inverse operator G − 1 defined in Section 3.3 and affects the statement and proof of Proposition 17. We provide a revised statement and proof. [ABSTRACT FROM AUTHOR]

The Birch and Swinnerton-Dyer conjecture – which is one of the seven million-dollar Clay Mathematics Institute Millennium Prize Problems – and its generalizations are a significant focus of number theory research. A 2017 article of Jetchev, Skinner and Wan proved a Birch and Swinnerton-Dyer formula at a prime p for certain rational elliptic curves of rank 1. We generalize and adapt that article's arguments to prove an analogous formula for certain modular forms. For newforms f of even weight higher than 2 with Galois representation V containing a Galois-stable lattice T , let W = V / T and let K be an imaginary quadratic field in which the prime p splits. Our main result is that under some conditions, the p -index of the size of the Shafarevich-Tate group of W with respect to the Galois group of K is twice the p -index of a logarithm of the Abel-Jacobi map of a Heegner cycle defined by Bertolini, Darmon and Prasanna. Significant original adaptations we make to the 2017 arguments are (1) a generalized version of a previous calculation of the size of the cokernel of a localization-modulo-torsion map, and (2) a comparison of different Heegner cycles. [ABSTRACT FROM AUTHOR]

Let F be a finite field, μ be a fixed additive character and s be an integer coprime to | F × |. For any a ∈ F , the corresponding Weil sum is defined to be W F , s (a) = ∑ x ∈ F μ (x s − a x). The Weil spectrum counts distinct values of the Weil sum as a runs through the invertible elements in the finite field. The value of these sums and the size of the Weil spectrum are of particular interest, as they link problems in coding and information theory to other areas of math such as number theory and arithmetic geometry. In the setting of Niho exponents, we examine the Weil sum, its bounds and its spectrum. As a consequence, we give a new proof to the Vanishing Conjecture of Helleseth (1971) on the presence of zero in the Weil spectrum in the case of Niho exponents. We also state a conjecture for when the Weil spectrum contains at least five elements, and prove it for a certain class of Weil sums. [ABSTRACT FROM AUTHOR]

We study the normal number of prime factors of a f (p) + a g (p) with p prime and f , g distinct Hecke eigenforms of weight two. Assuming a quasi-generalized Riemann hypothesis, we show that the normal order is log log p. We also obtain an estimate for the number of primes p for which a f (p) + a g (p) = 0. [ABSTRACT FROM AUTHOR]

In this paper we develop an arbitrary order Melnikov function to study limit cycles bifurcating from a periodic submanifold for autonomous piecewise smooth differential systems in R n with two zones separated by a hyperplane. This result not only extends some of the known results on the Melnikov theory in dimension and order but also compensates for some defects of the averaging theory in studying the limit cycle bifurcation of autonomous systems from a periodic submanifold. To demonstrate the application of our theoretical result and its superiority for some systems to the existing averaging theory, we study the maximum number of limit cycles bifurcating from an n -dimensional periodic submanifold caused by non-smooth centers of the fold-fold type, providing an upper bound for any order piecewise polynomial perturbations of degree m. Concerning the planar case of the unperturbed system, a piecewise Hamiltonian system, we obtain a better upper bound for piecewise polynomial Hamiltonian perturbations up to order two. The realizability of these upper bounds is also discussed. [ABSTRACT FROM AUTHOR]

We give an effective algorithm to determine the endomorphism ring of a Drinfeld module, both over its field of definition and over a separable or algebraic closure thereof. Using previous results we deduce an effective description of the image of the adelic Galois representation associated to the Drinfeld module, up to commensurability. We also give an effective algorithm to decide whether two Drinfeld modules are isogenous, again both over their field of definition and over a separable or algebraic closure thereof. Questions of efficiency are left completely out of consideration. [ABSTRACT FROM AUTHOR]

