pinecone/core-2020-05-10-deduplication huggingface dataset:
On the Value Distribution of Arithmetic Functions

AbstractIn part II of a series of articles on the least common multiple, the central object of investigation was a particular integer-valued arithmetic functiong1(n). The most interesting problem there was the value distribution ofg1(n). We proved that the counting function card{n⩽x:g1(n)⩽d} has orderod(x) for any fixedd. A characteristic feature ofg1(n) is its so-called super-periodicity which will be discussed here. An integer-valued arithmetic functiong(n) is calledsuper-periodic, if there is a sequence (rj) of positive integers withrj⩾2 (j⩾2) such that, settingRk≔∏kj=1rj,g(rRk+j)⩾g((r−1)Rk+j) for allk⩾1, 1⩽r<rk+1, and 1⩽j⩽Rk. In the present paper, we show that the above-mentioned property holds for a wide class of super-periodic functions, containing other interesting number-theoretical examples. The method is analytic and completely different from the one used in the earlier work
Class Group Relations from Burnside Ring Idempotents

AbstractWe show that for each finite cohomological Mackey functor on a finite groupGthere exist explicit relations in the category of finite abelian groups between the evaluations of the Mackey functor at all the subgroups, one for each conjugacy class of nonhypo-elementary subgroups ofG. Furthermore, we show that the class groups of the intermediate fields of a Galois extension of number fields form such a Mackey functor on the Galois group, thereby obtaining class group relations by using the presence of the structure of a cohomological Mackey functor
On the Sieve Parametersακandβκfor Largeκ

AbstractTwo parameters,ακandβκ, play a central role in the sieve method of Diamond, Halberstam, and Richert. For each value of the sieve dimensionκ>1,ακis the point beyond which the DHR upper sieve improves upon the upper bound sieve function of Ankeny and Onishi; and the sieving limitβκis the point beyond which the DHR lower sieve yields a nontrivial (i.e., positive) lower bound. We show that for all large enoughκ,ακlies within a bounded distance ofρκ, the largest positive zero of Iwaniec's functionqκ(·) and thatβκis smaller thanνκ, the sieving limit of the Ankeny–Onishi sieve
Traces in Arithmetic Fuchsian Groups

AbstractLet A be a quaternion algebra over an algebraic number fieldFof degreenover Q such that A⊗OR≌M2(R)×Hn−1, where H are the Hamiltonian quaternions. If O is an order in A, then O1={x∈O:N(x)=1} can be embedded inSL2(R) as a Fuchsian group. Let TO={t:t=Tr(γ),γ∈O1} be the set of traces of the elements in O1and TO(r)={t∈TO:0⩽t⩽2r}. The first purpose of this paper is to show thatlimr→∞|TO(r)|r=22n−1DF∏p|d(O)mp(O),whereDFis the discriminant ofF,d(O) is the (reduced) discriminant of O, andmp(O) only depends on the completion Opfr;of O for nonzero prime ideals p dividingd(O). We also calculate the invariantsmp(O) for important families of orders
On Infinite Sum-free Sets of Natural Numbers

AbstractA subset of the natural numbers isk-sum-free if it contains no solutions of the equationx1+…+xk=y, and stronglyk-sum-free when it is ℓ-sum-free for every ℓ=2,…,k. It is shown that everyk-sum-free set with upper density larger than 1/(k+1) is a subset of a periodick-sum-free set and that eachk-sum-free set with upper density larger than 2/(k+3) is subset of ak-sum-free arithmetic progression. In particular, nok-sum-free set has upper density larger than 1/ρ1(k), whereρ1(k)=min{i:i∤k−1}, as conjectured by Calkin and Erdős. Similar problems are studied also for stronglyk-sum-free sets
On the Invariant Factors of Kummer Orders in the Rings of Integers of p-adic Number Fields of Degreep2

