Jonathan Sondow's Home Page

 

   

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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.

Princeton University in 1965 (photos of old Fine Hall)

Advisor: John MILNOR

 

Mathematical Genealogy

B.A.-with-Honors 1962

 

Diploma and Calculus Prize 1959

Stuyvesant H.S.

MILNOR WINS THE 2004 STEELE PRIZE FOR EXPOSITION
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Milnor's Lectures on the h-Cobordism Theorem (PDF)

Poster of my November 2006 lectures on number theory at Keio University (PDF)

My Erdös number: 2

NEW & NOTEWORTHY 9 July  2008

* Publication of [28]

* 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]

* Link [xxiii] to Jesus Guillera's history of formulas for pi; he gives my faster product for pi/2 from [12]

* At long last, the MAA has put a citation of [3] in Dunham's book "Euler, the Master of Us All" - see [I] 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 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]

 

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

i. X. Gourdon and P. Sebah's

Numbers, Constants and Computation/GammaFormulas

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. _______________

I.

Euler, the Master of Us All

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)

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, 2003, pp. 109, 206 (a citation of [3] on the antisymmetric formula for gamma on p.109 is promised for the 2008 edition)

IV.

CRC Concise Encyclopedia of Mathematics

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

V.
 

VI.

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

by G. Boros and V. Moll, Cambridge Univ. Press, 2004, pp. 184, 234.

VII.

El Omnipresente Número "Pi"

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

My papers on the arXiv

MY PAPERS IN NUMBER THEORY

CiteSeer 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.

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 prime (2002, preprint), with . . .
 

8. A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant (2002, preprint) 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, preprint).

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

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, preprint).

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).

Abstract, PDF

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 (2005, preprint), to appear in Ramanujan J., with . . .

17. New Vacca-type rational series for Euler's constant and its "alternating" analog ln 4/pi (2005, preprint).

 

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

Jeffrey SHALLIT

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

My solution (PDF)

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

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.

My solution (PDF)

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

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, preprint).

(PDF)

25. Integrals Over Polytopes, Multiple Zeta Values and Polylogarithms, and Euler's Constant (2007, preprint), to appear in Mat. Zametki, with . . .

Sergey ZLOBIN

26. A counterexample to Havil's "reformulation" of the Riemann Hypothesis (2007, preprint).

Abstract, PDF

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 . . .

Kyle SCHALM

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