Goldston's Math Page

 Math on the Web: Everything related to Math Number Theory Net : Everything related to  number theory. American Institute of Mathematics:  A different kind of math institute Mathematics ArXiv  :  Preprints from all areas of mathematics.

My Publications  (click on title for PDF of paper)

 1982-1986 1987-1991 1992-1996 1997-2002 2003-2009 Preprints Notes 1981 Berkeley PhD Thesis

# What I am Doing in 2010 (2011 update soon)

I'm still working on gaps between primes. The big problem here is to prove that there are bounded gaps between primes, but I don't think my abilities are well suited for the direct attack on this problem, and I am hoping someone else will pull that chestnut out of the fire.

Speaking of pulling chestnuts out of the fire, thanks to a clever idea of Pintz, the long delayed and almost dead paper Tuples IV will finally make its appearance this year. The main result we prove is that a positive proportion of  the gaps between primes are short gaps between primes, where short is any given fraction of the average spacing. If things work out there may also be a Tuples V and Tuples VI in the future.

I am working on Jumping Champions for gaps between primes with Andrew Ledoan, and this year we hope to resolve how the Hardy-Littlewood Prime Tuple Conjecture determines the Jumping Champions. Dec 2010 Update: We did it.

Finally, Sid Graham, Janos Pintz, Cem Yildirim and I are  writing a book on small gaps between primes.

# List of Publications       PDF of Vita as of 2011

 1982 [1]  On a result of Littlewood concerning prime numbers, Acta Arithmetica XL vol 3 (1982), 263-271. 1983 [2]  On a result of Littlewood concerning prime numbers II,  Acta Arithmetica XLIII  (1983), 49-51. 1984 [3] (with D. R. Heath-Brown) A note on the difference between consecutive primes, Math. Annalen  266 (1984), 317-320. [4] The second moment for prime numbers, Quart. J. Math. Oxford (2)  35 (1984), 153-163. 1985 [5] Prime numbers and the pair correlation of zeros of the zeta function, Topics in Analytic Number Theory, University of Texas Press, Austin, 1985, 82-91. [6] (with J. B. Conrey, A. Ghosh, S. M. Gonek, and D. R. Heath-Brown) On the distribution of gaps between zeros of the zeta-function, Quart. J. Math. Oxford (2)  36 (1985), 43-51. 1987 [7] (with A. Y. Cheer) A moment method for primes in short intervals, C. R.Math. Rep. Acad. Sci. Canada, Vol XI, No. 2, April 1987, 101-106. [8] (with H. L. Montgomery) Pair correlation of zeros and primes in short intervals, Analytic Number Theory and Diophantine Problems, Birkhauser, Boston, Mass., 1987, 183-203. [9] The function S(T) in the theory of the Riemann zeta-function, Journal of Number Theory  27, May, 1987, 149-177. [10] (with A. Y. Cheer) Longer than average intervals containing no primes, Transactions of the AMS, 304  Dec. 1987, 469-486. 1988
 [11] (with Kevin McCurley) Sieving the positive integers by large primes,  Journal of Number Theory 28 Jan. 1988, 94-115. [12] On the pair correlation conjecture for zeros of the Riemann zeta-function, J. reine angew. Math.  385 (1988), 24-40. [13] (with Kevin McCurley) Sieving the positive integers by small primes, Transactions of the AMS, 307  May 1988, 51-62..
 [14] (with S. M. Gonek) A note on the number of primes in short intervals, Proceedings of the AMS, 108 No. 3, March 1990, 613-620. [15] (with A. Y. Cheer) A difference delay equation arising from the sieve of Eratosthenes, Mathematics of Computation, .55 No. 191, July 1990,  129-141.
 [16] Linnik's theorem on Goldbach numbers in short intervals, Glasgow Math. J.  32  (1990), 285-297.
 [17] On Bombieri and Davenport's theorem concerning small gaps between primes, Mathematika 39 (1992), 10-17.
 1993
 [18]  An exponential sum over primes, Number theory with an emphasis on the Markov spectrum, Lecture Noves in Pure and Applied Mathematics Vol. 147, Marcel Dekker, New York, 1993, 101-106. [19] On Hardy and Littlewood's contribution to the Goldbach problem,  Proceedings of the Symposium on Analytic Number Theory, Amalfi 1989, 115-155.
 [20] (with A. Y. Cheer) Simple zeros of the Riemann zeta-function,  Proceedings of the AMS,  118 No. 2., June 1993, 365-372.
 1995
 [21] (with J. B. Friedlander) Some singular series averages and the distribution of Goldbach numbers in short intervals,  Illinois J. of Math., 39 No. 1 Spring 1995, 158-180.
 [22]  A lower bound for the second moment of primes in short intervals,  Expo Math. 13 (1995), 366-376.
 1996
 [23]  (with A. Y. Cheer) Turan's pure power sum problem,  Mathematics of Computation, 65 No. 215, July 1996, 1349-1358. [24]  (with J. B. Friedlander) Variance of distribution of primes in residue classes,  Quart. J. Math. (Oxford) (2)  47 (1996) 313-336.
 [25] (with R. C. Vaughan) On the Montgomery-Hooley asymptotic formula, Sieve Methods, Exponential Sums, and their Application in Number Theory, Greaves, Harman, Huxley Eds., Cambridge University Press, (1996), 117-142.
 1997 [26] (with J. B. Friedlander) Sums of three or more primes, Transactions of the AMS,  349,  No. 1, January 1997, 287-310.
 1998
 [27] (with S. M. Gonek) Mean value theorems for long Dirichlet polynomials and tails of Dirichlet series, Acta Arithmetica  LXXXIV.2 (1998), 155-192.
 [28] (with C. Y. Yildirim)  Primes in short segments of arithmetic progressions,  Can. J. Math.. 50 (3) 1998, 563-580.
 1999 [29] (with J. B. Friedlander) Note on a variance in the distribution of primes, Proceedings of the Schinzel Conference, July 1997, Zakopane, Poland, de Gruyter, 841-828.
 2000
 [30] (with Gonek, Ozluk, Snyder) On the pair correlation of the zeros of the Riemann zeta-function,   Proc. London Math. Soc. (3) 80 (2000) 31-49. [31] The major arcs approximation for an exponential sum over primes,  Acta Arithmetica  XCII.2  (2000) 169-179.. 2001
 [32] (with S. M. Gonek and H. L. Montgomery) Mean values of the logarithmic derivative of the Riemann zeta-function with applications to primes in short intervals,  J. reine angew. Math., 537 (2001) 105-126. [33] (with C. Y. Yildirim) On the second moment for primes in an arithmetic progression,  Acta Arithmetica  .100 Issues 1-4, 2001, 85-104. 2003 [34] (with C. Y. Yildirim) Higher correlations of divisor sums related to primes: I: Triple correlations, Integers 3 (2003), A5, 66pp. (electronic). 2005 [35] Notes on pair correlation of zeros and prime numbers, Recent Perspectives in Random Matrix Theory and Number Theory, LMS Lecture Note Series 322, Edited by F. Mezzadri and N. C. Snaith, Cambridge University Press, 2005, 79-110. 2006 [36] (with Y. Motohashi, J. Pintz, and C. Y. Yildirim), Small gaps between primes exist, Proc. Japan Acad., 82A (2006), 61-65. [37] (with J. Pintz and C. Y. Yildirim), Primes in Tuples III, Functiones at Approximatio, XXXV (2006), 79-89. 2007 [38] (with S. M. Gonek), A note on S(t) and the zeros of the Riemann zeta function, Bull. London Math. Soc. 39 (2007) 482-486. [39] (with C. Y. Yildirim), Higher correlation of short divisor sums II: Triple correlations: Variations of the error term in the prime number theorem, Proc. London Math. Soc. (3) 95 (2007) 199-247. [40] (with C. Y. Yildirim) Higher correlation of short divisor sums III: Gaps between primes, Proc. London Math. Soc. (3) 95 (2007) 653-686. [41] (with J. Pintz, C. Y. Yildirim), The path to recent progress on small gaps between primes, Clay Mathematics Proceedings, Volume 7, 2007, 129-139. 2009 [42] (with S. W. Graham, J. Pintz, C. Y. Yildirim), Small gaps between products of two primes,  Proc. London Math. Soc. (3)  98  (2009), 741-774. [43] (with J. Pintz, C. Y. Yildirim), Primes in Tuples I, Annals of Mathematics, Annals of Mathematics, 170 (2009), 819--862. [44] (with S. W. Graham, J. Pintz, and C. Y. Yildirim), Small gaps between primes or almost primes,  Transactions of the AMS, 361 No. 10, (2009), 5285-5330. Papers to Appear [45] Are there infinitely many twin primes?, to appear in book of Bay Area Math Adventure (BAMA) Talks [46] (with J. Pintz and C. Y. Yildirim) Primes in Tuples II AIM Preprint 2007-76 2010 [47] (with S. W. Graham, J. Pintz and C. Y. Yildirim) Small gaps between almost primes, the parity problem and some conjectures of Erdos on consecutive integers    International Mathematics Research Notices 2010; doi: 10.1093/imrn/rnq124. [48] (with A. Ledoan) Jumping champions and gaps between consecutive primes.  ArXiv 09102960 [49] (with Tsz Ho Chan and C. Y. Yildirim) Pair correlation of zeros of L-functions, in preparation. Other publications [1] (with A. Y. Cheer) A note on using calculus to explain the decimal expansion of 1/81, Math. and Computer Education, 25, no. 3, Fall 1991, 283-284.

