On digit sums of multiples of an integer

Let g>1 be an integer and sg(m) be the sum of digits in base g of the positive integer m. In this paper, we study the positive integers n such that sg(n) and sg(kn) satisfy certain relations for a fixed, or arbitrary positive integer k. In the first part of the paper, we prove that if n is not a power of g, then there exists a nontrivial multiple of n say kn such that sg(n)=sg(kn). In the second part of the paper, we show that for any K>0 the set of the integers n satisfying sg(n)?Ksg(kn) for all k∈N is of asymptotic density 0. This gives an affirmative answer to a question of W.M. Schmidt.     

On the average orders of the error term in the Dirichlet divisor problem

Let Δ(x) be the error term in the Dirichlet divisor problem. The purpose of this paper is to study the difference between two kinds of mean value formulas of Δ(x), that is, the mean value formulas and ∑_n?xΔ(n)^k with a natural number k. In particular we study the case k=2 and 3 in detail.     

On arrangements of roots for a real hyperbolic polynomial and its derivatives

In this paper we count the number ?_n^(0,k), k?n−1, of connected components in the space Δ_n^(0,k) of all real degree n polynomials which a) have all their roots real and simple; and b) have no common root with their kth derivatives. In this case, we show that the only restriction on the arrangement of the roots of such a polynomial together with the roots of its kth derivative comes from the standard Rolle's theorem. On the other hand, we pose the general question of counting all possible root arrangements for a polynomial p(x) together with all its nonvanishing derivatives under the assumption that the roots of p(x) are real. Already the first nontrivial case n=4 shows that the obvious restrictions coming from the standard Rolle's theorem are insufficient. We prove a generalized Rolle's theorem which gives an additional restriction on root arrangements for polynomials.     

On the group of modular units and the ideal class group

J.-M. Kim, S. Bae and I.-S. Lee showed that there exists an isomorphism between the p-primary part of the ideal class group and p-primary part of the unit group modulo cyclotomic unit group in Q⁺(ζpn) for all sufficiently large n under some conditions. In the present paper, we shall give an analogue of their result for modular units.     

Dirichlet series associated with polynomials and applications

We consider the possibility of the analytic continuation of the Dirichlet series SP;Z(s) associated with a polynomial P(x) and auxiliary series Z(s). In fact, we derive a certain criterion for the analytic continuation for some class of polynomials and give examples such that SP;Z(s) cannot be continued meromorphically to the whole plane C. We also study the asymptotic behaviors of the sum MP(x)=_∑P(n₁,…,nk)?xΛ(n₁)?Λ(nk) considered first by M. Peter. Generalizations of this sum are also considered.     

The inversion formulae for automorphisms of polynomial algebras and rings of differential operators in prime characteristic

Let K be an arbitrary field of characteristic p>0. We find an explicit formula for the inverse of any algebra automorphism of any of the following algebras: the polynomial algebra P_n?K[x₁,…,x_n], the ring of differential operators D(P_n) on P_n, D(P_n)⊗P_m, the n’th Weyl algebra A_n or A_n⊗P_m, the power series algebra K[[x₁,…,x_n]], the subalgebra T_k1,…,kn of D(P_n) generated by P_n and the higher derivations , 0≤j<p^k_i, i=1,…,n (where k₁,…,k_n∈N), T_k1,…,kn⊗P_m or an arbitrary central simple (countably generated) algebra over an arbitrary field.     

Uniformity over primes of tamely ramified splittings

We fix a prime p and let f(X) vary over all monic integer polynomials of fixed degree n. Given any possible shape of a tamely ramified splitting of p in an extension of degree n, we prove that there exists a rational function φ(X)∈ℚ(X) such that the density of the monic integer polynomials f(X) for which the splitting of p has the given shape in ℚ[X]/f(X) is φ(p) (here reducible polynomials can be neglected). As a corollary, we prove that, for p≥n, the density of irreducible monic polynomials of degree n in ℤ _p [X] is the value at p of a rational function φ _n (X)∈ℚ(X). All rational functions involved are effectively computable. Received: 15 September 1998 / Revised version: 21 October 1999     

Computation of the Iwasawa invariants of certain real abelian fields

Let p be a prime number and k a finite extension of . It is conjectured that the Iwasawa invariants λ_p(k) and μ_p(k) vanish for all p and totally real number fields k. Some methods to verify the conjecture for each real abelian field k are known, in which cyclotomic units and a set of auxiliary prime numbers are used. We give an effective method, based on the previous one, to compute the exact value of the other Iwasawa invariant ν_p(k) by using Gauss sums and another set of auxiliary prime numbers. As numerical examples, we compute the Iwasawa invariants associated to in the range 1<f<200 and 5?p<10000.     

Limit sets of automatic sequences

We show that for every k-automatic sequence there exists a natural number p>0 such that the sequences of the form (k^pn+j)_n?0 with j=0,…,p−1 are scaling sequences for f. Moreover, we demonstrate that every limit set is the union of certain basic limit sets.     

