Sidney W. Graham

• Doctor of Philosophy
• Professor (Full) at Central Michigan University

We show that for any positive integer n, there is some fixed A such that d(x)=d(x+n)=A infinitely often where d(x) denotes the number of divisors of x. In fact, we establish the stronger result that both x and x+n have the same fixed exponent pattern for infinitely many x. Here the exponent pattern of an integer x>1 is the multiset of nonzero expon...

This paper is intended as a sequel to a paper arXiv:0803.2636 written by four of the coauthors here. In the paper, they proved a stronger form of the Erd\H{o}s-Mirksy conjecture which states that there are infinitely many positive integers $x$ such that $d(x)=d(x+1)$ where $d(x)$ denotes the number of divisors of $x$. This conjecture was first prov...

We recast the classical Lindelöf hypothesis as an estimate for the sums ∑ n ≤ x n − i t \sum _{n\leq x}n^{-it} . This leads us to propose that a more general form of the Lindelöf hypothesis may be true, one involving estimates for sums of the type ∑ n ≤ x n ∈ N n − i t , \begin{equation*} \sum _{ \substack {n\leq x \\ n\in \mathscr {N} }}n^{-it}, \...

The majority of people who suffer morbidity due to smoking may have initiated smoking during adolescent period. In this chapter we look at the prevalence and associated factors for cigarette smoking among schoolgoing adolescents in Lithuania. Data from the Global Youth Tobacco Survey (GYTS) 2005 were used to conduct this study. Data were analyzed u...

We survey various developments in Number Theory that were inspired by classical papers by Roth [On the gaps between squarefree numbers, J. London Math. Soc. 26 (1951) 263–268] and by Halberstam and Roth [On the gaps between consecutive k-free integres, J. London Math. Soc. 26 (1951) 268–273].

In a previous paper, the authors proved that in any system of three linear forms satisfying obvious necessary local conditions, there are at least two forms that infinitely often assume E2-values; that is, values that are products of exactly two primes. We use this result to prove that there are infinitely many positive integers x such that both x...

We show that using character sum estimates due to H. Iwaniec leads to an improvement of recent results about the distribution and finding RSA moduli $M=pl$, where $p$ and $l$ are primes, with prescribed bit patterns. We are now able to specify about $n$ bits instead of about $n/2$ bits as in the previous work. We also show that the same result of H...

Let qn denote the nth number that is a product of exactly two distinct primes. We prove that qn+1 − qn ≤ 6 infinitely often. This sharpens an earlier result of the authors, which had 26 in place of 6. More generally, we prove that if ν is any positive integer, then (qn+ν − qn) ≤ ν eν − γ (1+o(1)) infinitely often. We also prove several other relate...

Let $p_n$ denote the $n^{th}$ prime. Goldston, Pintz, and Yildirim recently proved that $ \liminf_{n\to \infty} \frac{(p_{n+1}-p_n)}{\log p_n} =0.$ We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let $q_n$ denote the $n^{th}$ number that is a product of exactly two distinct p...

We apply van der Corput's method of exponential sums to obtain explicit upper bounds for the Riemann zeta function on the line σ = 1/2. For example, we prove that if t ≥ e, then |ζ(1/2 + it)| ≤ 3t 1/6 log t. These results will be used in an application on primes to short intervals.

this paper is to give a further sharpening. Theorem. The number of n # t for which y n (x) is reducible is # t 2/3 . The first author's earlier work used the Tchebotarev Density Theorem, but the proof given here uses only elementary estimates. Our starting point is the Corollary to Lemma 2 in [1], which states that if (1) # # # p|n(n+1) p # # 2 # #...

Several results involving d(n!) are obtained, where d(m) denotes the number of positive divisors of m. These include estimates for d(n!)/d((n - 1)!), d(n!) - d((n - 1)!), as well as the least number K with d((n +K)!)/d(n!) # 2. 1

Let A be a set (finite or infinite) of natural numbers, and let a i denote a generic element of A. We say that A is a B h sequence if the sums of the form a 1 + … + a h (a 1 ≤ … ≤ a h ) are all distinct. We give an expository account of upper bounds for B h sequences. We also give a new upper bound for B 3 sequences.

We determine the distribution of 3−(q + 1,k,λ) designs, with k ϵ {4,5}, among the orbits of k-element subsets under the action of PSL(2,q), for q ϵ 3 (mod 4), on the projective line. As a consequence, we give necessary and sufficient conditions for the existence of a uniformly-PSL(2,q) large set of 3−(q + 1,k,λ) designs, with k ϵ {4,5} and q ≡ 3 (m...

It is shown that for two common broadcasting problems, a star graph performs better than a k -ary hypercube with a comparable number of nodes only in networks consisting of an impractically large numbers of nodes. This result is based on a comparison of the costs of known solutions to the one-to-all broadcast and the complete broadcast problems for...

Let k be a fixed positive integer, and let sn denote the nth k-free number. If k ≥ 3 and 0 ≤ γ < 2k − 2 + 4/(k + 1), then [formula].

This book is a self-contained account of the one- and two-dimensional van der Corput method and its use in estimating exponential sums. These arise in many problems in analytic number theory. It is the first cohesive account of much of this material and will be welcomed by graduates and professionals in analytic number theory. The authors show how...

Let p be a prime, and let n p denote the least positive integer n such that n is a quadratic non-residue mod p. In 1949, Fridlender [F] and Salié [Sa] independently showed that \( {n_p} = \Omega \left( {\log p} \right) \); in other words, there are infinitely many primes p such that \( {n_p} \geqslant c\log p \) for some absolute constant c. In 197...

In number theory, one often encounters sums of the form Where D is a bounded domain in R k and e(w) =e2πiw. We shall Refer to the case k = 1 as the one-dimensional case, k = 2 as the two- dimensional case, etc. Our objective here is to give an exposition of van der Corput’s method for estimating the sums in (1). The one- dimensional case is well un...

On presente un algorithme pour calculer inf θ ou θ(k,l)=ak+bl+c/dk+el+f, a,...,f etant des nombres reels, (k, l) une paire d'exposants, l'infinum etant pris sur tous les (k, l) produits par les processus A et B

We determine a class of real valued, integrable functions $f(x)$ and corresponding functions $M_f(x)$ such that $f(x) \leqslant M_f(x)$ for all $x$, the Fourier transform $\hat M_f(t)$ is zero when $|t| \geqslant 1$, and the value of $M_f(0)$ is minimized. Several applications of these functions to number theory and analysis are given.

Suppose 1 ≤ z1 ≤ z2 ≤ N, and let for i = 1, 2. We show that We then use this to improve a result of Barban-Vehov which has applications to zero-density theorems.

We study the number and nature of solutions of the equation φ(n) = φ(n + k), where φ denotes Euler's phi-function. We exhibit some families of solutions when k is even, and we conjecture an asymptotic formula for the number of solutions in this case. We show that our conjecture follows from a quantitative form of the prime k-tuples conjecture. We a...