AbstractLetK/kbe an extension of degreep2over a p-adic number fieldkwith the Galois groupG. We study the Galois module structure of the ring OKof integers inK. We determine conditions under which the invariant factors of Kummer orders OKin OKof two extensions coincide with each other and give two examples, one of which shows there exist Kummer extensionsKandLwithD(K)=D(L) such that OKand OLare not ZpG-isomorphic. The other shows the existence of extensionsFandKsuch that OFand OKare isomorphic over ZpGbut not over okG
CM-Fields with Cyclic Ideal Class Groups of 2-Power Orders

AbstractWe prove that there are only finitely many CM-fields N with cyclic ideal class groups of 2-power orders such that the complex conjugation is the square of some automorphism of N. Since their actual determination would be too difficult, we only content ourselves with the determination of the nonquadratic imaginary cyclic number fields of 2-power degrees with cyclic ideal class groups of 2-power orders. There are exactly 22 such number fields, 10 of them having class number one, 9 of them having class number two, and 3 of them having class number four. This present determination is a nice complement to the determination of all nonquadratic imaginary cyclic number fields of 2-power degrees with ideal class groups of exponents ⩽2 completed in an earlier paper
Solving Families of Simultaneous Pell Equations

AbstractIfaandbare distinct positive integers then a previous result of the author implies that the simultaneous Diophantine equationsx2−az2=y2−bz2=1possess at most 3 solutions in positive integers (x,y,z). On the other hand, there are infinite families of distinct integers (a,b) for which the above equations have at least 2 positive solutions. For each such family, we prove that there are precisely 2 solutions, with the possible exceptions of finitely many pairs (a,b). Since these families provide essentially the only pairs (a,b) for which the above equations are known to have more than a single solution (in positive (x,y,z)), this lends support to the conjecture that the number of such solutions to the above equations is ⩽2 in all cases
Orthogonal Sequences of Polynomials over Arbitrary Fields

AbstractLetfandgbe polynomials over some field, thought of as elements of the ring of one-sided Laurent series, and suppose that degf<degg. The quotientf/gisbadly approximableif all the partial quotients of the continued fraction expansion off/ghave degree 1. We investigate the set of polynomials which occur as the denominatorsgof badly approximable quotientsf/g. Such polynomials arise in stream cipher theory (part of cryptography) as the minimal polynomials of sequences with perfect linear complexity profile. They also occur in the theory of linear cellular automata and in the analysis of certain pseudorandom number generators
Rankin–Cohen Operators for Jacobi and Siegel Forms

AbstractFor any non-negative integer v we construct explicitly ⌊v/2⌋+1 independent covariant bilinear differential operators fromJk,m×Jk′,m′toJk+k′+v,m+m′. As an application we construct a covariant bilinear differential operator mappingS(2)k×S(2)k′toS(2)k+k′+v. HereJk,mdenotes the space of Jacobi forms of weightkand indexmandS(2)kthe space of Siegel modular forms of degree 2 and weightk. The covariant bilinear differential operators constructed are analogous to operators already studied in the elliptic case by R. Rankin and H. Cohen and we call them Rankin–Cohen operators
On Modular Relations for the Göllnitz–Gordon Functions with Applications to Partitions

AbstractIn a manuscript of Ramanujan, published with his Lost Notebook [21, pp. 236–237], there are forty identities involving the Rogers–Ramanujan functions. According to G. N. Watson, the beauty of these identities are comparable to that of the Rogers– Ramanujan identities. In the paper, we establish modular relations involving the Göllnitz–Gordon functions which are analogous to Ramanujan's forty identities. Furthermore, we extract interesting partition results from some of the modular relations
Local Units Modulo Gauss Sums

AbstractFor a prime numberpand a number fieldk, letk∞/kbe the cyclotomic Zp-extension. LetA∞be the projective limit of thep-part of the ideal class group of each intermediate field ofk∞/k. Whenkis totally real, it is conjectured thatA∞is finite, namely that the characteristic polynomial char(A∞) ofA∞as aΛ-module is 1. We give an interpretation of char(A∞) (and hence, of the conjecture) in terms ofp-adic behaviour of certain Gauss sums whenkis a real abelian field (satisfying some conditions). Whenk=Q(cos(2π/p)), similar results are already obtained by Coleman [3], Kaneko and the author [9]
An Equivariant Version of an Index Formula of Fröhlich and McCulloh

