Jonathan Sondow's Homepage

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A parabolic analog of the arbelos, from [54]

 

   

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Before

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Giving a talk at a meeting of the

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After

EDUCATION: Ph.D. 1965
Princeton University
Advisor:

Mathematical Genealogy

B.A.-with-Honors 1962

 
Diploma and Calculus Prize 1959

MILNOR WINS THE 2004 STEELE PRIZE FOR EXPOSITION
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FOOTNOTE TO MILNOR'S "DIFFERENTIAL TOPOLOGY 46..."
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My question on the Riemann Hypothesis at MathOverflow

2012 slide talk on primes and transcendental numbers at Keio and Hirosaki Universities

DANI WISE'S RESPONSE TO WINNING THE VEBLEN PRIZE
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My Erdös number: 2

A Fixed Point In Amsterdam

<- Four images from the Monthly's Facebook page ->

Five Platonic Solids

NEW AND NOTEWORTHY 30 OCTOBER 2014

* Link in [xixii] to Matiyasevich's paper on RH citing [6] and [17]; publication data for [39] and [49]; link in [xlxi] to Wiki page on my solution to Landau's problem on zeta(s); link in [xlx] to Waldschmidt's "Transcendental Number Theory: recent results and open problems"; publication data for [57] "Ramanujan, Robin, highly composite numbers, and the Riemann Hypothesis" and [7] "Formulas for pi(n) and the nth prime"; link to new paper [66] with Allouche; update of [xlii] and [II] for Finch's book; reference in [4] to Chinese translation; link in [xlix] to paper citing [37]; link in [XIV] to new book "Neverending Fractions, An Introduction to Continued Fractions"; corrected links in [xix] and [xlvi] to Stefan Kraemer's website on Euler's constant and Hamza Khelif's website on the parbelos [54]; expanded title of [47]; links in [32] and [48] to Zbl reviews; DOI for [62]; link at bottom to my papers on Microsoft Academic Search; link in [47] to version of "Gauss and the eccentric Halsted" with 8 portraits; link in [62] to new version; link in [xlviii] to Carlos Rivera's Prime Puzzles & Problems; link in [7] to new version; links in [35] and [54] to MR and Zbl reviews; links in [65] to sequences of ideal multigrades; links in [xlvi] and [xlvii] to Hamza Khelif's and Antonio Oller's work related to the parbelos [54]; link in [XIII] to citations in Ramanujan's Lost Notebook: Part IV

* Link in [xxxv] to Paul Loya's book "Amazing and Aesthetic Aspects of Analysis." My symmetric formula for pi (see below) is a "beautiful expression" and "quite astonishing!"

* Belated publication of my solution [21] to my Monthly Problem 11222: An Infinite Product Based on a Base; the editor comments "The proposer's elegant solution covers all cases simultaneously and efficiently."

* Link [xxvi] to Michel Waldschmidt's Bombay lectures on irrationality; he mentions my work on Euler's constant [5] and gives my geometric proof that e is irrational and my irrationality measure for e [14]

* At long last, the MAA has put a citation of [3] in Dunham's book "Euler, the Master of Us All", and PUP has cited [3] in Havil's book "Gamma: Exploring Euler's Constant" - see [I] and [III] below

* Link [xviii] to John Baez's web page This Week's Finds in Mathematical Physics. He displays the infinite products for e, e^gamma, and pi/2 shown below, calls them "eerily similar" and a "mystery thrown down to us by the math gods, like a bone from on high", and cites my Monthly note "A faster product for pi ..." [12].

* Link [VII] to A. V. Zhukov's book El Omnipresente Número "Pi". My symmetric formula for pi (see below) is an exhibit in his "museum of elegant mathematics."