On the representation of integers as sums of triangular numbers

Summary In this survey article we discuss the problem of determining the number of representations of an integer as sums of triangular numbers. This study yields several interesting results. Ifn 0 is a non-negative integer, then thenth triangular number isT _n =n(n + 1)/2. Letk be a positive integer. We denote by _k (n) the number of representations ofn as a sum ofk triangular numbers. Here we use the theory of modular forms to calculate _k (n). The case wherek = 24 is particularly interesting. It turns out that, ifn 3 is odd, then the number of points on the 24 dimensional Leech lattice of norm 2n is 2¹²(2¹² – 1) ₂₄(n – 3). Furthermore the formula for ₂₄(n) involves the Ramanujan(n)-function. As a consequence, we get elementary congruences for(n). In a similar vein, whenp is a prime, we demonstrate ₂₄(p ^k – 3) as a Dirichlet convolution of ₁₁(n) and(n). It is also of interest to know that this study produces formulas for the number of lattice points insidek-dimensional spheres.     

Regularity of patterns in the factorization of n!

Consider the multiplicities ep₁(n),ep₂(n),…,epk(n) in which the primes p₁,p₂,…,pk appear in the factorization of n!. We show that these multiplicities are jointly uniformly distributed modulo (m₁,m₂,…,mk) for any fixed integers m₁,m₂,…,mk, thus improving a result of Luca and St?nic? [F. Luca, P. St?nic?, On the prime power factorization of n!, J. Number Theory 102 (2003) 298-305]. To prove the theorem, we obtain a result regarding the joint distribution of several completely q-additive functions, which seems to be of independent interest.     

The Rubin-Stark conjecture for a special class of function field extensions

We prove a strong form of the Brumer-Stark Conjecture and, as a consequence, a strong form of Rubin's integral refinement of the abelian Stark Conjecture, for a large class of abelian extensions of an arbitrary characteristic p global field k. This class includes all the abelian extensions K/k contained in the compositum k_p∞?k_p·k_∞ of the maximal pro-p abelian extension k_p/k and the maximal constant field extension k_∞/k of k, which happens to sit inside the maximal abelian extension k^ab of k with a quasi-finite index. This way, we extend the results obtained by the present author in (Comp. Math. 116 (1999) 321-367).     

Split orders and convex polytopes in buildings

As part of his work to develop an explicit trace formula for Hecke operators on congruence subgroups of SL₂(Z), Hijikata (1974) [13] defines and characterizes the notion of a split order in M₂(k), where k is a local field. In this paper, we generalize the notion of a split order to Mn(k) for n>2 and give a natural geometric characterization in terms of the affine building for SLn(k). In particular, we show that there is a one-to-one correspondence between split orders in Mn(k) and a collection of convex polytopes in apartments of the building such that the split order is the intersection of all the maximal orders representing the vertices in the polytope. This generalizes the geometric interpretation in the n=2 case in which split orders correspond to geodesics in the tree for SL₂(k) with the split order given as the intersection of the endpoints of the geodesic.     

Arithmetic of the Ramanujan-Göllnitz-Gordon continued fraction

Text

We extend the results of Chan and Huang [H.H. Chan, S.-S. Huang, On the Ramanujan-Göllnitz-Gordon continued fraction, Ramanujan J. 1 (1997) 75-90] and Vasuki, Srivatsa Kumar [K.R. Vasuki, B.R. Srivatsa Kumar, Certain identities for Ramanujan-Göllnitz-Gordon continued fraction, J. Comput. Appl. Math. 187 (2006) 87-95] to all odd primes p on the modular equations of the Ramanujan-Göllnitz-Gordon continued fraction v(τ) by computing the affine models of modular curves X(Γ) with Γ=Γ₁(8)∩Γ₀(16p). We then deduce the Kronecker congruence relations for these modular equations. Further, by showing that v(τ) is a modular unit over Z we give a new proof of the fact that the singular values of v(τ) are units at all imaginary quadratic arguments and obtain that they generate ray class fields modulo 8 over imaginary quadratic fields.

Video

For a video summary of this paper, please visit http://www.youtube.com/watch?v=FWdmYvdf5Jg.     

Strong Sperner property of the subgroup lattice of an Abelianp-group

Letn andk be arbitrary positive integers,p a prime number and L_(k ⁿ)(p) the subgroup lattice of the Abelianp-group (Z/p ^k ) ⁿ . Then there is a positive integerN(n,k) such that whenp N(n,k),L _(k ^N )(p) has the strong Sperner property.     

Some irreducibility results for truncated binomial expansions

For positive integers n>k, let be the polynomial obtained by truncating the binomial expansion of n(1+x) at the kth stage. These polynomials arose in the investigation of Schubert calculus in Grassmannians. In this paper, the authors prove the irreducibility of Pn,k(x) over the field of rational numbers when 2≤2k≤n<³(k+1).     

Arithmetic properties of bipartitions with even parts distinct

In this work, we consider various arithmetic properties of the function ped _?2(n) that denotes the number of bipartitions of n with even parts distinct. We prove two infinite families of congruences for p _?2(n) modulo 3. We also give characterizations of ped _?2(n) modulo 2 and 4. Furthermore, for a fixed positive integer k, we show that ped _?2(n) is divisible by 2 ^k for almost all n.     

Galois module structure for dihedral extensions of degree 8: Realizable classes over the group ring

Let k be a number field with ring of integers O_k, and let Γ be the dihedral group of order 8. For each tame Galois extension N/k with group isomorphic to Γ, the ring of integers O_N of N determines a class in the locally free class group Cl(O_k[Γ]). We show that the set of classes in Cl(O_k[Γ]) realized in this way is the kernel of the augmentation homomorphism from Cl(O_k[Γ]) to the ideal class group Cl(O_k), provided that the ray class group of O_k for the modulus 4O_k has odd order. This refines a result of the second-named author (J. Algebra 223 (2000) 367-378) on Galois module structure over a maximal order in k[Γ].