August 20, 2017


 Dr. Jesús Guillera

University of Zaragoza - Departament of Mathematics (Colaborator)

(jguillera [at] gmail [dot] com)

THESIS (PDF) - Extraordinary Prize of doctorate - Mathematical Genealogy

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 Coloquios Cine...Máticos  (Vimeo).


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

03) Generators of some Ramanujan formulas (arXiv 1104.0392). The Ramanujan J. 11, 41 - 48.
04) A new method to obtain series for 1/pi and 1/pi. Exp. Math. 15, 83 - 89. (Project Euclid PDF)
05) A class of conjectured series representations for 1/pi and 1/pi2 . Exp. Math. 15, 409 - 414. (Project Euclid PDF)

06) Historia de las fórmulas y algoritmos para pi. La Gaceta de la RSME, 10, 159 - 178. PDF
07) Construction of binomial sums for pi and polylogarithmic constants inspired in BBP formulas, with  B. Gourevitch. Appl. Math. E. Notes, 7, 237 -  246 PDF                              

08) Hypergeometric identities for 10 extended Ramanujan-type series. (arXiv 1104.0396). The Ramanujan J. 15, 219 - 234.  
09) Double integrals and infinite products for some classical constants, with J. Sondow (arXiv 0506319).  The Ramanujan J, 16, 247 - 270.  
10) Easy proofs of some Borwein's algorithms for pi. (arXiv 0803.0991). The Amer. Math. Monthly, 115, 850 - 854. 
11) On WZ-pairs which prove Ramanujan series. (arXiv 0904.0406). The Ramanujan J. 22, 249 - 259.  

12) History of the formulas and algorithms for pi. (arXiv 0807.0872). Gems in Experimental Mathematics: Contemp. Math. 517, 173 - 178. 
13) A matrix form of Ramanujan-type series for 1/pi. (arXiv 0907.1547). Gems in Experimental Mathematics: Contemp. Math. 517, 189 - 206.  

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

15) "Divergent" Ramanujan-type supercongruences, with W. Zudilin (arXiv 1004.4337). Proc. of the Amer. Math. Soc. 140, 765 - 777.
16) Mosaic supercongruences of Ramanujan-type. (arXiv 1007.2290). Exp. Math. 21,  65 - 68. 
17) Ramanujan-like series for 1/pi2 and String Theory, with Gert Almkvist (arXiv 1009.5202). Exp. Math. 21, 223 - 234. 

18) Ramanujan-Sato-like series, with G. Almkvist (arXiv 1201.5233). NT & related fields: Springer Proceedings in Mathematics  & Statistics, 43, 55 - 74, in memory of Alf van der Poorteen. 
19) 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.   
20) More hypergeometric identities related to Ramanujan-type series. (arXiv 1104.1994). The Ramanujan J. 32, 5 - 22. 
21) Ramanujan-type formulae for 1/pi: the art of translation, with W. Zudilin (arXiv 1302.0548).,in The Legacy of S. Ramanujan, R. Balasubramanian et al. (eds). Ramanujan Math. Soc. Lecture Notes series 20, 181 - 195.  
22) Ramanujan series upside-down, with M. Rogers (arXiv 1206.3981). Journal of the Australian Math. Society, 97, 78 - 106.
23) Mahler measure and the WZ-algorithm, with M. Rogers (arXiv 1006.1654). Proceedings of the AMS, 143, 2873 - 2886.  
24) A family of Ramanujan-Orr formulas for 1/pi. (arXiv 1501.06413). Integral Transforms and Special Functions, 26, 531 - 538.

25) New proofs of Borwein-type algorithms for Pi. (arXiv  1604.00193). Integral Transforms and Special Functions (published online: June 29).