AbstractBy Iwasawa's index formula, the order of the minus class groupCl(Z[ζp])−equals the index ofI−in ZG−, whereG=Gal(Q(ζp)/Q) andIis the Stickelberger ideal in ZG. Fröhlich and McCulloh proved a very similar theorem concerning the order ofCl[ZΓ]−, whereΓis any finite elementary abelian group. The proof uses Iwasawa's formula. We show how the Main Conjecture of Iwasawa theory implies aG-invariant strengthening of Iwasawa's formula. This allows us to prove the main result of this note: a sharpened equivariant version of the above-mentioned result of Fröhlich and McCulloh
Power Bases for Cyclotomic Integer Rings

AbstractLetpbe an odd prime and Opbe the ring of integers in the cyclotomic fieldQ(ζ), whereζis a primitivepth root of unity. Then Op=Z[α] ifα=ζ, 1/(1+ζ), or one of the conjugates of these two elements. In 1988, Bremner [3] conjectured that up to integer translation there are no further generators for Opand proved that this is indeed the case whenp=7. We establish a criterion for verifying Bremner's conjecture for a given regular primepand use it to prove the conjecture forp⩽23,p≠17. A key step in the proof of the criterion is a determinant formula for the relative class numberh−ofQ(ζ)
On a Partition Theorem of Göllnitz and Quartic Transformations

AbstractFori=1, 2, 3, 4, letQi(n) denote the number of partitions of n into distinct parts ≢ (mod4). New weighted identities in three free parameters are established connectingQi(n) with partitions whose parts differ by ⩾4 and such that consecutive members of the arithmetic progression ≡i(mod4) cannot occur as parts. By the use of suitable quartic transformations, these weighted identities are shown to be reformulations of a deep partition theorem of Göllnitz. Applications include new relations for partitions of the Göllnitz-Gordon type, a new proof of Jacobi's triple product identity and a remarkable congruence modulo powers of 2 forQ2(n)
Well Distribution of Sidon Sets in Residue Classes

AbstractA setAof non-negative integers is aSidon setif the sumsa+b(a,b∈A,a⩽b) are distinct. Assume thata⊆[1,n] and that |A|=(1+o(1))n1/2. Letm⩾2 be an integer. In Theorem 1 I prove that asymptotically 1/mof all elements inAfall into each residue class modulom. Whenm=2 I prove a sharper result in Theorem 2. Assume that |A|⩾n1/2. Then the difference between the number of odd and the number of even elements inAisO(n3/8). If the interval [1,n] is divided intomequal parts and the number of elements fromAin each part is counted, then similar results hold for these counts
Asymptotic Estimates for a Class of Summatory Functions

AbstractWe establish explicit expressions for bothPandEin ∑n⩽xa(n)=P(x)+E(x)=“principal term”+“error term”, when the (complex) arithmetical function a has a generating function of the formζ(s)Z(s), whereζis the Riemann zeta function, and whereZhas a representation as a Dirichlet series having an abscissa of absolute convergence smaller than 1 (and satisfying some other conditions). We obtainO-estimates (and in some casesΩ-estimates) onE. We also obtain asymptotic expressions for ∑n⩽xnβa(n) when the real numberβis not too small, and for ∫x1E(t)dtand ∑n⩽xE(n). This can be applied to a number of arithmetical functions a which have been studied in the literature with various methods. In most cases what we obtain improves on, or extends, the existing results
On a Linear Diophantine Problem of Frobenius: Extending the Basis

