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We give some corrections of our paper "Primes in arithmetic progressions to large moduli"[BFI]. The corrections do not affect the statements of any of the theorems in the paper. The contents of our two sequel papers [BFI2, BFI3] also remain unchanged.

We fix a gap in our proof of an upper bound for the number of positive integers $n\le x$ for which the Euler function $\varphi(n)$ has all prime factors at most $y$. While doing this we obtain a stronger, likely best-possible result.

We give a new proof of a theorem of B.M. Bredihin which was originally proved by extending Linnik's solution, via his dispersion method, of a problem of Hardy and Littlewood.

This paper was written, apart from one technical correction, in July and August of 2013. The, then very recent, breakthrough of Y. Zhang \cite{Z} had revived in us an intention to produce a second edition of our book "Opera de Cribro", one which would include an account of Zhang's result, stressing the sieve aspects of the method. A complete and connected version of the proof, in our style but not intended for journal publication, seemed a natural first step in this project. Following the further spectacular advance given by J. Maynard (arXiv:1311.4600, Nov 20, 2013), we have had to re-think our position. Maynard's method, at least in its current form, proceeds from GPY in quite a different direction than does Zhang's, and achieves numerically superior results. Consequently, although Zhang's contribution to the distribution of primes in arithmetic progressions certainly retains its importance, the fact remains that much of the material in this paper would no longer appear in a new edition of our book. Because this paper contains some innovations that we do not wish to become lost, we have decided to make the work publicly available in its current form.

Fixing a nontrivial automorphism of a number field K, we associate to ideals in K an invariant (with values in {0,1,-1}) that we call the "spin" and for which the associated L-function does not possess Euler products. We are nevertheless able, using the techniques of bilinear forms, to handle spin value distribution over primes, obtaining stronger results than the analogous ones which follow from the technology of L-functions in its current state. The initial application of our theorem is to the arithmetic statistics of Selmer groups of elliptic curves., Comment: Minor revisions. To appear in Inventiones Mathematicae

Science On a Sphere (SOS) is a large visualization system, developed by the National Oceanic and Atmospheric Administration (NOAH), that uses computers running Redhat Linux and four video projectors to display animated data onto the outside of a sphere. Said another way, SOS is a stationary globe that can show dynamic, animated images in spherical form. Visualization of cylindrical data maps show planets, their atmosphere, oceans, and land, in very realistic form. The SOS system uses 4 video projectors to display images onto the sphere. Each projector is driven by a separate computer, and a fifth computer is used to control the operation of the display computers. Each computer is a relatively powerful PC with a high-end graphics card. The video projectors have native XGA resolution. The projectors are placed at the corners of a 30' x 30' square with a 68" carbon fiber sphere suspended in the center of the square. The equator of the sphere is typically located 86" off the floor. SOS uses common image formats such as JPEG, or TIFF in a very specific, but simple form; the images are plotted on an equatorial cylindrical equidistant projection, or as it is commonly known, a latitude/longitude grid, where the image is twice as wide as it is high (rectangular). 2048x] 024 is the minimum usable spatial resolution without some noticeable pixelation. Labels and text can be applied within the image, or using a timestamp-like feature within the SOS system software. There are two basic modes of operation for SOS: displaying a single image or an animated sequence of frames. The frame or frames can be setup to rotate or tilt, as in a planetary rotation. Sequences of images that animate through time produce a movie visualization, with or without an overlain soundtrack. After the images are processed, SOS will display the images in sequence and play them like a movie across the entire sphere surface. Movies can be of any arbitrary length, limited mainly by disk space and can be animated at frame rates up to 30 frames per second. Transitions, special effects, and other computer graphics techniques can be added to a sequence through the use of off-the-shelf software, like Final Cut Pro. However, one drawback is that the Sphere cannot be used in the same manner as a flat movie screen; images cannot be pushed to a "side", a highlighted area must be viewable to all sides of the room simultaneously, and some transitions do not work as well as others. We discuss these issues and workarounds in our poster.