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Pi and the AGM
Polynomials and Polynomial Inequalities
Chaptern 1 Introduction and Basic Properties.- 2 Some Special Polynomials.- 3 Chebyshev and Descartes Systems.- 4 Denseness Questions.- 5 Basic Inequalities.- 6 Inequalities in Muntz Spaces.-Expand
Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity
Complete Elliptic Integrals and the Arithmetic-Geometric Mean Iteration. Theta Functions and the Arithmetic-Geometric Mean Iteration. Jacobi's Triple-Product and Some Number Theoretic Applications.Expand
Modular Equations and Approximations to π
If we suppose that $$(1 + {e^{ - \pi Nn}})(1 + {e^{ - 3\pi Nn}})(1 + {e^{ - b\pi Nn}}) \ldots = {2^{\frac{1}{4}}}{c^{ - \pi Nn/24}}{G_n} \ldots \ldots \ldots $$ (1) and $$(1 - {e^{ - \piExpand
On the rapid computation of various polylogarithmic constants
These algorithms can be easily implemented, require virtually no memory, and feature run times that scale nearly linearly with the order of the digit desired make it feasible to compute the billionth binary digit of log(2) or π on a modest work station in a few hours run time. Expand
A cubic counterpart of Jacobi’s identity and the AGM
We produce exact cubic analogues of Jacobi's celebrated theta function identity and of the arithmetic-geometric mean iteration of Gauss and Legendre. The iteration in question is $a_n+1 := a_n + 2b_nExpand
Computational Excursions in Analysis and Number Theory
Preface.- Introduction.- LLL and PSLQ.- Pisot and Salem Numbers.- Rudin-Shapiro Polynomials.- Fekete Polynomials.- Products of Cyclotomic Polynomials.- Location of Zeros.- Maximal Vanishing.-Expand
Some cubic modular identities of Ramanujan
There is a beautiful cubic analogue of Jacobi's fundamental theta function identity: θ⁴₃ = θ⁴₄ + θ⁴₂. It is $(\sum_{n,m=-\infty}^{\infty} q^{n^2+nm+m^2})³ = (\sum_{n,m=-\infty}^{\infty}Expand
Giuga's Conjecture on Primality
G. Giuga conjectured that if an integer n satisfies \sum\limits_{k=1}^{n-1} k^{n-1} \equiv -1 mod n, then n must be a prime. We survey what is known about this interesting and now fairly oldExpand
The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike
to the Riemann Hypothesis.- Why This Book.- Analytic Preliminaries.- Algorithms for Calculating ?(s).- Empirical Evidence.- Equivalent Statements.- Extensions of the Riemann Hypothesis.- Assuming theExpand