AbstractLetXk={a1,a2,…,ak},k>1, be a subset of N such that gcd(Xk)=1. We shall say that a natural numbernisdependent(onXk) if there are nonnegative integersxisuch thatnhas a representationn=∑ki=1xiai, elseindependent. The Frobenius numberg(Xk) ofXkis the greatest integer withnosuch representation. Selmer has raised the problem of extendingXkwithout changing the value ofg. He showed that under certain conditions it is possible to add an elementc=a+kdto the arithmetic sequencea,a+d,a+2d,…, a+(k−1)d, gcd(a,d)=1, without alteringg. In this paper, we give the setCof all independent numberscsatisfyingg(A,c)=g(A), whereAcontains the elements of the arithmetic sequence. Moreover, ifa>kthen we give as an application, a setBof maximal cardinality such thatg(A,B)=g(A) and each element ofA∪Bis independent of the other ones
Small Solutions of the Legendre Equation

AbstractHolzer, using a deep theorem, proved that if the Legendre equationax2+by2−cz2=0 is in normal form and has a nonzero solution then there is a solution with|x|⩽bc, |y|⩽acand|z|⩽ab. The first elementary proof of this result was given by Mordell with a small gap filled by Williams. We give a new elementary proof of this theorem. We show in fact that the lattice of solutions of the congruenceax2+by2−cz2≡0(modabc) that is constructed in most textbook proofs of Legendre's theorem always contains such a solution
Deformations for Function Fields

AbstractWe consider a question of describing the one-dimensionalP-adic representations that lift a given representation over a finite field of the absolute Galois group of a function field. In this case, the characterization of abelianp-power extensions of fields of characteristicpcan be extended to abelian pro-p-extensions, and refined to allow only restricted ramification at the places ofK, and can be a tool for analyzing one-dimensionP-adic representations. We then turn to the problem of classifying those representations which can be realized as the action of the Galois group on the division points of a rank one Drinfeld module, discussing both results and a conjecture about the form of the representations that arise in this manner
On the Class Group Problem for Function Fields

AbstractLetGbe a finite abelian group, it is a difficult and unsolved problem to find a number fieldFwhose ideal class group is isomorphic toG. In [WAS], Corollary 3.9 and in [COR], Theorem 2, it is proved that every finite abelian group is isomorphic to a factor group of the ideal class group of some number field. Furthermore, O. Yahagi [YAH] has proved that, if l is a prime number, every finite abelian ℓ-group is isomorphic to the ℓ-Sylow subgroup of the ideal class group of some number field. In this paper, we prove similar results for the function field case
Sumsets in Vector Spaces over Finite Fields

AbstractWe determine explicitly the least possible size of the sumset of two subsetsA, B⊂(Z/pZ)Nwith fixed cardinalities, thereby generalizing both Cauchy–Davenport's theorem (caseN=1) and Yuzvinsky's theorem(casep=2). The solution involves a natural generalization of the well-known Hopf–Stiefel–Pfister function. The corresponding problem for more than two summands is also considered and solved. We then consider restricted sumsets, formed by taking sums of distinct elements only. We determine almost completely the least possible size of the restricted sumset of two subsets in (Z/pZ)Nwith fixed cardinalities. Our result generalizes the recent solution(s) of the Erdo&#x030B;s–Heilbronn conjecture dealing with the restricted sumsets of two equal subsets inZ/pZ
Local and Global Zeta-Functions of Singular Algebraic Curves

AbstractLetXbe a complete singular algebraic curve defined over a finite field ofqelements. To each local ring O ofXthere is associated a zeta-functionζO(s) that encodes the numbers of ideals of given norms. It splits into a finite sum of partial zeta-functions, which are rational functions inq−s. We provide explicit formulae for the partial zeta-functions and prove that the quotient of the zeta-functions of O and its normalization O is a polynomial inq−sof degree not larger than the conductor degree of O. The global zeta-functionζOX(s), defined by encoding the numbers of coherent ideal sheaves of given degrees, satisfies the global functional equation if and only ifXis a Gorenstein curve. We introduce a modified zeta-function, which always satisfies the functional equation and which in the Gorenstein case coincides withζOX(s). We prove that the two global zeta-functions have the same residue ats=0, and that this residue determines the number of the rational points of the compactified Jacobian ofX
A Family of Markov Shifts (Almost) Classified by Periodic Points

