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| Before |
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| Giving a talk at a meeting of the |
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| After |
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| MILNOR WINS THE 2004 STEELE PRIZE FOR EXPOSITION |
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| FOOTNOTE TO MILNOR'S "DIFFERENTIAL TOPOLOGY 46..." |
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NEW AND NOTEWORTHY 17 MAY 2013
UPCOMING TALK ``Ramanujan, Robin, the Riemann Hypothesis, and Recent Results'' at CANT 2013 on May 23 at 10:50 AM
* New co-author of [57], with new version to follow
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* 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!"
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* 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."
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* 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]
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* 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
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* 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].
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* 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."
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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] |
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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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| 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
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i. X. Gourdon and P. Sebah's
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Numbers, Constants and Computation/GammaFormulas(PDF)
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ii. Matthew Watkins'
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Inexplicable Secrets of Creation/RHreformulations
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iii. Eric Weisstein's
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iv. _____________
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v. _____________
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vi. _____________
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vii. ____________
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viii. Philippe Biane, Jim Pitman and Marc Yor's
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ix. Wadim Zudilin's
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Zeta Values on the Web
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x. Michel Waldschmidt's
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xi. American Scientist's
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xii. Keith Matthews'
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xiii. ____________
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xiv. Neil J. Sloane's
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xv. ____________
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xvi. ___________
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xvii. ___________
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xviii. John Baez's
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xix. Stefan Kraemer's
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xx. Michel Waldschmidt's
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xxi. _______________
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xxii. _______________
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xxiii. Jesus Guillera's
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xxiv. Michel Waldschmidt's
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xxv. _______________
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xxxi. Thomas and Joseph Dence's
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xxxii. Jean-Paul Allouche's
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xxxiii. Doron Zeilberger's
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xxxvii. Yuri Matiyasevich, Filip Saidak, and Peter Zvengrowski's
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xxxviii. Robert P. Schneider's
xxxix. Aliza Steurer and Thomas Hagedorn's
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BOOKS CITING MY WORK IN NUMBER THEORY
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I.
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Euler, the Master of Us All
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by William Dunham, MAA, 1999 (8th printing, 2010), p. 36 (my antisymmetric
formula for Euler's constant exhibits "a delightful symmetry")
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Book of the Year 1967, p. 503
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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.
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2. The Riemann Hypothesis, simple zeros, and the asymptotic convergence degree of improper Riemann sums,
Proc. Amer. Math. Soc. 126 (1998) 1311-1314.
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3. An antisymmetric formula for Euler's constant, Math. Mag. 71 (1998) 219-220.
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4. Zeros of the alternating zeta function on the line R(s)=1, Amer. Math. Monthly 110 (2003) 435-437.
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5. Criteria for irrationality of Euler's constant, Proc. Amer. Math. Soc. 131 (2003) 3335-3344.
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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.
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7. Formulas for pi(n) and the nth prime (2002, e-print), with . . .
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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 . . .
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Sergey ZLOBIN
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9. Euler's constant, q-logarithms, and formulas of Ramanujan and Gosper, Ramanujan J. 12 (2006) 225-244,
with . . .
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Wadim ZUDILIN
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10. An irrationality measure for Liouville numbers and conditional measures for Euler's constant (2003,
e-print).
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11. An infinite product for e^gamma via hypergeometric formulas for Euler's constant, gamma (2003, e-print).
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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).
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13. Irrationality measures, irrationality bases, and a theorem of Jarnik (2004, e-print).
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Abstract, PDF
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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).
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15. Corrigendum: On the irrationality of some alternating series, Studia Univ. Babes-Bolyai Math. 49
no. 1 (2004) 105-106, with . . .
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Jozsef SANDOR
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16. Double integrals and infinite products for some classical constants via analytic continuations of Lerch's
transcendent, Ramanujan J. 16 (2008) 247-270, with . . .
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17. New Vacca-type rational series for Euler's constant and its "alternating" analog ln 4/pi,
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18. Summation of series defined by counting blocks of digits, J. Number Theory 123 (2007) 133-143, with
. . .
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Jeffrey SHALLIT
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19. Problem 11026 (see image above): An identity involving harmonic numbers, Amer. Math. Monthly 110
(2003) 636 (proposal), 112 (2005) 367-369 (solution).
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My solution
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20. Problem 11132 (see image above): Choice bounds, Amer. Math. Monthly 112 (2005) 180 (proposal), 114
(2007) 359-360 (solution).
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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).
