Dr. Jesús Guillera

Last update:   September 25, 2014           

Mathematical genealogy      Extraordinary prize of doctorate 2007  (number theory)  THESIS  PDF  


Here is the fastest of my proved formulas for pi (3 digits per term) written in a popular form. I proved it in 2002 by the WZ-method.

And here is the fastest of all my formulas for pi (5 digits per term). It has not been proved yet. I found it in 2003 using the PSLQ algorithm.

In 2013 I have obtained a smooth representation of the function of Mangoldt as a sum over all the non-trivial zeros of the Riemann zeta function. Below we write the formula assuming the RH:

where gamma denotes the imaginary parts of the zeros of zeta and x<pi tending to pi . Here we show the graphic when we take x=3.14 and sum over the 10000 first zeros of zeta.

PAPERS (Arxiv)
Some binomial series obtained by the WZ-method. (arXiv 0503345)
(Accepted by D. Zeilberger). Adv. in Appl. Math. 29, 599 - 603. (Open Archive, 7 PDF)
About a new kind of Ramanujan-type series. 
Exp. Math. 12, 507 - 510. PDF (Project Euclid PDF)

Generators of some Ramanujan formulas. (arXiv 1104.0392)
The Ramanujan J. 11, 41 - 48.

A new method to obtain series for 1/pi and 1/pi2
Exp. Math. 15, 83 - 89. (Project Euclid PDF 

A class of conjectured series representations for 1/pi and 1/pi2
Exp. Math. 15, 409 - 414. (Project Euclid PDF)

Historia de las fórmulas y algoritmos para pi.  
La Gaceta de la RSME, 10, 159 - 178. PDF

Construction of binomial sums for pi and polylogarithmic constants inspired in BBP formulas. 
(with  B. Gourevitch). Appl. Math. E. Notes, 7, 237 -  246 PDF                              

Hypergeometric identities for 10 extended Ramanujan-type series. (arXiv 1104.0396) 
The Ramanujan J. 15, 219 - 234.  

Double integrals and infinite products for some classical constants. (arXiv 0506319)
(with J. Sondow). The Ramanujan J, 16, 247 - 270.  

Easy proofs of some Borwein's algorithms for pi. (arXiv 0803.0991)
The Amer. Math. Monthly, 115, 850 - 854. 

On WZ-pairs which prove Ramanujan series. (arXiv 0904.0406
The Ramanujan J. 22, 249 - 259.  

History of the formulas and algorithms for pi. (arXiv 0807.0872)
Gems in Experimental Mathematics: Contemp. Math. 517, 173 - 178. 

A matrix form of Ramanujan-type series for 1/pi. (arXiv 0907.1547)
Gems in Experimental Mathematics: Contemp. Math. 517, 189 - 206.  

A new Ramanujan-like series for 1/pi2(arXiv 1003.1915)
The Ramanujan J.  26, 369 - 374.   

"Divergent" Ramanujan-type supercongruences. (arXiv 1004.4337)
(with W. Zudilin), Proc. of the Amer. Math. Soc. 140, 765 - 777.

Mosaic supercongruences of Ramanujan-type. (arXiv 1007.2290)
Exp. Math. 21,  65 - 68. 

Ramanujan-like series for 1/pi2 and String Theory. (arXiv 1009.5202)
(with Gert Almkvist). Exp. Math. 21, 223 - 234. 

Ramanujan-Sato-like series. (arXiv 1201.5233)
(with G. Almkvist), Number Theory & related fields: Springer Proceedings in Mathematics  & Statistics, 43, 55 - 74, in memory of Alf van der Poorteen. 
WZ-Proofs of "Divergent" Ramanujan-Type Series. (arXiv 1012.2681) 
Advances in Combinatorics, (in memory of Herbert S. Wilf), I. Kotsireas and E.V. Zima (eds.), Springer, 187-195.   

More hypergeometric identities related to Ramanujan-type series. (arXiv 1104.1994)
The Ramanujan J. 32, 5 - 22. 

Ramanujan-type formulae for 1/pi: the art of translation.  (arXiv 1302.0548)
(with W. Zudilin), in The Legacy of Srinivasa Ramanujan, R. Balasubramanian et al. (eds)
Ramanujan Math. Soc. Lecture Notes series 20, 181 - 195.  
Ramanujan series upside-down. (arXiv 1206.3981)
(with M. Rogers), Journal of the Australian Math. Society, 97, 78 - 106.
2014 -   (To appear)
Mahler measure and the WZ-algorithm. (arXiv 1006.1654)
(with M. Rogers), Proceedings of the AMS. 

Kind of proofs of Ramanujan-like series. (arXiv 1203.1255)

About a class of Calabi-Yau differential equations. (arXiv 1310.6658)
(with Gert Almkvist and Michael Bogner).  

Some sums over the non-trivial zeros of the Riemann zeta function. (arXiv 1307.5723)

The Amer. Math. Monthly115 - 7, (2008) p. 665. Problem 11381. See it in the web of J. Sondow  (coauthor)
Siam ProblemsA new formula for pi related to series of Ramanujan. Classical Analysis. Sequences and Series.

(ideas, formulas, conjectures, etc)

My pi formulas PDF
Series closely related to Ramanujan formulas for pi. PDF
Tables of Ramanujan series with rational values of z. PDF
Chains of series for 1/pi associated to WZ-pairs. PDF
Expansions related to Ramanujan series and alike. PDF
Collection of Ramanujan-like series for 1/pi2. PDF
Some challenging formulas for pi. PDF

El método WZ y las series de tipo Ramanujan para pi. PDF
Seminario Rubio de Francia. Univ de Zaragoza (11 de Marzo de 2004).
WZ-method proofs of some Ramanujan-type series for 1/pi and new series for 1/pi2. PDF
Journées Aritmet. XXIV. Marseille (July 5, 2005).
Series de Ramanujan: Generalizaciones y conjeturas. PDF
Thesis presentation. Univ de Zaragoza (2 de Julio de 2007).
Seminario Teoría de Números. Univ. Autónoma de Madrid (22 de Noviembre de 2007).
Ramanujan-like series for 1/pi2 and String Theory PDF
Centenario de la RSME. Palacio de congresos de Ávila. (4 de Febrero de 2011).
K-Theory, Quadratic Forms and Number Th. Seminar. School of Math. Sci. Univ. College Dublin. (Feb. 23, 2011).
Seminario Rubio de Francia. Univ de Zaragoza. (17 de Marzo de 2011)
Seminario Teoría de Números. Univ. del País Vasco, Bilbao (12 de Mayo de 2011).
Ramanujan series upside-down  PDF (30 minutes) PDF (50 minutes)
Seminario Rubio de Francia. Univ de Zaragoza (21 de Marzo de 2013).
Seminario Gama. Univ. Carlos III de Madrid (11 de Abril de 2013).
Quintas Jornadas de Teoría de Números. Univ de Sevilla (8-12 de Julio de 2013).
Arithmetical functions and zeros of zeta PDF
Seminario Rubio de Francia. Univ de Zaragoza (6 de Febrero de 2014).
Proofs of some Ramanujan series by the WZ-method PDF
Rutgers Experimental Mathematics Seminar (September 18, 2014).