AbstractLetGbe a finite group, and letXG={x=(x(s, t))∈GZ2∣x(s, t)=x(s, t−1)·x(s+1, t−1)for all (s, t)∈Z2}.The compact zero-dimensional setXGcarries a natural shift Z2-actionσGand the pairσG=(XG, σG) is a two-dimensional topological Markov shift. Using recent work by Crandall, Dilcher and Pomerance on the Fermat quotient, we show the following: ifGis abelian, and the order ofGis not divisible by 1024, nor by the square of any Wieferich prime larger than 4×1012, andHis any abelian group for whichΣGhas the same periodic point data asΣH, thenGis isomorphic toH. This result may be viewed as an example of the “rigidity” properties of higher-dimensional Markov shifts with zero entropy
Observations on Fermat Motives ofK3-Type

AbstractIn [8, 9], Zarhin introduced the notion of varieties ofK3-typein even dimension over finite fields. Zarhin showed that ordinary abelian surfaces, ordinaryK3 surfaces, and ordinary cubic fourfolds are examples of such varieties. As Zarhin already points out, it is easy to extend his method to definemotivesofK3-type. In this note, we extend Zarhin's work by giving this definition and then finding and studying a large number of examples of such motives. Our examples are Fermat motives, i.e., they arise from the Fermat varietyX:Xm0+Xm1+…+Xmn+1=0⊂Pnof degreemand dimensionn=2d. Such motives ofK3-type need not be ordinary; on the other hand, they are never supersingular (essentially by definition—see [1], Chapter 3). Hence, ourK3-type motives give rise to transcendental cycles. We show that being ofK3-type is (resp. is not) hereditary with respect to thetype I inductive structure(resp. thetype II inductive structure) of Fermat motives. When the degreemtends to infinity, the type I inductive structure gives rise to infinitely many Fermat motives ofK3-type (see Remark (3.9)). Following a suggestion of Zarhin, we also consider powers of Fermat motives ofK3-type. Powers of motives ofK3-type are no longer ofK3-type, and hence they can give rise to classes of algebraic cycles. Thus, it is of interest to establish the validity of Tate's conjecture for powers of Fermat motives ofK3-type. This is done in the Section 4, extending Theorem (6.1) of Zarhin [10]. Finally, in Section 5 we discuss theBrauer numbersof Fermat motives ofK3-type, especially, of even composite degreem. This is an attempt to extend the results about Brauer numbers of Fermat motives (of prime or odd prime power degree) in Gouvêa and Yui [1, 2], Miki [4], Shoida [5], and Yui [7]. Throughout, we retain Zarhin's assumptions that the dimensionnis even and that we are working over a finite field
Power Reciprocity for Binomial Cyclotomic Integers

AbstractWe give an explicit expression for the inversion factor (α/β)l(β/α)−1lof thelth power residue symbol over the cyclotomic field of l th roots of unity, whenαandβare binomial cyclotomic integersx+yζnrelatively prime to each other and tol. Herelis an odd prime number,ζa primitivelth root of unity andx, y∈Z We note that Eisenstein's reciprocity law extends to the case where primary binomial integers replace rational integers. As an application, we obtain necessary and sufficient congruence conditions for a rational integer to be anlth power residue modulo some prime numbers of the form (xl+1)/(x+1)
A Pollard Type Result for Restricted Sums

