Ramanujan, modular equations, and approximations to Pi or how to compute one billion digits of Pi

@article{Borwein1989RamanujanME,
  title={Ramanujan, modular equations, and approximations to Pi or how to compute one billion digits of Pi},
  author={Jonathan M. Borwein and Peter B. Borwein and David H. Bailey},
  journal={American Mathematical Monthly},
  year={1989},
  volume={96},
  pages={201-219}
}
The year 1987 was the centenary of Ramanujan’s birth. He died in 1920 Had he not died so young, his presence in modern mathematics might be more immediately felt. Had he lived to have access to powerful algebraic manipulation software. such as MACSYMA, who knows how much more spectacular his already astonishing career might have been. 
The Life of π: From Archimedes to ENIAC and Beyond
The desire to understand pi, the challenge, and originally the need, to calculate ever more accurate values of pi, the ratio of the circumference of a circle to its diameter, has challengedExpand
History of the formulas and algorithms for pi
Throughout more than two millennia many formulas have been obtained, some of them beautiful, to calculate the number pi. Among them, we can find series, infinite products, expansions as continuedExpand
New proofs of Borwein-type algorithms for Pi
ABSTRACT We use a method of translation to recover Borweins' quadratic and quartic iterations. Then, by using the WZ-method, we obtain some initial values which lead to the limit . We use neither theExpand
Birth, growth and computation of pi to ten trillion digits
The universal real constant pi, the ratio of the circumference of any circle and its diameter, has no exact numerical representation in a finite number of digits in any number/radix system. It hasExpand
THE $w$-MODULAR FUNCTION AND THE EVALUATION OF ROGERS RAMANUJAN CONTINUED FRACTION
Abstract: In previous article we have considered attached to elliptic singular moduli kr, the parameter wr, in which if one knows its value then can evaluate explicit the elliptic singular moduli andExpand
On the rapid computation of various polylogarithmic constants
TLDR
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
Kind of proofs of Ramanujan-like series
We make a summary of the different types of proofs adding some new ideas. In addition we conjecture some relations which could be necessary in "modular type proofs" (not still found) of theExpand
Rational analogues of Ramanujan's series for 1/π†
  • H. Chan, Shaun Cooper
  • Mathematics
  • Mathematical Proceedings of the Cambridge Philosophical Society
  • 2012
Abstract A general theorem is stated that unifies 93 rational Ramanujan-type series for 1/π, 40 of which are believed to be new. Moreover, each series is shown to have a companion identity, therebyExpand
Evaluation of Fifth Degree Elliptic Singular Moduli
In this article we extract solutions in radicals and elementary functions of the fifth degree modular equation. More precisely we evaluate the values of $k_{25^nr_0}$, when we know only twoExpand
Pi Day Is Upon Us Again and We Still Do Not Know if Pi Is Normal
TLDR
The history of this venerable constant is reviewed, some recent research on the question of whether π is normal, or whether its digits are statistically random in a specific sense are described. Expand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 64 REFERENCES
The Computation of π to 29,360,000 Decimal Digits Using Borweins’ Quartically Convergent Algorithm
Paper 7: David H. Bailey, “The computation of pi to 29,360,000 decimal digits using Borweins’ quartically convergent algorithm,” Mathematics of Computation, vol. 50 (1988), p. 283–296. Reprinted byExpand
Transcendental Number Theory
First published in 1975, this classic book gives a systematic account of transcendental number theory, that is those numbers which cannot be expressed as the roots of algebraic equations havingExpand
Numerical results on the transcendence of constants involving pi, e, and Euler's constant
The existence of simple polynomial equations (integer relations) for the constants e/pi, e + pi, log pi, gamma (Euler's constant), e exp gamma, gamma/e, gamma/pi, and log gamma is investigated byExpand
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
Lehrbuch der Algebra
A “Treatise on Algebra” is rarely found to fulfil the promise of its title. It is too often a mere collection of problems and examples, thrown together without much regard to order or method; suchExpand
Computation of π Using Arithmetic-Geometric Mean
A new formula for π is derived. It is a direct consequence of Gauss’ arithmetic-geometric mean, the traditional method for calculating elliptic integrals, and of Legendre’s relation for ellipticExpand
OF CALCULATIONS PAST AND PRESENT: THE ARCHIMEDEAN ALGORITHM
was a forerunner of a certain type of algorithms used on today's computers. Stimulated by an article of G. M. Phillips [49], which recently appeared in this monthly, we trace the evolution ofExpand
The computational complexity of algebraic and numeric problems
Thank you for downloading the computational complexity of algebraic and numeric problems elsevier computer science library theory of computation series 1. As you may know, people have search numerousExpand
Dihedral quartic approximations and series for π
Abstract Imaginary quadratic fields with class groups that have C(4) as a subgroup are analyzed in depth, and the units of associated dihedral quartic fields are thereby evaluated using Epstein zetaExpand
III. A memoir on the transformation of elliptic functions
  • A. Cayley
  • Mathematics
  • Proceedings of the Royal Society of London
  • 1874
The theory of Transformation in Elliptic Functions was established by Jacobi in the ‘Fundamenta Nova’ (1829); and he has there developed, transcendentally, with an approach to completeness, theExpand
...
1
2
3
4
5
...