New zero free regions for the derivatives of the Riemann zeta function

The main aim of this paper is twofold. First we generalize, in a novel way, most of the known non-vanishing results for the derivatives of the Riemann zeta function by establishing the existence of an infinite sequence of regions in the right half-plane where these derivatives cannot have any zeros; and then, in the rare regions of the complex plane that do contain zeros of the k-th derivative of the zeta function, we describe a unexpected phenomenon, which implies great regularities in their zero distributions. In particular, we prove sharp estimates for the number of zeros in each of these new critical strips, and we explain how they converge, in a very precise, periodic fashion, to their central, critical lines, as k increases. This not only shows that the zeros are not randomly scattered to the right of the line Re(s)=1, but that, in many respects, their two-dimensional distribution eventually becomes much simpler and more predictable than the one-dimensional behavior of the zeros of the zeta function on the line Re(s)=1/2

Compositions with the Euler and Carmichael Functions

http://www.math.missouri.edu/~bbanks/papers/index.htmlLet ' and _ be the Euler and Carmichael functions, respectively. In this paper, we establish lower and upper bounds for the number of positive integers n ≤ x such that '(_(n)) = _('(n)). We also study the normal order of the function '(_(n))/_('(n))

Values of Arithmetical Functions Equal to a Sum of Two Squares

http://www.math.missouri.edu/~bbanks/papers/index.htmlLet '(n) denote the Euler function. In this paper, we determine the order of growth for the number of positive integers n ≤ x for which '(n) is the sum of two square numbers. We also obtain similar results for the Dedekind function (n) and the sum of divisors function _(n)

On the Prime Number Lemma of Selberg

The key result needed in almost all elementary proofs of the Prime Number Theorem is a prime number lemma proved by Atle Selberg in 1948.Without restricting ourselves to purely elementary techniques we show how the error term in Selberg’s fundamental lemma relates to the error term in the Prime Number Theorem. In spite of all the interest in this topic over the last sixty years this particular question seems to have been overlooked in the past

Bounds for Recurrences on Ranked Posets

This note considers an extension of the concept of linear recurrence to recurrences on ranked posets. Some results on growth rates in the linear case are then extended to this generalized scenario. The work is motivated by recent results on multi-dimensional recurrences which have had applications for obtaining bounds for complex multidimensional generating functions. Some further connections to Möbius functions for binary relations and inverses of {0, 1} triangular matrices are also discussed