Difference between revisions of "Bounded gaps between primes"
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* [https://math.mit.edu/~primegaps/MaynardMathematicaNotebook.txt Mathematica Notebook for optimising M_k], James Maynard | * [https://math.mit.edu/~primegaps/MaynardMathematicaNotebook.txt Mathematica Notebook for optimising M_k], James Maynard | ||
* Some [[notes on polytope decomposition]] | * Some [[notes on polytope decomposition]] | ||
− | * [https://math.mit.edu/~drew/ompadm_v0. | + | * [https://math.mit.edu/~drew/ompadm_v0.4.tar Multi-threaded admissibility testing for very large tuples], Andrew Sutherland |
=== Tuples applet === | === Tuples applet === |
Revision as of 13:58, 18 May 2014
This is the home page for the Polymath8 project, which has two components:
- Polymath8a, "Bounded gaps between primes", was a project to improve the bound H=H_1 on the least gap between consecutive primes that was attained infinitely often, by developing the techniques of Zhang. This project concluded with a bound of H = 4,680.
- Polymath8b, "Bounded intervals with many primes", is an ongoing project to improve the value of H_1 further, as well as H_m (the least gap between primes with m-1 primes between them that is attained infinitely often), by combining the Polymath8a results with the techniques of Maynard.
Contents
World records
Current records
This table lists the current best upper bounds on [math]H_m[/math] - the least quantity for which it is the case that there are infinitely many intervals [math]n, n+1, \ldots, n+H_m[/math] which contain [math]m+1[/math] consecutive primes - both on the assumption of the Elliott-Halberstam conjecture (or more precisely, a generalization of this conjecture, formulated as Conjecture 1 in [BFI1986]), without this assumption, and without EH or the use of Deligne's theorems. The boldface entry - the bound on [math]H_1[/math] without assuming Elliott-Halberstam, but assuming the use of Deligne's theorems - is the quantity that has attracted the most attention. The conjectured value [math]H_1=2[/math] for [math]H_1[/math] is the twin prime conjecture.
[math]m[/math] | Conjectural | Assuming EH | Without EH | Without EH or Deligne |
---|---|---|---|---|
1 | 2 | 6 (on GEH)
12 (on EH only) |
246 | 246 |
2 | 6 | 270 | 395,106 | 474,266 |
3 | 8 | 52,116 | 24,462,654 | 32,285,928 |
4 | 12 | 474,266 | 1,440,495,268 | 2,082,729,956 |
5 | 16 | 4,137,854 | 78,807,316,822 | 124,840,189,042 |
[math]m[/math] | [math]\displaystyle (1+o(1)) m \log m[/math] | [math]\displaystyle O( m e^{2m} )[/math] | [math]O( m \exp((4 - \frac{52}{283}) m) )[/math] | [math]O( m \exp((4 - \frac{4}{43}) m) )[/math] |
We have been working on improving a number of other quantities, including the quantity [math]H_m[/math] mentioned above:
- [math]H = H_1[/math] is a quantity such that there are infinitely many pairs of consecutive primes of distance at most [math]H[/math] apart. Would like to be as small as possible (this is a primary goal of the Polymath8 project).
- [math]k_0[/math] is a quantity such that every admissible [math]k_0[/math]-tuple has infinitely many translates which each contain at least two primes. Would like to be as small as possible. Improvements in [math]k_0[/math] lead to improvements in [math]H[/math]. (The relationship is roughly of the form [math]H \sim k_0 \log k_0[/math]; see the page on finding narrow admissible tuples.) More recent improvements on [math]k_0[/math] have come from solving a Selberg sieve variational problem.
- [math]\varpi[/math] is a technical parameter related to a specialized form of the Elliott-Halberstam conjecture. Would like to be as large as possible. Improvements in [math]\varpi[/math] lead to improvements in [math]k_0[/math], as described in the page on Dickson-Hardy-Littlewood theorems. In more recent work, the single parameter [math]\varpi[/math] is replaced by a pair [math](\varpi,\delta)[/math] (in previous work we had [math]\delta=\varpi[/math]). These estimates are obtained in turn from Type I, Type II, and Type III estimates, as described at the page on distribution of primes in smooth moduli.
