Maximal Collections of Intersecting Arithmetic Progressions

Let N _t (k) be the maximum number of k-term arithmetic progressions of real numbers, any two of which have t points in common. We determine N ₂(k) for prime k and all large k, and give upper and lower bounds for N _t (k) when t 3.* Research supported in part by NSF grant DMS-0070618.     

Conjectures de Greenberg et extensions¶pro-p-libres d'un corps de nombres

For an algebraic number field k and a prime number p (if p=2, we assume that μ₄⊂k), we study the maximal rank ρ _p of a free pro-p-extension of k. This problem is related to deep conjectures of Greenberg in Iwasawa theory. We give different equivalent formulations of these conjectures and we apply them to show that, essentially, ρ _k =r ₂(k)+1 if and only if k is a so-called p-rational field. Received: 29 April 1999 / Revised version: 31 January 2000     

Digital Expansions with Respect to Different Bases

We consider the g-ary expansion N=∑ _k b _k (N, g)g ^k of non-negative integers N and prove various results on the distribution and the mean value of the k-th digit b _k (N, g) if g varies in an interval of the form 2≤g≤N ^η. As an application we also consider the average value of the sum-of-digits function s(N, g)=∑ _k b _k (N, g). Received 5 November 2001 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

Comportement moyen du cardinal de certains ensembles de facteurs premiers

Letn be a positive integer andS _n a particular set of prime divisors ofn. We establish the average order off(n) wheref(n) stands for the cardinality ofS _n . Thek-ary,k-free, semi-k-ary prime factors ofn are some of the classes of prime divisors studied in this paper.

     

Sums of two k-th powers: an Omega estimate for the error term

The arithmetic function r _k (n) counts the number of ways to write a natural number n as a sum of two k-th powers (k ≧ 2 fixed). The investigation of the asymptotic behaviour of the Dirichlet summatory function of r _k(n) leads in a natural way to a certain error term P _k(t). In this article, we establish an Ω-estimate for P _k(t) (k τ; 2 arbitrary) which is essentially as sharp as the best known one in the classic case k=2. This article is part of a research project supported by the Austrian Science Foundation (Nr. P 9892-PHY).     

On the Function w( x)=|{1= s= k : x= a s (mod n s)}|

For a finite system of arithmetic sequences the covering function is w(x) = |{1 s k : x a_s (mod n_s)}|. Using equalities involving roots of unity we characterize those systems with a fixed covering function w(x). From the characterization we reveal some connections between a period n₀ of w(x) and the moduli n₁, . . . , n_k in such a system A. Here are three central results: (a) For each r=0,1, . . .,n_k/(n₀,n_k)–1 there exists a Jc{1, . . . , k–1} such that . (b) If n₁ ···n_k–l <n_k–l+1 =···=n_k (0 < l < k), then for any positive integer r < n_k/n_k–l with r 0 (mod n_k/(n₀,n_k)), the binomial coefficient can be written as the sum of some (not necessarily distinct) prime divisors of n_k. (c) max_(xw(x) can be written in the form where m₁, . . .,m_k are positive integers.The research is supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, and the National Natural Science Foundation of P. R. China.     

On the distribution of square-full and cube-full numbers

There are many results on the distribution of square-full and cube-full numbers. In this article the distribution of these numbers are studied in more detail. Suchk-full numbers (k=2,3) are considered which are at the same time 1-free (1k+2). At first an asymptotic result is given for the numberN _k,1(x) ofk-full and 1-free numbers not exceedingx. Then the distribution of these numbers in short intervals is investigated. We obtain different estimations of the differenceN _k,1(x+h)–N_k,1(x) in the casesk=2, 1=4,5,6,7,18 andk=3, 1=5,6,7, 18.     

Ramsey partitions of integers and pair divisions

Ifk ₁ andk ₂ are positive integers, the partitionP = (₁,₂,..., _n ) ofk ₁+k ₂ is said to be a Ramsey partition for the pairk ₁,k ₂ if for any sublistL ofP, either there is a sublist ofL which sums tok ₁ or a sublist ofP –L which sums tok ₂. Properties of Ramsey partitions are discussed. In particular it is shown that there is a unique Ramsey partition fork ₁,k ₂ having the smallest numbern of terms, and in this casen is one more than the sum of the quotients in the Euclidean algorithm fork ₁ andk ₂.An application of Ramsey partitions to the following fair division problem is also discussed: Suppose two persons are to divide a cake fairly in the ratiok ₁k ₂. This can be done trivially usingk ₁+k ₂-1 cuts. However, every Ramsey partition ofk ₁+k ₂ also yields a fair division algorithm. This method yields fewer cuts except whenk ₁=1 andk ₂=1, 2 or 4.     