# Unpublished Notes and Manuscripts

I have a number of papers which never were completed or published (yet), but I have not totally given up on them. I will be adding these notes and new ones here in the future. You can freely make use of them. They will be removed from here if they ever make it to publication or I give up on them entirely.  I also include here a few early versions of papers later published in a different form which still contain something that might be interesting.

1. A note on twin primes  (Written sometime in 2004.) This note is  not to be taken seriously but as a comment related to the arxiv paper Gaps Between Primes I,  which was published in revised form as Higher correlation of short divisor sums III: Gaps between primes,  It shows that extending the range of validity of the formulas in Gaps I would have significant applications but is probably very difficult. Until I decide whether this idea has any value except as a warning it will remain here.

2. Gaps Between Primes II  This is the original manuscript of Goldston-Pintz-Yildirim first distributed privately on Feb.8, 2005.  If you are interested in how our work looked in its original form, you can see that here. The content of this paper is what eventually was published in  Primes in Tuples I.

3. Here is a 3 page summary of Pintz, Yildirim and my results on small gaps between primes which I wrote for an Oberwolfach report in March 2008. There is nothing here that isn't in Primes in Tuples I,, but if you only have 10 minutes and want to know something about our method, this should do the trick.

4.  Here is a note I wrote in 2006 on an idea due to Pintz, who showed that in the GPY result on small gaps between primes one does not actually need to average over all tuples in an interval and then use Gallagher's singular series average result. Instead you can just throw enough extra stuff into your tuples to pick up the needed epsilon log N. No one has expressed much enthusiasm for this idea so it will probably remain unpublished until our book to be written by GGPY comes out, which is delayed because we have not started writing it yet.

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