Timeline of bounds
A table of bounds as a function of time may be found at timeline of prime gap bounds. In this table, infinitesimal losses in [math]\delta,\varpi[/math] are ignored.
Polymath threads
- I just can’t resist: there are infinitely many pairs of primes at most 59470640 apart, Scott Morrison, 30 May 2013. Inactive
- The prime tuples conjecture, sieve theory, and the work of Goldston-Pintz-Yildirim, Motohashi-Pintz, and Zhang, Terence Tao, 3 June 2013. Inactive
- Polymath proposal: bounded gaps between primes, Terence Tao, 4 June 2013. Inactive
- Online reading seminar for Zhang’s “bounded gaps between primes”, Terence Tao, 4 June 2013. Inactive
- More narrow admissible sets, Scott Morrison, 5 June 2013. Inactive
- The elementary Selberg sieve and bounded prime gaps, Terence Tao, 8 June 2013. Inactive
- A combinatorial subset sum problem associated with bounded prime gaps, Terence Tao, 10 June 2013. Inactive
- Further analysis of the truncated GPY sieve, Terence Tao, 11 June 2013. Inactive
- Estimation of the Type I and Type II sums, Terence Tao, 12 June 2013. Inactive
- Estimation of the Type III sums, Terence Tao, 14 June 2013. Inactive
- A truncated elementary Selberg sieve of Pintz, Terence Tao, 18 June, 2013. Inactive
- Bounding short exponential sums on smooth moduli via Weyl differencing, Terence Tao, 22 June, 2013. Inactive
- The distribution of primes in densely divisible moduli, Terence Tao, 23 June, 2013. Inactive
- Bounded gaps between primes (Polymath8) – a progress report, Terence Tao, 30 June 2013. Inactive
- The quest for narrow admissible tuples, Andrew Sutherland, 2 July 2013. Inactive
- The distribution of primes in doubly densely divisible moduli, Terence Tao, 7 July 2013. Inactive.
- An improved Type I estimate, Terence Tao, 27 July 2013. Inactive
- Polymath8: writing the paper, Terence Tao, 17 August 2013. Inactive
- Polymath8: writing the paper, II, Terence Tao, 2 September 2013. Inactive
- Polymath8: writing the paper, III, Terence Tao, 22 September 2013. Inactive
- Polymath8: writing the paper, IV, Terence Tao, 15 October 2013. Inactive
- Polymath8: Writing the first paper, V, and a look ahead, Terence Tao, 17 November 2013. Inactive
- Polymath8b: Bounded intervals with many primes, after Maynard, Terence Tao, 19 November 2013. Inactive
- Polymath8b, II: Optimising the variational problem and the sieve Terence Tao, 22 November 2013. Inactive
- Polymath8b, III: Numerical optimisation of the variational problem, and a search for new sieves, Terence Tao, 8 December 2013. Inactive
- Polymath8b, IV: Enlarging the sieve support, more efficient numerics, and explicit asymptotics, Terence Tao, 20 December 2013. Inactive
- Polymath8b, V: Stretching the sieve support further, Terence Tao, 8 January 2014. Inactive
- Polymath8b, VI: A low-dimensional variational problem, Terence Tao, 17 January 2014. Inactive
- Polymath8b, VII: Using the generalised Elliott-Halberstam hypothesis to enlarge the sieve support yet further, Terence Tao, 28 January 2014. Inactive
- “New equidistribution estimates of Zhang type, and bounded gaps between primes” – and a retrospective, Terence Tao, 7 February 2013. Active
- Polymath8b, VIII: Time to start writing up the results?, Terence Tao, 9 February 2014. Inactive
- Polymath8b, IX: Large quadratic programs, Terence Tao, 21 February 2014. Inactive
- Polymath8b, X: Writing the paper, and chasing down loose ends, Terence Tao, 14 April 2014. Inactive
- Polymath 8b, XI: Finishing up the paper, Terence Tao, 17 May 2014. Active
Writeup
- Files for the submitted paper for the Polymath8a project may be found in this directory.
- The compiled PDF for this paper is available here.
- The paper is now on the arXiv as "New equidistribution estimates of Zhang type, and bounded gaps between primes".
