Military Marches: Song

https://digitalcommons.library.umaine.edu/mmb-vp/2909/thumbnail.jp

Say It Again

https://digitalcommons.library.umaine.edu/mmb-vp/2424/thumbnail.jp

Pitter - Patter

https://digitalcommons.library.umaine.edu/mmb-vp/4033/thumbnail.jp

I Saved A Waltz For You

Illustration of clouds and a rainbowhttps://scholarsjunction.msstate.edu/cht-sheet-music/1457/thumbnail.jp

Exponential Sums over Mersenne Numbers

© Foundation Compositio Mathematica 2004. Cambridge Journals. doi: 10.1112/S0010437X03000022.We give estimates for exponential sums of the form Σn≤N Λ(n) exp(2πiagn/m), where m is a positive integer, a and g are integers relatively prime to m, and Λ is the von Mangoldt function. In particular, our results yield bounds for exponential sums of the form Σ p≤N exp(2πiaMp/m), where Mp is the Mersenne number; Mp = 2p −1 for any prime p.We also estimate some closely related sums, including Σn≤N μ(n) exp(2πiagn/m) and Σn≤N μ2(n) exp(2πiagn/m), where μ is the Möbius function

Multiplicative Structure of Values of the Euler Function

This is a preprint of a book chapter published in High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams, Fields Institute Communications, AMS (2004). © American Mathematical Society.We establish upper bounds for the number of smooth values of the Euler function. In particular, although the Euler function has a certain “smoothing” effect on its integer arguments, our results show that, in fact, most values produced by the Euler function are not smooth. We apply our results to study the distribution of “strong primes”, which are commonly encountered in cryptography. We also consider the problem of obtaining upper and lower bounds for the number of positive integers n ≤ x for which the value of the Euler function φ (n) is a perfect square and also for the number of n ≤ x such that φ (n) is squarefull. We give similar bounds for the Carmichael function λ (n)

Incomplete exponential sums and Diffie-Hellman triples

http://www.math.missouri.edu/~bbanks/papers/index.htmlLet p be a prime and 79 an integer of order t in the multiplicative group modulo p. In this paper, we continue the study of the distribution of Diffie-Hellman triples (V-x, V-y, V-xy) by considering the closely related problem of estimating exponential sums formed from linear combinations of the entries in such triples. We show that the techniques developed earlier for complete sums can be combined, modified and developed further to treat incomplete sums as well. Our bounds imply uniformity of distribution results for Diffie-Hellman triples as the pair (x, y) varies over small boxes

The fundamental pro-groupoid of an affine 2-scheme

A natural question in the theory of Tannakian categories is: What if you don't remember \Forget? Working over an arbitrary commutative ring $R$, we prove that an answer to this question is given by the functor represented by the \'etale fundamental groupoid \pi_1(\spec(R)), i.e.\ the separable absolute Galois group of $R$ when it is a field. This gives a new definition for \'etale \pi_1(\spec(R)) in terms of the category of $R$-modules rather than the category of \'etale covers. More generally, we introduce a new notion of "commutative 2-ring" that includes both Grothendieck topoi and symmetric monoidal categories of modules, and define a notion of $\pi_1$ for the corresponding "affine 2-schemes." These results help to simplify and clarify some of the peculiarities of the \'etale fundamental group. For example, \'etale fundamental groups are not "true" groups but only profinite groups, and one cannot hope to recover more: the "Tannakian" functor represented by the \'etale fundamental group of a scheme preserves finite products but not all products.Comment: 46 pages + bibliography. Diagrams drawn in Tik