Arithmetical identities of the Brauer-Rademacher type

Summary LetA be a regular arithmetical convolution andk a positive integer. LetA _k (r) = {d: d ^k A(r ^k )}, and letf A _k g denote the convolution of arithmetical functionsf andg with respect toA _k . A pair (f, g) of arithmetical functions is calledadmissible if(f A _k g)(m) 0 for allm and if the functions satisfy an arithmetical functional equation which generalizes the Brauer—Rademacher identity. Necessary and sufficient conditions are found for a pair (f, g) of multiplicative functions to be admissible, and it follows that, if(f A _k g)(m) 0 f(m) for allm, then (f, g) is admissible if and only if itsdual pair (f A _k g, g ^–1 ) is admissible.Iff andg ^–1 areA _k -multiplicative (a condition stronger than being multiplicative), and(f A _k g)(m) 0 for allm, then (f, g) is admissible, calledCohen admissible. Its dual pair is calledSubbarao admissible. If (f A _k g) ^–1 (m) 0 itsinverse pair (g ^–1 , f ^–1 ) is also Cohen admissible.Ifg is a multiplicative function then there exists a multiplicative functionf such that the pair (f, g) is admissible if and only if for everyA _k -primitive prime powerp ⁱ either (i)g(p ⁱ ) 0 or (ii)g(p ) = 0 for allp havingA _k -type equal tot. There is a similar kind of characterization of the multiplicative functions which are first components of admissible pairs of multiplicative functions. IfA _k is not the unitary convolution, then there exist multiplicative functionsg which satisfy (i) and are such that neitherg norg ^–1 isA _k -multiplicative: hence there exist admissible pairs of multiplicative functions which are neither Cohen admissible nor Subbarao admissible.An arithmetical functionf is said to be anA _k -totient if there areA _k -multiplicative functionsf _T andf _V such thatf = f _T A _k f _V ^-1 Iff andg areA _k -totients with(f A _k g)(m) 0 for allm, and iff _V = g _T , then the pair (f, g) is admissible. The class of such admissible pairs includes many pairs which are neither Cohen admissible nor Subbarao admissible. If (f, g) is a pair in this class, and iff(m), (f A _k g) ^–1 (m), g ^–1 (m),f ^–1 (m) andg(m) are all nonzero for allm, then its dual, its inverse, the dual of its inverse, the inverse of its dual and the inverse of the dual of its inverse are also admissible, and in many cases these six pairs are distinct.A number of related results, and many examples, are given.     

Squares of Primes and Powers of 2

 As an extension of the Linnik-Gallagher results on the “almost Goldbach” problem, we prove, among other things, that there exists a positive integer k ₀ such that every large even integer is a sum of four squares of primes and k ₀ powers of 2. (Received 7 September 1998; in revised form 3 May 1999)     

Upper bounds for prime <Emphasis Type="Italic">k</Emphasis>-tuples of size log <Emphasis Type="Italic">N</Emphasis> and oscillations

We prove the estimate for the number E_k(N) of k-tuples (n + a₁,..., n + a_k) of primes not exceeding N, for k of size c₁ log N and N sufficiently large. A bound of this strength was previously known in the special case < only, (Vaughan, 1973). For general a_i this is an improvement upon the work of Hofmann and Wolke (1996). The number of prime tuples of this size has considerable oscillations, when varying the prime pattern. Received: 20 December 2002     

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.     

On Sums of Two Coprime k-th Powers

For fixed k3, let It is known that the asymptotic formula holds for some constant c_k. Let E_k(x)=R_k(x)–c_kx^2/k. We cannot improve the exponent 1/k at present if we do not have further knowledge about the distribution of the zeros of the Riemann Zeta function (s). In this paper, we shall prove that if the Riemann Hypothesis (RH) is true, then E_k(x)=O(x^4/15+), which improves the earlier exponent 5/18 due to Nowak. A mean square estimate of E_k(x) for k6 is also obtained, which implies that E_k(x)=(x^1/k–1/k2) for k6 under RH.     

On the Iwasawa invariants of certain non-cyclotomic ℤ2-extensions

Let k be an imaginary quadratic field in which the prime 2 splits. We consider the Iwasawa invariants of a certain non-cyclotomic ℤ₂-extension of k and give some sufficient conditions for the vanishing of λ- and μ-invariants.     

On the law of the iterated logarithm for the discrepancy of 〈n k x〉

By a well known result of Philipp (1975), the discrepancy D _N (ω) of the sequence (n _k ω) _k≥1 mod 1 satisfies the law of the iterated logarithm under the Hadamard gap condition n _{k + 1}/n _k ≥ q > 1 (k = 1, 2, …). Recently Berkes, Philipp and Tichy (2006) showed that this result remains valid, under Diophantine conditions on (n _k ), for subexpenentially growing (n _k ), but in general the behavior of (n _k ω) becomes very complicated in the subexponential case. Using a different norming factor depending on the density properties of the sequence (n _k ), in this paper we prove a law of the iterated logarithm for the discrepancy D _N (ω) for subexponentially growing (n _k ) without number theoretic assumptions. C. Aistleitner, Research supported by FWF grant S9603-N13. I. Berkes, Research supported by FWF grant S9603-N13 and OTKA grants K 61052 and K 67961. Authors’ addresses: C. Aistleitner, Institute of Mathematics A, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria; I. Berkes, Institute of Statistics, Graz University of Technology, Steyrergasse 17/IV, 8010 Graz, Austria     

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).     

Stacked lattice boxes

We study the number of solutions of the Diophantine equationn=x ₁ x ₂+x ₂ x ₃+x ₃ x ₄+...+x _k x _k+1 The combinatorial interpretation of this equation provides the name stacked lattices boxes. The study of these objects unites three separate threads in number theory: (1) the Liouville methods, (2) MacMahon's partitions withk different parts, (3) the asymptotics of divisor sums begun by Ingham.Partially supported by National Science Foundation Grant DMS-9206993, USA.     

On generalized gamma functions related to the Laurent coefficients of the Riemann zeta function

Summary We study a class of generalized gamma functions _k (z) which relate to the generalized Euler constants _k (basically the Laurent coefficients of(s)) as (z) does to the Euler constant. A new series expansion for _k is derived, and the constant term in the asymptotic expansion for log _k (z) is studied in detail. These and related constants are numerically computed for 1 k 15.     

On Finite Pseudorandom Binary Sequences, V.On (nα) and (n 2α) Sequences

 Let k be a positive integer and α be a real number, and for if the fractional part of is <1/2 and e _n =−1 if it is ≥1/2. The pseudorandom properties of the sequence are studied. As measures of pseudorandomness, the regularity of the distribution relative to arithmetic progressions and the correlation are used. Here the special cases k=1 and k=2 are studied (while the case k>2 will be studied in the sequel). (Received 23 April 1999)