- Files for the draft paper for the Polymath8 retrospective may be found in this directory.
- The compiled PDF for this paper is available here.
- Files for the draft paper for the Polymath8b project may be found in this directory.
- The compiled PDF for this paper is available here.
Here are the Polymath8 grant acknowledgments.
Code and data
- Hensely-Richards sequences, Scott Morrison
- A mathematica notebook for finding k_0, Scott Morrison
- python implementation, Avi Levy
- k-tuple pattern data, Thomas J Engelsma
- Sifted sequences, Vit Tucek
- Other sifted sequences, Christian Elsholtz
- Size of largest admissible tuples in intervals of length up to 1050, Clark and Jarvis
- C implementation of the "greedy-greedy" algorithm, Andrew Sutherland
- Dropbox for sequences, pedant
- Spreadsheet for admissible sequences, Vit Tucek
- Java code for optimising a given tuple V1.1, xfxie
- Mathematica Notebook for optimising M_k, James Maynard
- Some notes on polytope decomposition
- Multi-threaded admissibility testing for very large tuples, Andrew Sutherland
Tuples applet
Here is a small javascript applet that illustrates the process of sieving for an admissible 632-tuple of diameter 4680 (the sieved residue classes match the example in the paper when translated to [0,4680]).
The same applet can also be used to interactively create new admissible tuples. The default sieve interval is [0,400], but you can change the diameter to any value you like by adding “?d=nnnn” to the URL (e.g. use https://math.mit.edu/~primegaps/sieve.html?d=4680 for diameter 4680). The applet will highlight a suggested residue class to sieve in green (corresponding to a greedy choice that doesn’t hit the end points), but you can sieve any classes you like.
You can also create sieving demos similar to the k=632 example above by specifying a list of residues to sieve. A longer but equivalent version of the “?ktuple=632″ URL is
The numbers listed after “r=” are residue classes modulo increasing primes 2,3,5,7,…, omitting any classes that do not require sieving — the applet will automatically skip such primes (e.g. it skips 151 in the k=632 example).
A few caveats: You will want to run this on a PC with a reasonably wide display (say 1360 or greater), and it has only been tested on the current versions of Chrome and Firefox.
Errata
Page numbers refer to the file linked to for the relevant paper.
- Errata for Zhang's "Bounded gaps between primes"
- Page 5: In the first display, [math]\mathcal{E}[/math] should be multiplied by [math]\mathcal{L}^{2k_0+2l_0}[/math], because [math]\lambda(n)^2[/math] in (2.2) can be that large, cf. (2.4).
- Page 14: In the final display, the constraint [math](n,d_1=1[/math] should be [math](n,d_1)=1[/math].
- Page 35: In the display after (10.5), the subscript on [math]{\mathcal J}_i[/math] should be deleted.
- Page 36: In the third display, a factor of [math]\tau(q_0r)^{O(1)}[/math] may be needed on the right-hand side (but is ultimately harmless).
- Page 38: In the display after (10.14), [math]\xi(r,a;q_1,b_1;q_2,b_2;n,k)[/math] should be [math]\xi(r,a;k;q_1,b_1;q_2,b_2;n)[/math].
- Page 42: In (12.3), [math]B[/math] should probably be 2.
- Page 47: In the third display after (13.13), the condition [math]l \in {\mathcal I}_i(h)[/math] should be [math]l \in {\mathcal I}_i(sh)[/math].
- Page 49: In the top line, a comma in [math](h_1,h_2;,n_1,n_2)[/math] should be deleted.
- Page 51: In the penultimate display, one of the two consecutive commas should be deleted.
- Page 54: Three displays before (14.17), [math]\bar{r_2}(m_1+m_2)q[/math] should be [math]\bar{r_2}(m_1+m_2)/q[/math].
- Errata for Motohashi-Pintz's "A smoothed GPY sieve", version 1. Update: the errata below have been corrected in the most recent arXiv version of the paper.