26) Crouching AGM, Hidden Modularity, with S. Cooper, A. Straub, and W. Zudilin (arXiv 1604.01106). in Frontiers in Orthogonal Polynomials and q-Series, M. Zuhair Nashed and Xin Li (eds.) (2018), World Scientific, 169–187 
27) Self-replication and Borwein-like algorithms (arXiv 1702.05378). The Ramanujan J. (Published online: August 16)
28) Dougall's 5F4 sum and the WZ-algorithm (arXiv 1611.04385)
29) The Mangoldt function and the non-trivial zeros of the Riemann zeta function (arXiv 1705.06367)


A1) Kind of proofs of Ramanujan-like series. (arXiv 1203.1255)
A2) About a class of Calabi-Yau differential equations, with Gert Almkvist and Michael Bogner (arXiv 1310.6658)
A3) Some sums over the non-trivial zeros of the Riemann zeta function. (arXiv 1307.5723)
A4) Bilateral sums related to Ramanujan-like series (arXiv 1610.04839)


Problem 1: An infinite product for e^x. Problem 11381. See it in the web of J. Sondow LINK (coauthor). The Amer. Math. Monthly. 115 - 7, (2008) p. 665.
Problem 2: A new formula for pi related to series of Ramanujan. Problem 11-003 LINK . Solution with M. Rogers, Siam Problems and solutions. CA, Sequences and Series (2015).
Problem 3: Generalization of a Ramanujan series. LINK. Researchgate. Contributions. Other research. (2015)


My Pi formulas. PDF
Series closely related to Ramanujan formulas for pi. 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

Talks and the papers in which they are based on

El método WZ y las series de tipo Ramanujan para pi. (01-02-03)
Seminario Rubio de Francia. Univ de Zaragoza (11 de Marzo de 2004)  LINK

WZ-method proofs of some Ramanujan-type series for 1/pi and new series for 1 / pi2. (01-02-03)
Journées Aritmetiques XXIV. Marseille (July 5, 2005)  LINK

Series de Ramanujan: Generalizaciones y conjeturas. (01-02-03-04-05-08) 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 (13-17) PDF
Centenario de la RSME Palacio de congresos de Ávila. (4 de Febrero de 2011)  LINK
K-Theory, Quadratic Forms and Number Th. Seminar. School of Math. Sci. Univ. College Dublin. (Feb. 23, 2011)  LINK
Seminario. Rubio de Francia Univ de Zaragoza. (17 de Marzo de 2011)  LINK
Seminario Teoría de Números. Univ. del País Vasco, Bilbao (12 de Mayo de 2011).

Ramanujan series upside-down (22) PDF
Seminario Rubio de Francia. Univ de Zaragoza (21 de Marzo de 2013)  LINK
Seminario Gama. Univ. Carlos III de Madrid (11 de Abril de 2013)  LINK
Quintas Jornadas de Teoría de Números. Univ de Sevilla (8-12 de Julio de 2013)  LINK

Arithmetical functions and zeros of zeta (A3) PDF
Seminario Rubio de Francia. Univ de Zaragoza (6 de Febrero de 2014)  LINK

Proofs of some Ramanujan series by the WZ-method (01-02-03-08-14) PDF
Rutgers Experimental Mathematics Seminar (September 18, 2014) LINK

Una familia de series para 1 / pi2  relacionada con la teoría de Calabi-Yau (13-17-18) PDF
Seminario Martes Cuántico. Univ de Zaragoza  (21 de Abril de 2015) LINK

Ramanujan-Orr type series (24) PDF
Sextas Jornadas de Teoría de Números. Univ de Valladolid (30 de Junio de 2015 - conferencia plenaria) LINK
Seminario Rubio de Francia. Univ de Zaragoza (12 de Noviembre de 2015) LINK

El número Pi (06-12) PDF
Seminario Rubio de Francia. Univ de Zaragoza (26 de Noviembre de 2015) LINK
Seminario de Actualización en Matemáticas. Univ. de La Rioja (16 de Marzo de 2016) LINK

Self-replication and Borwein-type algorithmi
Séptimas Jornadas de Teoría de Números. Universitat de Lleida (26-29 de Junio de 2017) LINK


I am very grateful to the authors of the following book and articles for their comments about my formulas in the indicated pages or section:

- Los Números. Autores: Javier Cilleruelo y Antonio Córdoba. CSIC, 2010 (páginas 105 y 106).
- Ramanujan and Pi. Author: Jonathan Borwein. In Notices of the Amer. Math. Society (2012): Srinivasa Ramanujan: Going Strong at 125, Part I PDF (pp. 1535-1537).
- Ramanujan series for 1/pi: A survey. Authors: Nayandeep Deka Baruah, Bruce C. Berndt, and Heng Huat Chan, In Amer. Math. Monthly 116 (2009) PDF (Sect 10).
- Arithmetic Hypergeometric Series. Author: Wadim Zudilin, Russian Math Surveys (2011) PDF (Sect. 2.5).