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My solution
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22. Problem 88 (see image above): A symmetric formula for pi, Math Horizons 5 (Sept., 1997) 32, 34.
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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 . . .
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Petros HADJICOSTAS
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24. The Taylor series for e and the primes 2, 5, 13, 37, 463, ...: a surprising connection (2006,
e-print).
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PDF
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25. Integrals over polytopes, multiple zeta values and polylogarithms, and Euler's constant, Math. Notes
(in English) 84 (2008) pp. 568-583; Mat. Zametki (in Russian) 84:4 (2008) pp. 609-626; with . . .
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Sergey ZLOBIN
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26. A simple counterexample to Havil's ``reformulation'' of the Riemann Hypothesis, Elemente der Mathematik
67 (2012) 61-67.
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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, with . . .
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Kyle SCHALM
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28. Infinite products with strongly B-multiplicative exponents, Annales Univ. Sci. Budapest, Sect. Comp.
28 (2008) 35-53 (article), 32 (2010) 253 (errata), with . . .
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29. Primes, pi, and irrationality measure (2007, e-print).
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Abstract, PDF
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30. Problem 11322 (see image above): A double integral, Amer. Math. Monthly 114 (2007) 835 (proposal),
116 (2009) 650 (solution).
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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 . . .
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32. Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly 116 (2009) 630-635.
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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 . . .
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34. A monotonicity property of Riemann's xi function and a reformulation of the Riemann Hypothesis,
Periodica Mathematica Hungarica 60 (2010) 37-40, with . . .
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35. Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae
et Informaticae 37 (2010) 151-164, with . . .
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36. How not to prove/disprove the Riemann Hypothesis (in preparation), with . . .
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37. New Wallis- and Catalan-type infinite products for pi, e, and sqrt(2+sqrt(2)), Amer. Math. Monthly
117 (2010) 912-917, with . . .
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38. Another quadratic residue race (submitted for publication), with . . .
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39. Lerch quotients, Lerch primes, Fermat-Wilson quotients, and the Wieferich-non-Wilson primes 2, 3,
14771, to appear in Proceedings of CANT 2011.
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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 . . .
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41. Proofs
of power sum and binomial coefficient congruences via Pascal's identity, Amer. Math. Monthly 118 (2011)
549-551, with . . .
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42.
Divisibility of power sums and the generalized Erdos-Moser equation, Elemente der Mathematik 67 (2012) 182–186,
with . . .
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43. Ramanujan
primes: bounds, runs, twins, and gaps, J. Integer Seq. 14 (2011) article 11.6.2, with . . .
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44. Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers
11 (2011) article A33, with . . .
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Jean-Louis NICOLAS
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45.
The Schanuel subset conjecture implies Gelfond's power tower conjecture (submitted
for publication), with . . .
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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 . . .
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47. From the Monthly 100 years ago: Gauss and the eccentric Halsted, accepted by Amer. Math. Monthly.
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48. Evaluation of Tachiya's algebraic infinite products involving Fibonacci and Lucas numbers,
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49. Generalized Ramanujan primes, to appear in Proceedings of CANT 2011, ｗith Olivia BECKWITH, Ryan
RONAN, and . . .
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50. On SA, CA, and GA numbers, Ramanujan J. 29 (2012) 359-384, with . . .
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Jean-Louis NICOLAS
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51. Translation of D. Zimin’s Dynasty Foundation and Pierre Deligne Contests for Young Mathematicians,
Russian Math. Surveys 62:1 (2007) 213–216.
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52. Rebuttal of Kowalenko's paper as concerns the irrationality of Euler's
constant, Acta Applicandae Mathematicae 121 (2012) 1-3, with . . .
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Online Preview
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53. Universal parabolic constant, MathWorld and Online Encyclopedia of Integer Sequences, 2005, with
. . .
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54. The parbelos, a parabolic analog of the arbelos, accepted by Amer. Math. Monthly.
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55. New approximations to Euler's constant (in preparation), with . . .
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56. On variations of the arbelos (in preparation), with . . .
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57. Ramanujan, Robin, the Riemann Hypothesis, and recent
results, to appear in Proceedings of RAMA125, with
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Jean-Louis NICOLAS
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58. Divisibility of power sums and the generalized Erdos-Moser equation (in preparation), with . . .
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59. Universal equilateral hyperbolic constant, Online Encyclopedia of Integer Sequences, 2013, with . . .
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60. More circles in the generalized arbelos (in preparation), with . . .
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FOR MY PAPERS IN TOPOLOGY & NUMBER THEORY ON Google Scholar,
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