- Page 31: The estimation of (5.14) by (5.15) does not appear to be justified. In the text, it is claimed that the second summation in (5.14) can be treated by a variant of (4.15); however, whereas (5.14) contains a factor of [math](\log \frac{R}{|D|})^{2\ell+1}[/math], (4.15) contains instead a factor of [math](\log \frac{R/w}{|K|})^{2\ell+1}[/math] which is significantly smaller (K in (4.15) plays a similar role to D in (5.14)). As such, the crucial gain of [math]\exp(-k\omega/3)[/math] in (4.15) does not seem to be available for estimating the second sum in (5.14).
- Errata for Pintz's "A note on bounded gaps between primes", version 1. Update: the errata below have been corrected in subsequent versions of Pintz's paper.
- Page 7: In (2.39), the exponent of [math]3a/2[/math] should instead be [math]-5a/2[/math] (it comes from dividing (2.38) by (2.37)). This impacts the numerics for the rest of the paper.
- Page 8: The "easy calculation" that the relative error caused by discarding all but the smooth moduli appears to be unjustified, as it relies on the treatment of (5.14) in Motohashi-Pintz which has the issue pointed out in 2.1 above.
Other relevant blog posts
- Marker lecture III: “Small gaps between primes”, Terence Tao, 19 Nov 2008.
- The Goldston-Pintz-Yildirim result, and how far do we have to walk to twin primes ?, Emmanuel Kowalski, 22 Jan 2009.
- Number Theory News, Peter Woit, 12 May 2013.
- Bounded Gaps Between Primes, Emily Riehl, 14 May 2013.
- Bounded gaps between primes!, Emmanuel Kowalski, 21 May 2013.
- Bounded gaps between primes: some grittier details, Emmanuel Kowalski, 4 June 2013.
- Bound on prime gaps bound decreasing by leaps and bounds, Christian Perfect, 8 June 2013.
- Narrowing the Gap, Brie Finegold, AMS Blog on Math Blogs, 14 June 2013.
- Bounded gaps between primes: the dawn of (some) enlightenment, Emmanuel Kowalski, 22 June 2013.
- A ternary divisor variation, Emmanuel Kowalski, 25 June 2013.
- Conductors of one-variable transforms of trace functions, Emmanuel Kowalski, 9 September 2013.
- Polymath 8 – a Success!, Gil Kalai, 20 September 2013.
- James Maynard, auteur du théorème de l’année, Emmanuel Kowalski, 24 October 2013.
- Reflections on reading the Polymath8(a) paper, Emmanuel Kowalski, 8 December 2013.
MathOverflow
- Philosophy behind Yitang Zhang’s work on the Twin Primes Conjecture, 20 May 2013.
- A technical question related to Zhang’s result of bounded prime gaps, 25 May 2013.
- How does Yitang Zhang use Cauchy’s inequality and Theorem 2 to obtain the error term coming from the [math]S_2[/math] sum, 31 May 2013.
- Tightening Zhang’s bound, 3 June 2013.
- Does Zhang’s theorem generalize to 3 or more primes in an interval of fixed length?, 3 June 2013.
Wikipedia and other references
- Bessel function
- Bombieri-Vinogradov theorem
- Brun-Titchmarsh theorem
- Dispersion method
- Prime gap
- Second Hardy-Littlewood conjecture
- Twin prime conjecture
Recent papers and notes
- On the exponent of distribution of the ternary divisor function, Étienne Fouvry, Emmanuel Kowalski, Philippe Michel, 11 Apr 2013.
- On the optimal weight function in the Goldston-Pintz-Yildirim method for finding small gaps between consecutive primes, Bálint Farkas, János Pintz and Szilárd Gy. Révész, To appear in: Paul Turán Memorial Volume: Number Theory, Analysis and Combinatorics, de Gruyter, Berlin, 2013. 23 pages. arXiv
- The existence of small prime gaps in subsets of the integers, Jacques Benatar, 2 May, 2013.
- Bounded gaps between primes, Yitang Zhang, to appear, Annals of Mathematics. Released 21 May, 2013.
- Polignac Numbers, Conjectures of Erdös on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture, Janos Pintz, 27 May 2013.
- A poor man's improvement on Zhang's result: there are infinitely many prime gaps less than 60 million, T. S. Trudgian, 28 May 2013.
