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This paper studies explicit and theoretical bounds for several interesting quantities in number theory, conditionally on the Generalized Riemann Hypothesis. Specifically, we improve the existing explicit bounds for the least qua-dratic non-residue and the least prime in an arithmetic progression. We also refine the classical conditional bounds of Littlewood(More)
We investigate the distribution of the well-studied Shapley–Shubik values in weighted voting games where the agents are stochastically determined. The Shapley–Shubik value measures the voting power of an agent, in typical collective decision making systems. While easy to estimate empirically given the parameters of a weighted voting game, the Shapley values(More)
Modular Forms, Partitions, and Mock Theta Functions Ramanujan's work on modular forms, partitions, and mock theta functions is rich with tantalizing examples of deeper mathematical structures. Indeed , his tau-function, his partition congruences, and his mock theta functions are prototypes that have helped to shape the modern mathematical landscape.(More)
Although the sequence of primes is very well distributed in the reduced residue classes [Formula: see text], the distribution of pairs of consecutive primes among the permissible ϕ(q)(2) pairs of reduced residue classes [Formula: see text] is surprisingly erratic. This paper proposes a conjectural explanation for this phenomenon, based on the(More)
When numbers are added in the usual way carries occur along the route. These carries cause a mess and it is natural to seek ways to minimize them. This paper proves that balanced arithmetic minimizes the proportion of carries. It also positions carries as cocycles in group theory and shows that if coset representatives for a finite-index normal subgroup H(More)
Dedicated to my grandmother, Marianne Bodenheimer ii ACKNOWLEDGEMENTS I would like to thank my advisor, Christopher Skinner, for many useful conversations and for his generosity in sharing his ideas. I have learned much from him that will continue to be of use in my future research. I am grateful to him for suggesting the problem that led to the topic of(More)
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