FORMULAS AND PLOTS FROM MY WORK
(SCROLL DOWN FOR CITATIONS AND LINKS TO PAPERS)

Conjectured by Knopp and first proved by Hasse
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A global series rediscovered in [1], [12]; cited in [iii], [viii], [III], [IV], [VIII], [IX]

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Antisymmetric formula for Euler's constant [3], cited in [i], [iv], [I], [II], [III], [VI]

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Symmetric formula for pi [22], cited in [vi], [xxxv], [VII]

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Series with the number of 1s and 0s in the binary expansion of n, from [17], [18], cited in [xxxii]

 

The product below for e is due to

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Infinite products from [11], [12], cited in [iv], [xviii], [xxiii]; the 2nd found earlier by J. Ser

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Partial products of Wallis's product (left) and my faster product for pi/2, from [12]

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Catalan-type infinite products, from [37] with Yi HUANG

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Hypergeometric formula for Euler's constant [8], [11]

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Series showing ln(4/pi) is an "alternating Euler constant" [6], [17], [23], cited in [iv], [xxxii]

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Integrals from [6], [12], cited in [i], [iv], [XI]. The 3rd is a special case of Kummer's integral.
 

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Fibonacci- & Lucas-number products from [48], where phi is the golden ratio

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Wallis-type infinite products, from [37] with Yi HUANG

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Monthly Problem: a new binomial coefficient identity [19], [5]

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Monthly Problem: inequalities for generalized binomial coefficients, from [9] and [20]

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Monthly Problem: a new base-B generalization of the Woods-Robbins product (B=2), from [21] and [28]

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Monthly Problem: Beukers-type double integral = Nesterenko-type series [5], [8], [10], [30], [VI]

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Monthly Problem: An infinite product for e^x [31], from [16]

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Monthly Problem: the highest power of 2 dividing a power sum [46], from [42], cited in [xxxiii]

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The continued fraction L is a Liouville number with irrationality base one [10]

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A number with irrationality base k/h > 1, and an inequality for all integers p,q with q large [13]

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A geometric proof that e is irrational, and a new measure of its irrationality [14]

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Transcendental numbers if Schanuel's Conjecture holds, from [40] with Diego MARQUES, cited in [xlii]

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T has irrationality base 2 and S has irrationality base infinity [10], [13]

FOR THE THREE PLOTS BELOW I THANK

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If <{log S(n)}> does NOT tend to zero as n increases, then Euler's constant is irrational [5], [8]

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If F(n) -> 0, then Euler's constant is irrational, but is not a super Liouville number [10]

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If the graph -> 0 as n increases, then ln(pi) is irrational, but isn't a super Liouville number [13]

WEB PAGES AND ARTICLES CITING MY WORK IN NUMBER THEORY AND GEOMETRY

i. X. Gourdon and P. Sebah's

Numbers, Constants and Computation/GammaFormulas(PDF)

ii. Matthew Watkins'

Inexplicable Secrets of Creation/RHreformulations

iii. Eric Weisstein's
iv. _____________
v. _____________
vi. _____________
vii. ____________
viii. Philippe Biane, Jim Pitman and Marc Yor's
ix. Wadim Zudilin's

Zeta Values on the Web

x. Michel Waldschmidt's
xi. American Scientist's
xii. Keith Matthews'
xiii. ____________
xiv. Neil J. Sloane's
xv. ____________
xvi. ___________
xvii. ___________
xviii. John Baez's
xix. Stefan Kraemer's
xx. Michel Waldschmidt's
xxi. _______________
xxii. _______________
xxiii. Jesus Guillera's
xxiv. Michel Waldschmidt's
xxv. _______________
xxvi. _______________
xxvii Wikipedia's
xxviii. _________
xxix. Benoît Rittaud's
xxx. Nuit Blanche's
xxxi. Thomas and Joseph Dence's

