On a theorem of Graham

AbstractA strengthened form of Gurevich's conjecture was proved by R. L. Graham, which says that for any α > 0 and any pair of non-parallel lines L1 and L2, in any partition of the plane into finitely many classes, some class contains the vertices of a triangle which has area α and two sides parallel to the lines Li. In this note, using the main idea of Graham, we present a shorter proof of the result

On an error term related to the Jordan totient function Jk(n)

AbstractWe investigate the error terms Ek(x)=∑n⩽xJk(n)−xk+1(k+1)ζ(k+1) for k⩾2, where Jk(n) = nkΠp|n(1 − 1pk) for k ≥ 1. For k ≥ 2, we prove ∑n⩽xEk(n)∼xk+12(k+1)ζ(k+1). Also, lim infn→∞Ek(x)xk⩽Dζ(k+1), where D = .7159 when k = 2, .6063 when k ≥ 3. On the other hand, even though lim infn→∞Ek(x)xk⩽−12ζ(k+1), Ek(n) > 0 for integers n sufficiently large

On weighted zero-sum sequences

Let G be a finite additive abelian group with exponent exp(G)=n>1 and let A be a nonempty subset of {1,...,n-1}. In this paper, we investigate the smallest positive integer $m$, denoted by s_A(G), such that any sequence {c_i}_{i=1}^m with terms from G has a length n=exp(G) subsequence {c_{i_j}}_{j=1}^n for which there are a_1,...,a_n in A such that sum_{j=1}^na_ic_{i_j}=0. When G is a p-group, A contains no multiples of p and any two distinct elements of A are incongruent mod p, we show that s_A(G) is at most $\lceil D(G)/|A|\rceil+exp(G)-1$ if |A| is at least (D(G)-1)/(exp(G)-1), where D(G) is the Davenport constant of G and this upper bound for s_A(G)in terms of |A| is essentially best possible. In the case A={1,-1}, we determine the asymptotic behavior of s_{{1,-1}}(G) when exp(G) is even, showing that, for finite abelian groups of even exponent and fixed rank, s_{{1,-1}}(G)=exp(G)+log_2|G|+O(log_2log_2|G|) as exp(G) tends to the infinity. Combined with a lower bound of $exp(G)+sum{i=1}{r}\lfloor\log_2 n_i\rfloor$, where $G=\Z_{n_1}\oplus...\oplus \Z_{n_r}$ with 1<n_1|... |n_r, this determines s_{{1,-1}}(G), for even exponent groups, up to a small order error term. Our method makes use of the theory of L-intersecting set systems. Some additional more specific values and results related to s_{{1,-1}}(G) are also computed.Comment: 24 pages. Accepted version for publication in Adv. in Appl. Mat

On a question regarding visibility of lattice points—III

AbstractFor a positive integer m, let ω(m) denote the number of distinct prime factors of m. Let h(n) be a function defined on the set of positive integers such that h(n)→∞ as n→∞ and let En(h)={d:disapositiveinteger,d⩽n,ω(d)⩾h(n)}. Writing Δn={(x,y):x,yareintegers,1⩽x,y⩽n}, in the present paper we show that one can give explicit description of a set Xn⊂Δn such that Δn is visible from Xn with at most 100|En(h)|2 exceptional points and for all sufficiently large n, one has|Xn|⩽800h(n)loglogh(n).As a corollary it follows that one can give explicit description of a set Yn⊂Δn such that for large n's, Δn is visible except for at most 100n2/(loglogn)2 exceptional points from Yn where Yn satisfies|Yn|=O((loglogn)(loglogloglogn))

Remarks on monochromatic configurations for finite colorings of the plane

Gurevich had conjectured that for any finite coloring of the Euclidean plane, there always exists a triangle of unit area with monochromatic vertices. Graham ([5], [6]) gave the first proof of this conjecture; a much shorter proof has been obtained recently by Dumitrescu and Jiang [4]. A similar result in the case of a trapezium, claimed by the present authors in [3] does not hold due to an error and a weaker result is recovered for quadrilaterals in this paper. We also take up the original question of triangle

Monochromatic configurations for finite colourings of the plane

A strengthened form of Gurevich's conjecture was proved by R.L.Graham ([2],[3]),Which says that for any \alpha >0 and any pair of non-parallel lines $L_1$ and $L_2$, in any partition of the plane into finitely many classes, some class contains the vertices of a triangle which has area $\alpha$ and two sides parallel to the lines $L_i$.  Later, a shorter proof, using the main idea of Graham, was presented in [1]. Following some questions raised by Graham [2] and by suitable modifications of methods therein, here we establish a similiar in the case of vertices of a trapezium

On a question of regarding visibility of lattice points

Let Δn = {(x, y): x, y are integers 1 ≤ x, y ≤ n} be the n x n square array of integer lattice points in the plane

Partition function congruences: Some flowers and seeds from ‘Ramanujan's garden’

AbstractThe discovery of some partition function congruences by Ramanujan, and subsequent research motivated by these congruences as well as some of his questions and conjectures, have brought forth a beautiful bower in ‘Ramanujan's Garden’.In this short expository article, starting from Ramanujan's pioneering work in this area, we have some glimpses of contributions of many of the later researchers like Atkin, Watson, Newman, Winquist, Zuckerman, Dyson, Andrews, Garvan, Schinzel, Wirsing, Nicolas, Ruzsa, Sárközy, Serre, Berndt and Ono. While we dwell mainly on the question of parity of p(n) and related topics, we try to mention other important achievements in the area