Gamma-positive polynomials frequently appear in finite geometries, algebraic combinatorics and number theory. Sagan and Tirrell (2020) [34] stumbled upon some unimodal sequences, which turn out to be alternating gamma-positive instead of gamma-positive. Motivated by this work, we first show that one can derive alternatingly γ -positive polynomials from γ -positive polynomials. We then prove the alternating γ -positivity and Hurwitz stability of several polynomials associated with the Narayana polynomials of types A and B. In particular, by introducing the definition of colored 2 × n Young diagrams, we provide combinatorial interpretations for three identities related to the Narayana numbers of type B. Finally, we present several identities involving the Eulerian polynomials of types A and B. [ABSTRACT FROM AUTHOR]

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

In recent years, there has been a lot of progress in obtaining non-trivial bounds for bilinear forms of Kloosterman sums in Z / m Z for arbitrary integers m. These results have been motivated by a wide variety of applications, such as improved asymptotic formulas for moments of L -functions. However, there has been very little work done in this area in the setting of rational function fields over finite fields. We remedy this and provide a number of new non-trivial bounds for bilinear forms of Kloosterman and Gauss sums in this setting, based on new bounds on the number of solutions to certain modular congruences in F q [ T ]. These improve upon some results of Macourt and Shparlinski (2019). [ABSTRACT FROM AUTHOR]

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

We close a gap in the literature by showing that the full covering groups of the Mathieu group M 22 and of its automorphism group Aut (M 22) occur as the Galois groups of Q -regular Galois extensions of Q (t). [ABSTRACT FROM AUTHOR]

We consider, for the first time, the exploration of dynamic graphs of arbitrary unknown topology. We study the number of agents necessary and sufficient to explore such graphs under the fully synchronous (Fsync) and the semi-synchronous (Ssync) activation schedulers. We prove that, under the minimal assumption on the dynamics, temporal connectivity , the number of agents sufficient to perform exploration depends on a parameter we call evanescence of the graph, and this number is tight. We then consider the stronger well-known assumption of 1-interval connectivity when the number of edges missing at each time is bounded. We provide tight bounds also in this setting, proving the existence of a difference between Fsync and Ssync , as well as between anonymous and non-anonymous agents. [ABSTRACT FROM AUTHOR]

A folklore conjecture in number theory states that the only integers whose expansions in base 3 , 4 and 5 contain solely binary digits are 0 , 1 and 82000. In this paper, we present the first progress on this conjecture. Furthermore, we investigate the density of the integers containing only binary digits in their base 3 or 4 expansion, whereon an exciting transition in behaviour is observed. Our methods shed light on the reasons for this, and relate to several well-known questions, such as Graham's problem and a related conjecture of Pomerance. Finally, we generalise this setting and prove that the set of numbers in [ 0 , 1 ] who do not contain some digit in their b -expansion for all b ≥ 3 has zero Hausdorff dimension. [ABSTRACT FROM AUTHOR]

In this paper we investigate the factorization behaviour of the binomial polynomials ( x n ) = x (x − 1) ⋯ (x − n + 1) n ! and their powers in the ring of integer-valued polynomials Int (Z). While it is well-known that the binomial polynomials are irreducible elements in Int (Z) , the factorization behaviour of their powers has not yet been fully understood. We fill this gap and show that the binomial polynomials are absolutely irreducible in Int (Z) , that is, ( x n ) m factors uniquely into irreducible elements in Int (Z) for all m ∈ N. By reformulating the problem in terms of linear algebra and number theory, we show that the question can be reduced to determining the rank of, what we call, the valuation matrix of n. A main ingredient in computing this rank is the following number-theoretical result for which we also provide a proof: If n > 10 and n , n − 1 , ..., n − (k − 1) are composite integers, then there exists a prime number p > 2 k that divides one of these integers. [ABSTRACT FROM AUTHOR]

We characterize all infinite-dimensional graded virtual modules for Thompson's sporadic simple group whose graded traces are weight 3 2 weakly holomorphic modular forms satisfying certain special properties. We then use these modules to detect the non-triviality of Mordell–Weil, Selmer, and Tate-Shafarevich groups of quadratic twists of certain elliptic curves. [ABSTRACT FROM AUTHOR]