Quadratic transformations and Guillera’s formulas for 1/π2

We prove two new series of Ramanujan type for 1/π².     

On theorems of Gelfond and Selberg concerning integral-valued entire functions

For each sN define the constant θ_s with the following properties: if an entire function g(z) of type t(g)<θ_s satisfies then g is a polynomial; conversely, for any δ>0 there exists an entire transcendental function g(z) satisfying the display conditin and t(g)<θ_s+δ. The result θ₁=log2 is known due to Hardy and Pólya. We provide the upper bound θ_sπs/3 and improve earlier lower bounds due to Gelfond (1929) and Selberg (1941).     

Diversity of phage-displayed libraries of peptides during panning and amplification

The amplification of phage-displayed libraries is an essential step in the selection of ligands from these libraries. The amplification of libraries, however, decreases their diversity and limits the number of binding clones that a screen can identify. While this decrease might not be a problem for screens against targets with a single binding site (e.g., proteins), it can severely hinder the identification of useful ligands for targets with multiple binding sites (e.g., cells). This review aims to characterize the loss in the diversity of libraries during amplification. Analysis of the peptide sequences obtained in several hundred screens of peptide libraries shows explicitly that there is a significant decrease in library diversity that occurs during the amplification of phage in bacteria. This loss during amplification is not unique to specific libraries: it is observed in many of the phage display systems we have surveyed. The loss in library diversity originates from competition among phage clones in a common pool of bacteria. Based on growth data from the literature and models of phage growth, we show that this competition originates from growth rate differences of only a few percent for different phage clones. We summarize the findings using a simple two-dimensional "phage phase diagram", which describes how the collapse of libraries, due to panning and amplification, leads to the identification of only a subset of the available ligands. This review also highlights techniques that allow elimination of amplification-induced losses of diversity, and how these techniques can be used to improve phage-display selection and enable the identification of novel ligands.     

Ramanujan-type supercongruences

We present several supercongruences that may be viewed as p-adic analogues of Ramanujan-type series for 1/π and 1/π², and prove three of these examples.     

Euler’s constant, q-logarithms, and formulas of Ramanujan and Gosper

The aim of the paper is to relate computational and arithmetic questions about Euler’s constant γ with properties of the values of the q-logarithm function, with natural choice of~q. By these means, we generalize a classical formula for γ due to Ramanujan, together with Vacca’s and Gosper’s series for γ, as well as deduce irrationality criteria and tests and new asymptotic formulas for computing Euler’s constant. The main tools are Euler-type integrals and hypergeometric series. 2000 Mathematics Subject Classification Primary—11Y60; Secondary—11J72, 33C20, 33D15 The work of the second author is supported by an Alexander von Humboldt research fellowship Dedication: To Leonhard Euler on his 300th birthday.     

A refinement of Nesterenko’s linear independence criterion with applications to zeta values

We refine (and give a new proof of) Nesterenko’s famous linear independence criterion from 1985, by making use of the fact that some coefficients of linear forms may have large common divisors. This is a typical situation appearing in the context of hypergeometric constructions of \mathbbQ{\mathbb{Q}}-linear forms involving zeta values or their q-analogs. We apply our criterion to sharpen previously known results in this direction.     

Approximations to -, di- and tri-logarithms

We propose hypergeometric constructions of simultaneous approximations to polylogarithms. These approximations suit for computing the values of polylogarithms and satisfy 4-term Apéry-like (polynomial) recursions.     

One of the Eight Numbers ζ(5),ζ(7),...,ζ(17),ζ(19) Is Irrational

Mathematical Notes -