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| Giving a talk at a meeting of the |
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| MILNOR WINS THE 2004 STEELE PRIZE FOR EXPOSITION |
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Milnor's Lectures on the h-Cobordism Theorem (PDF)
NEW & NOTEWORTHY 9 July 2008
* 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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* Link [xxiii] to Jesus Guillera's history of formulas for pi; he gives my faster product for pi/2 from
[12]
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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" - see [I] 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 PAPERS)
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| A globally convergent series [1], [12], cited in [iii], [viii], [III], [IV], proved first by Hasse |
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| Symmetric formula for pi [22], cited in [vi], [V] |
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| Series involving the number of 1s and 0s in the binary expansion of n, from [17], [18] |
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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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| A series-integral for Euler's constant from [25] with Sergey ZLOBIN |
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| Antisymmetric formula for Euler's constant [3], cited in [i], [iv], [I], [II], [III], [VI] |
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| Hypergeometric formula for Euler's constant [8], [11] |
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| Series showing that ln(4/pi) is an "alternating Euler constant" [6], cited in [iv] |
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| Integrals from [6], [12], cited in [i], [iv]. The 3rd is a special case of an integral due to Kummer |
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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: a Beukers-type integral equals a Nesterenko-type series [5], [8], [10], [30], [VI] |
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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 any rational number k/h > 1, and an inequality for all integers p,q with q > 0 large
[13]
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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] |
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| A geometric proof that e is irrational, and a new measure of its irrationality [14] |
WEB PAGES AND BOOKS CITING MY WORK
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i. X. Gourdon and P. Sebah's
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Numbers, Constants and Computation/GammaFormulas
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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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I.
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Euler, the Master of Us All
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by William Dunham, MAA, 1999, p. 36 (my antisymmetric formula for Euler's constant exhibits "a delightful
symmetry"; a citation of [3] is promised for the 8th printing - see the sticker on p.37 in the 7th printing)
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III.
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Gamma: Exploring Euler's Constant
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by Julian Havil, Princeton Univ. Press, 2003, pp. 109, 206 (a citation of [3] on the antisymmetric
formula for gamma on p.109 is promised for the 2008 edition)
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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, preprint), with . . .
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8. A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant
(2002, preprint) 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, preprint).
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11. An infinite product for e^gamma via hypergeometric formulas for Euler's constant, gamma (2003, preprint).
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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, preprint).
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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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Abstract, PDF
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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 (2005, preprint), to appear in Ramanujan J., with . . .
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17. New Vacca-type rational series for Euler's constant and its "alternating" analog ln 4/pi (2005, preprint).
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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: An identity involving harmonic numbers, Amer. Math. Monthly 110 (2003) 636 (proposal), 112
(2005) 367-369 (solution) - see image above.
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My solution (PDF)
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20. Problem 11132: Choice bounds, Amer. Math. Monthly 112 (2005) 180 (proposal), 114 (2007) 359-360 (solution)
- see image above.
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21. Problem 11222: An Infinite Product Based on a Base , Amer. Math. Monthly 113 (2006) 459 (proposal),
115 (2008) 465-466 (solution) - see image above.
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My solution (PDF)
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22. Problem 88: A symmetric formula for pi, Math Horizons 5 (Sept., 1997) 32, 34 - see image above.
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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,
preprint).
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(PDF)
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25. Integrals Over Polytopes, Multiple Zeta Values and Polylogarithms, and Euler's Constant (2007, preprint),
to appear in Mat. Zametki, with . . .
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Sergey ZLOBIN
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26. A counterexample to Havil's "reformulation" of the Riemann Hypothesis (2007, preprint).
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Abstract, PDF
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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), v. 1, Tapas in Experimental Mathematics: AMS Special Session Jan. 5, 2007, New Orleans, LA, Contemporary
Mathematics 457, 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, with . . .
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