AbstractLet F be an arbitrary field. Letpbe the characteristic of F in case of finite characteristic and ∞ if F has characteristic 0. LetAbe a finite subset of F. By ∧2Awe denote the set {a+b|a, b∈Aanda≠b}. Forc∈∧2A, letν(R)cbe one-half of the cardinality of the set of pairs (a, b) satisfyinga≠banda+b=c. Denote byμ(R)ithe cardinality of the set {c∈∧2A|ν(R)c⩾i}. We prove that, fort=1, …, ⌊|A|/2⌋, ∑ti=1μ(R)i⩾tmin{p, 2(|A|−t)−1}. For F=Zpandt=1 we get the Erdős–Heilbronn conjecture, first proved by J. A. Dias da Silva and Y. O. Hamidoune (Bull. London Math. Soc.26, 1994, 140–146)
Ramification of Some Automorphisms of Local Fields

AbstractLetkbe a perfect field of characteristicpand letγ∈Aut(k((t))). Define the ramification numbers ofγbyim=vt(γpm(t)−t)−1. We give a characterization of the sequences (im) which are the sequences of ramification numbers of infinite order automorphisms of formal power series fields over finite fields. Then, given a perfect field k, we give sufficient conditions on (im) to be the sequence of ramification numbers of an autormorphismγ∈Autk(k((t))) and we investigate these sequences (im) in the case where there existsσ∈Autk(k((t))) such thatσγ=γσwithσ≠γνfor allν∈Zp
Cyclotomic Integers of Prescribed Absolute Value and the Class Group

AbstractWe obtain a new method for the study of class groups of cyclotomic fields by investigating cyclotomic integers of prescribed absolute value. Explicit subgroups of the classgroupCmodulo the class groupC+of the maximal real subfield are exhibited and lower bounds on their orders are derived. For themth cyclotomic fieldKm, wherem=pam′, (p, m′)=1, andpis a prime, we determine the structure ofC+CP/C+CQup to a binary parameter; hereCP,CQare the subgroups ofCgenerated by the classes [Pi] respectively [Qi], wherepfactors inKmas ∏Qi,Qi=Pϕ(pa)i, and thePiare prime ideals
Square-Classes in Lucas Sequences Having Odd Parameters

AbstractTwo or more terms of a sequence are said to be in the same square-class if the squarefree parts of the terms are identical. Let {Un(P,Q)} and {Vn(P,Q)} denote the Lucas sequence and companion Lucas sequence, respectively, with parametersPandQ. For all odd relatively prime values ofPandQwith discriminantP2−4Q>0, we show that {Un(P,Q)} and {Vn(P,Q)} have only finitely many non-trivial square-classes and each square-class contains at most three terms. The square-classes are explicitly determined in “most” case, and an effectively computable bound on the number of square-classes, depending onPandQ, is obtained in the remaining cases
On Finite Pseudorandom Binary Sequences II. The Champernowne, Rudin–Shapiro, and Thue–Morse Sequences, A Further Construction

AbstractSpecial finite binary sequences are tested for pseudorandomness. As measures of pseudorandomness, well-distribution relative to arithmetic progressions and small (auto)correlation are used. These properties of the Champernowne, Thue–Morse, and Rudin–Shapiro sequences are studied and it is shown that although each of them possesses certain pseudorandom properties, none of them can be considered completely pseudorandom. Finally, by using the Legendre symbol and permutation polynomials, a nearly ideally pseudorandom sequence is constructed
On the Parity of Additive Representation Functions

AbstractLet A be a set of positive integers,p(A,n) be the number of partitions ofnwith parts in A, andp(n)=p(N,n). It is proved that the number ofn⩽Nfor whichp(n) is even is ⪢N, while the number ofn⩽Nfor whichp(n) is odd is ⩾N1/2+o(1). Moreover, by using the theory of modular forms, it is proved (by J.-P. Serre) that, for allaandmthe number ofn, such thatn≡a(modm), andn⩽Nfor whichp(n) is even is ⩾cNfor any constantc, andNlarge enough. Further a set A is constructed with the properties thatp(A,n) is even for alln⩾4 and its counting functionA(x) (the number of elements of A not exceedingx) satisfiesA(x)⪢x/logx. Finally, we study the counting function of sets A such that the number of solutions ofa+a′=n,a,a′∈A,a<a′ is never 1 for largen