xxxii. Jean-Paul Allouche's

xxxiii. Doron Zeilberger's

xxxiv. Peter Hegarty's

xxxv. Paul Loya's

xxxvi. PlanetMath's

xxxvii. Yuri Matiyasevich, Filip Saidak, and Peter Zvengrowski's

   
xxxviii. Robert P. Schneider's
xxxix. Aliza Steurer and Thomas Hagedorn's
xl. Keith Matthews'
xli. Jeffrey Lagarias'
xlii. Steven Finch's
xliii. Emmanuel Tsukerman's
xliv. B. Berndt, S. Kim, and A. Zaharescu's
xlv. Jenda Vondra's
xlvi. Hamza Khelif's
xlvii. Antonio M. Oller-Marcén's
xlviii. Carlos Rivera's
xlix. I. Ben-Ari, D. Hay, and A. Roitershtein's
xlx. Michel Waldschmidt's
xlxi. Wikipedia's
xlxii. Yuri Matiyasevich's

BOOKS CITING MY WORK IN NUMBER THEORY

I. 

Euler, the Master of Us All

by William Dunham, MAA, 1999 (8th printing, 2010), p. 36 (my antisymmetric formula for Euler's constant exhibits "a delightful symmetry")

II.

Mathematical Constants

by Steven Finch, Camb. Univ. Press, 2003, pp. 35, 37

III.

Gamma: Exploring Euler's Constant

by Julian Havil, Princeton Univ. Press, 2009, pp. 109, 257

IV.

CRC Concise Encyclopedia of Mathematics

by Eric Weisstein, 2nd ed., CRC Press, 2002, p. 2562

V.

Book of the Year 1967, p. 503

VI. 

Irresistible Integrals: Symbolics, Analysis, and Experiments in the Evaluation of Integrals

by George Boros and Victor Moll, Cambridge Univ. Press, 2004, pp. 234, 293-4, 300, 306

VII. 

El Omnipresente Número "Pi"

by A. V. Zhukov, Serie de Divulgación Científica Matemática, N11, Moscú : URSS, 2005, p. 159

VIII.

Tauberian Theory: A Century of Developments

by Jacob Korevaar, Springer, 2004, p. 326

IX.

Number Theory: Volume II: Analytic and Modern Tools

by Henri Cohen, Springer, 2007, pp. 140, 260

X.

Excursions in Classical Analysis

by Hongwei Chen, Math. Assoc. of America, 2010, pp. 252

XI.

Zeta and q-Zeta Functions and Associated Series and Integrals

by H. M. Srivastava and J. Choi, Elsevier, 2012, pp. 15 (my double integral for Euler's constant is "elegant"), 239, 241, 648

XII. 

Numbers and Functions: From a Classical-Experimental Mathematician's Point of View

by Victor Moll, American Math. Society, 2012, pp. xxii, 278, 371, 453-4, 485, 488-9, 501

XIII. 

Ramanujan's Lost Notebook: Part IV

by George Andrews and Bruce Berndt, Springer, 2013, pp. 157, 164, 424, 433

XIV. 
by Jonathan Borwein, Alf van der Poorten, Jeffrey Shallit, Wadim Zudilin, CUP, 2014, pp. 159, 205

 

My papers on the arXiv

MY WORK IN NUMBER THEORY AND GEOMETRY

NOTE: links below to MR reviews do NOT require a subscription to MathSciNet

CiteSeerX Citations

1. Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series, Proc. Amer. Math. Soc. 120 (1994) 421-424.

2. The Riemann Hypothesis, simple zeros, and the asymptotic convergence degree of improper Riemann sums, Proc. Amer. Math. Soc. 126 (1998) 1311-1314.

3. An antisymmetric formula for Euler's constant, Math. Mag. 71 (1998) 219-220.

4. Zeros of the alternating zeta function on the line R(s)=1, Amer. Math. Monthly 110 (2003) 435-437. Translated into Chinese and published in: Mathematical Advance in Translation, Chinese Academy of Sciences, 33 #2 (2014) 188-189.

5. Criteria for irrationality of Euler's constant, Proc. Amer. Math. Soc. 131 (2003) 3335-3344.

6. Double integrals for Euler's constant and ln 4/pi and an analog of Hadjicostas's formula, Amer. Math. Monthly 112 (2005) 61-65.

