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William D. Banks

Publications and source records attributed to William D. Banks.

45 records · Page 3Linked to original sources
arXiv2007

The Nicolas and Robin inequalities with sums of two squares

In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality $σ(n) 5040$, where $σ(n)$ is the sum of divisors function, and $γ$ is the Euler-Mascheroni constant. We exhibit a broad class of subsets $\cS$ of the natural numbers such that the Robin inequality holds for all but finitely many $n\in\cS$. As a special case, we determine the finitely many numbers of the form $n=a^2+b^2$ that do not satisfy the Robin inequality. In fact, we prove our assertions with the Nicolas inequality $n/ϕ(n) 1$ our results for the Robin inequality follow at once.

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arXiv2007

Prime numbers with Beatty sequences

A study of certain Hamiltonian systems has lead Y. Long to conjecture the existence of infinitely many primes of the form $p=2[αn]+1$, where $1<α<2$ is a fixed irrational number. An argument of P. Ribenboim coupled with classical results about the distribution of fractional parts of irrational multiples of primes in an arithmetic progression immediately imply that this conjecture holds in a much more precise asymptotic form. Motivated by this observation, we give an asymptotic formula for the number of primes $p=q[αn+β]+a$ with $n\le N$, where $α,β$ are real numbers such that $α$ is positive and irrational of finite type (which is true for almost all $α$) and $a,q$ are integers with $0\le a 0$ depends only on $α$. We also prove a similar result for primes $p=[αn+β]$ such that $p\equiv a\pmod q$.

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arXiv2006

Character sums with Beatty sequences on Burgess-type intervals

We estimate multiplicative character sums taken on the values of a non-homogeneous Beatty sequence $\{[αn + β] : n =1,2,... \}$, where $α,β\in\R$, and $α$ is irrational. Our bounds are nontrivial over the same short intervals for which the classical character sum estimates of Burgess have been established.

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arXiv2006

Integers with a large smooth divisor

We study the function $Θ(x,y,z)$ that counts the number of positive integers $n\le x$ which have a divisor $d>z$ with the property that $p\le y$ for every prime $p$ dividing $d$. We also indicate some cryptographic applications of our results.

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arXiv2005

Uniform Distribution of Fractional Parts Related to Pseudoprimes

We estimate exponential sums with the Fermat-like quotients $$ f_g(n) = \frac{g^{n-1} - 1}{n} \mand h_g(n)=\frac{g^{n-1}-1}{P(n)}, $$ where $g$ and $n$ are positive integers, $n$ is composite, and P(n) is the largest prime factor of $n$. Clearly, both $f_g(n)$ and $h_g(n)$ are integers if $n$ is a Fermat pseudoprime to base $g$, and if $n$ is a Carmichael number this is true for all $g$ coprime to $n$. Nevertheless, our bounds imply that the fractional parts $\{f_g(n)\}$ and $\{h_g(n)\}$ are uniformly distributed, on average over $g$ for $f_g(n)$, and individually for $h_g(n)$. We also obtain similar results with the functions ${\widetilde f}_g(n) = gf_g(n)$ and ${\widetilde h}_g(n) = gh_g(n)$.

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arXiv2004

Noncototients and Nonaliquots

Let $ϕ(\cdot)$ and $σ(\cdot)$ denote the Euler function and the sum of divisors function, respectively. In this paper, we give a lower bound for the number of positive integers $m\le x$ for which the equation $m=n-ϕ(n)$ has no solution. We also give a lower bound for the number of $m\le x$ for which the equation $m=σ(n)-n$ has no solution. Finally, we show the set of positive integers $m$ not of the form $(p-1)/2-ϕ(p-1)$ for some prime number $p$ has a positive lower asymptotic density.

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arXiv2004

Almost all palindromes are composite

We study the distribution of palindromic numbers (with respect to a fixed base $g\ge 2$) over certain congruence classes, and we derive a nontrivial upper bound for the number of prime palindromes $n\le x$ as $x\to\infty$. Our results show that almost all palindromes in a given base are composite.

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