- The Friedlander-Iwaniec sum, É. Fouvry, E. Kowalski, Ph. Michel., May 2013.
- Zhang's Theorem on Bounded Gaps Between Primes, Dan Goldston, May? 2013.
- Notes on Zhang's prime gaps paper, Terence Tao, 1 June 2013.
- Bounded prime gaps in short intervals, Johan Andersson, 3 June 2013.
- Bounded length intervals containing two primes and an almost-prime II, James Maynard, 5 June 2013.
- A note on bounded gaps between primes, Janos Pintz, 6 June 2013.
- Lower Bounds for Admissible k-tuples, Avishay Tal, 15 June 2013.
- Notes in a truncated elementary Selberg sieve (Section 1, Section 2, Section 3), Janos Pintz, 18 June 2013.
- Lecture notes: bounded gaps between primes, Gergely Harcos, 1 Oct 2013.
- New bounds on gaps between primes, Andrew Sutherland, 17 Oct 2013.
- Bounded gaps between primes, Andrew Granville, 29 Oct 2013.
- Primes in intervals of bounded length, Andrew Granville, 19 Nov 2013.
- Small gaps between primes, James Maynard, 19 Nov 2013.
- A note on the theorem of Maynard and Tao, Tristan Freiberg, 21 Nov 2013.
- Consecutive primes in tuples, William D. Banks, Tristan Freiberg, and Caroline L. Turnage-Butterbaugh, 27 Nov 2013.
- Close encounters among the primes, John Friedlander, Henryk Iwaniec, 10 Dec 2013.
- A Friendly Intro to Sieves with a Look Towards Recent Progress on the Twin Primes Conjecture, David Lowry-Duda, 25 Jan, 2014.
- The twin prime conjecture, Yoichi Motohashi, 26 Jan 2014.
- Bounded gaps between primes in Chebotarev sets, Jesse Thorner, 26 Jan 2014.
- Bounded gaps between primes, Ben Green, 19 Feb 2014.
- Bounded gaps between primes of the special form, Hongze Li, Hao Pan, 19 Mar 2014.
- Bounded gaps between primes in number fields and function fields, Abel Castillo, Chris Hall, Robert J. Lemke Oliver, Paul Pollack, Lola Thompson, 23 Mar 2014.
- Bounded gaps between primes with a given primitive root, Paul Pollack, 15 Apr 2014.
- On limit points of the sequence of normalized prime gaps, William D. Banks, Tristan Freiberg, and James Maynard, 21 Apr 2014.
- Dense clusters of primes in subsets, James Maynard, 11 May 2014.
Media
- First proof that infinitely many prime numbers come in pairs, Maggie McKee, Nature, 14 May 2013.
- Proof that an infinite number of primes are paired, Lisa Grossman, New Scientist, 14 May 2013.
- Unheralded Mathematician Bridges the Prime Gap, Erica Klarreich, Simons science news, 20 May 2013.
- The article also appeared on Wired as "Unknown Mathematician Proves Elusive Property of Prime Numbers".
- The Beauty of Bounded Gaps, Jordan Ellenberg, Slate, 22 May 2013.
- Game of proofs boosts prime pair result by millions, Jacob Aron, New Scientist, 4 June 2013.
- L'union fait la force des mathématiciens, Philippe Pajot, Le Monde, 24 June, 2013.
- Primal Madness: Mathematicians’ Hunt for Twin Prime Numbers, Amir Aczel, Discover Magazine, 10 July, 2013.
- The Twin Prime Hero, Michael Segal, Nautilus, Issue 005, 2013.
- Prime Time, Casey Hamilton, Australian National University, 19 November 2013.
- Together and Alone, Closing the Prime Gap, Erica Klarreich, Quanta, 19 November 2013.
- The article also appeared on Wired as "Sudden Progress on Prime Number Problem Has Mathematicians Buzzing".
- Mathematicians Team Up To Close the Prime Gap, Slashdot, 20 November 2013.
- Ein großer Schritt zum Beweis der Primzahlzwillingsvermutung, Hans Engler, Spektrum, 13 December 2013.
- An old mathematical puzzle soon to be unraveled?, Benjamin Augereau, Phys.org, 15 January 2014.