7. Formulas for pi(n) and the nth primeInternational Journal of Mathematics and Computer Science 9 (2014), no. 2, 95–98, with . . .
 

8. A hypergeometric approach, via linear forms involving logarithms, to criteria for irrationality of Euler's constant, Math. Slovaca 59 (2009) 1-8, with an Appendix by . . .
 

Sergey ZLOBIN

9. Euler's constant, q-logarithms, and formulas of Ramanujan and Gosper, Ramanujan J. 12 (2006) 225-244, with . . .

Wadim ZUDILIN

10. An irrationality measure for Liouville numbers and conditional measures for Euler's constant (2003, e-print).

11. An infinite product for e^gamma via hypergeometric formulas for Euler's constant, gamma (2003, e-print).

12. A faster product for pi and a new integral for ln pi/2, Amer. Math. Monthly 112 (2005) 729-734 (article), 113 (2006) 670 (addendum).

13. Irrationality measures, irrationality bases, and a theorem of Jarnik (2004, e-print).

Abstract, PDF

14. A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641 (article), 114 (2007) 659 (addendum).

15. Corrigendum: On the irrationality of some alternating series, Studia Univ. Babes-Bolyai Math. 49 no. 1 (2004) 105-106, with . . .

Jozsef SANDOR

16. Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J. 16 (2008) 247-270, with . . .

17. New Vacca-type rational series for Euler's constant and its "alternating" analog ln 4/pi,

   
 

18. Summation of series defined by counting blocks of digits, J. Number Theory 123 (2007) 133-143, with . . .

Jeffrey SHALLIT

19. Problem 11026 (see image above): An identity involving harmonic numbers, Amer. Math. Monthly 110 (2003) 636 (proposal), 112 (2005) 367-369 (solution).

My solution

20. Problem 11132 (see image above): Choice bounds, Amer. Math. Monthly 112 (2005) 180 (proposal), 114 (2007) 359-360 (solution).

 

21. Problem 11222 (see image above): An infinite product based on a base , Amer. Math. Monthly 113 (2006) 459 (proposal), 115 (2008) 954-955 (solution).

My solution

 

22. Problem 88 (see image above): A symmetric formula for pi, Math Horizons 5 (Sept., 1997) 32, 34.

23. The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl. 332 (2007) 292-314, with . . .
 

Petros HADJICOSTAS

24. The Taylor series for e and the primes 2, 5, 13, 37, 463: a surprising connection (2006, e-print).

PDF

25. Integrals over polytopes, multiple zeta values and polylogarithms, and Euler's constant, Math. Notes (in English) 84 (2008) pp. 568-583, Erratum p. 887; Mat. Zametki (in Russian) 84:4 (2008) pp. 609-626; with . . .

Sergey ZLOBIN

26. A simple counterexample to Havil's ``reformulation'' of the Riemann Hypothesis, Elemente der Mathematik 67 (2012) 61-67.

27. Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463). Part II (includes corrected version of Part I), with . . .

Kyle SCHALM

28. Infinite products with strongly B-multiplicative exponents, Annales Univ. Sci. Budapest, Sect. Comp. 28 (2008) 35-53 (article), 32 (2010) 253 (errata), with . . .

29. Primes, pi, and irrationality measure (2007, e-print).

Abstract, PDF

30. Problem 11322 (see image above): A double integral, Amer. Math. Monthly 114 (2007) 835 (proposal), 116 (2009) 650 (solution).

31. Problem 11381 (see image above): An infinite product for the exponential, Amer. Math. Monthly, 115 (2008) 665 (proposal), 117 (2010) 283-284 (solution), with . . .

32. Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly 116 (2009) 630-635.

33. Reducing the Erdos-Moser equation 1^n + 2^n + ... + k^n = (k+1)^n modulo k and k^2, Integers 11 (2011) article A34, with . . .

34. A monotonicity property of Riemann's xi function and a reformulation of the Riemann Hypothesis, Periodica Mathematica Hungarica 60 (2010) 37-40, with . . .

35. Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164, with . . .
 

36. How not to prove/disprove the Riemann Hypothesis (in preparation), with . . .
 

37. New Wallis- and Catalan-type infinite products for pi, e, and sqrt(2+sqrt(2)), Amer. Math. Monthly 117 (2010) 912-917, with . . .

38. Another quadratic residue race (submitted for publication), with . . .

39. Lerch quotients, Lerch primes, Fermat-Wilson quotients, and the Wieferich-non-Wilson primes 2, 3, 14771

40. Schanuel's conjecture and algebraic powers z^w and w^z with z and w transcendental, East-West Journal of Mathematics 12, no. 1 (2010) 75-84, with . . .

41. Proofs of power sum and binomial coefficient congruences via Pascal's identity, Amer. Math. Monthly 118 (2011) 549-551, with . . .

42. Divisibility of power sums and the generalized Erdos-Moser equation, Elemente der Mathematik 67 (2012) 182–186, with . . .

43. Ramanujan primes: bounds, runs, twins, and gaps, J. Integer Seq. 14 (2011) article 11.6.2, with . . .

44. Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011) article A33, with . . .
 

Jean-Louis NICOLAS

45. The Schanuel subset conjecture implies Gelfond's power tower conjecture (2012, e-print), with . . .

46. Problem 11546 (see image above): 2-adic Valuation of Bernoulli-style Sums, Amer. Math. Monthly 118 (2011) 84 (proposal), 119 (2012) 886-887 (solution), with . . .

47. From the Monthly 100 years ago: Gauss and the eccentric Halsted, to appear in Amer. Math. Monthly.

48. Evaluation of Tachiya's algebraic infinite products involving Fibonacci and Lucas numbers,

49. Generalized Ramanujan primes, with Olivia BECKWITH, Ryan RONAN, and . . .
 

50. On SA, CA, and GA numbers, Ramanujan J. 29 (2012) 359-384, with . . .
 

Jean-Louis NICOLAS

 

51. Translation of D. Zimin’s Dynasty Foundation and Pierre Deligne Contests for Young Mathematicians, Russian Math. Surveys 62:1 (2007) 213–216.

52. Rebuttal of Kowalenko's paper as concerns the irrationality of Euler's constant, Acta Applicandae Mathematicae 121 (2012) 1-3, with . . .

Online Preview

53. Universal parabolic constant, MathWorld and Online Encyclopedia of Integer Sequences, 2005, with . . .

54. The parbelos, a parabolic analog of the arbelos, Amer. Math. Monthly 120 (2013) 929-935.

55. New approximations to Euler's constant (in preparation), with . . . 

56. On variations of the arbelos (in preparation), with . . .

57. Ramanujan, Robin, highly composite numbers, and the Riemann Hypothesis, Proceedings of RAMA125, Contemporary Mathematics 627 (2014) 145-156with . . .

Jean-Louis NICOLAS

58. The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers (2014, e-print), with

Abstract, PDF

59. Universal equilateral hyperbolic constant, Online Encyclopedia of Integer Sequences, 2013, with . . .

60. More circles in the generalized arbelos (in preparation), with . . .

61. Primitive antiharmonic numbers, Online Encyclopedia of Integer Sequences, 2013, with . . .

62. On the congruence 1^m + 2^m + … + m^m == n (mod m) with n m, Monatshefte für Mathematik, DOI 10.1007/s00605-014-0660-0, with
 

63. On primary pseudoperfect numbers (in preparation), with . . .

64. On Pascal's triangle (in preparation), with . . .

65. Ideal multigrades, symmetric and non-symmetric: the Prouhet-Tarry-Escott problem, Online Encyclopedia of Integer Sequences, 2014.

66. Summation of rational series twisted by strongly B-multiplicative coefficients, with . . .

MY PAPERS IN TOPOLOGY, NUMBER THEORY & GEOMETRY: