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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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Milnor's Lectures on the h-Cobordism Theorem (PDF)
NEW & NOTEWORTHY 2 May 2009
* PDF of [32] "Ramanujan primes and Bertrand's postulate"
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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" - 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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| Monthly Problem 11381: an infinite product for e^x [31], from [16] |
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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 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, 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, 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 (2005, preprint)
to appear in proceedings of CANT 2005
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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,
preprint).
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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 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), ''Tapas in Experimental Mathematics'' (T. Amdeberhan and V. Moll, eds.), Contemporary Mathematics,
vol. 457, Amer. Math. Soc., Providence, RI, 2008, pp. 273-284, 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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29. Primes, Pi, and Irrationality Measure (2007, preprint).
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Abstract, PDF
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30. Problem 11322 (see image above): A Beukers-type double integral equals a Nesterenko-type series, Amer. Math.
Monthly 114 (2007) 835.
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31. Problem 11381 (see image above): An infinite product for e^x, Amer. Math. Monthly 115 (2008) 665, with
...
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32. Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly 116 (2009) 630-635.
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33. The Erdos-Moser Diophantine equation and a related congruence (2008, preprint), with . . .
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34. A monotonicity property of the xi function and a reformulation of the Riemann Hypothesis (2008, preprint),
with . . .
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