- Neuer Durchbruch auf dem Weg zur Primzahlzwillingsvermutung, Christoph Poppe, Spektrum, 30 January 2014.
- Yitang Zhang: A prime-number proof and a world of persistence, Leslie Katz, CNET, February 12, 2014.
- Maths isn’t standing still, Adam Smith and Vicky Neale, Pod Academy, March 3, 2014.
Bibliography
Additional links for some of these references (e.g. to arXiv versions) would be greatly appreciated.
- [BFI1986] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. Acta Math. 156 (1986), no. 3-4, 203–251. MathSciNet Article
- [BFI1987] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. II. Math. Ann. 277 (1987), no. 3, 361–393. MathSciNet Article
- [BFI1989] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. III. J. Amer. Math. Soc. 2 (1989), no. 2, 215–224. MathSciNet Article
- [B1995] Jörg Brüdern, Einführung in die analytische Zahlentheorie, Springer Verlag 1995
- [CJ2001] Clark, David A.; Jarvis, Norman C.; Dense admissible sequences. Math. Comp. 70 (2001), no. 236, 1713–1718 MathSciNet Article
- [FI1981] Fouvry, E.; Iwaniec, H. On a theorem of Bombieri-Vinogradov type., Mathematika 27 (1980), no. 2, 135–152 (1981). MathSciNet Article
- [FI1983] Fouvry, E.; Iwaniec, H. Primes in arithmetic progressions. Acta Arith. 42 (1983), no. 2, 197–218. MathSciNet Article
- [FI1985] Friedlander, John B.; Iwaniec, Henryk, Incomplete Kloosterman sums and a divisor problem. With an appendix by Bryan J. Birch and Enrico Bombieri. Ann. of Math. (2) 121 (1985), no. 2, 319–350. MathSciNet JSTOR Appendix
- [GPY2009] Goldston, Daniel A.; Pintz, János; Yıldırım, Cem Y. Primes in tuples. I. Ann. of Math. (2) 170 (2009), no. 2, 819–862. arXiv MathSciNet
- [GR1998] Gordon, Daniel M.; Rodemich, Gene Dense admissible sets. Algorithmic number theory (Portland, OR, 1998), 216–225, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998. MathSciNet Article
- [GR1980] S. W. Graham, C. J. Ringrose, Lower bounds for least quadratic nonresidues. Analytic number theory (Allerton Park, IL, 1989), 269–309, Progr. Math., 85, Birkhäuser Boston, Boston, MA, 1990. MathSciNet Article
- [HB1978] D. R. Heath-Brown, Hybrid bounds for Dirichlet L-functions. Invent. Math. 47 (1978), no. 2, 149–170. MathSciNet Article
- [HB1986] D. R. Heath-Brown, The divisor function d3(n) in arithmetic progressions. Acta Arith. 47 (1986), no. 1, 29–56. MathSciNet Article
- [HR1973] Hensley, Douglas; Richards, Ian, On the incompatibility of two conjectures concerning primes. Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), pp. 123–127. Amer. Math. Soc., Providence, R.I., 1973. MathSciNet Article
- [HR1973b] Hensley, Douglas; Richards, Ian, Primes in intervals. Acta Arith. 25 (1973/74), 375–391. MathSciNet Article
- [MP2008] Motohashi, Yoichi; Pintz, János A smoothed GPY sieve. Bull. Lond. Math. Soc. 40 (2008), no. 2, 298–310. arXiv MathSciNet Article
- [MV1973] Montgomery, H. L.; Vaughan, R. C. The large sieve. Mathematika 20 (1973), 119–134. MathSciNet Article
- [M1978] Hugh L. Montgomery, The analytic principle of the large sieve. Bull. Amer. Math. Soc. 84 (1978), no. 4, 547–567. MathSciNet Article
- [R1974] Richards, Ian On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem. Bull. Amer. Math. Soc. 80 (1974), 419–438. MathSciNet Article
- [S1961] Schinzel, A. Remarks on the paper "Sur certaines hypothèses concernant les nombres premiers". Acta Arith. 7 1961/1962 1–8. MathSciNet Article
- [S2007] K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yıldırım. Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 1, 1–18. MathSciNet Article arXiv