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What's New ? |
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April, 2008: 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.
April, 2008: Cousin Hiromi visited the day after we got back from the desert.
April, 2008: Here is a 3 page summary of Pintz, Yildirim and my results on small gaps between primes which I wrote for an Oberwolfach report. 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.
March, 2008: Here are some Goldston family pictures from a short trip to the Mojave Desert at the end of March. Someday after our kids finally get to go to Disneyland or Sea World or Hawaii they are going to be really mad that this was even considered as a suitable place for a vacation.
March, 2008: The 23 page paper Small gaps between almost primes, the parity problem and some conjectures of Erdos on consecutive integers, with S. W. Graham, J. Pintz, and C. Y. Yildirim is now on Arxiv. This paper gives some applications of our method to some well-known conjectures of Erdos. The paper is essentially elementary, and perhaps there are further applications we haven't found yet.
December, 2007: Here are a few pictures of our family horseback riding on a trip to the San Luis Obispo area of California.
November, 2005: The 5 page paper A note on S(T) and the zeros of the Riemann zeta-function with S. M. Gonek is now on Arxiv and in the AIM preprint series. This paper assumes the Riemann hypothesis and proves that | S(T) | < (1/2 +o(1))log T/log log T. Except for the constant this is an old result of Littlewood, but maybe the constant 1/2 we obtain here is optimal given one uses an explicit formula and no deep information about primes.
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November, 2005: Here are some 2005 Goldston family pictures.
Mako, Cousin Hiromi, Aiko at the Jelly Belly Factory, May 2005
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August, 2005: The paper Primes in Tuples I (joint with J. Pintz, and C. Y. Yildirim) is now on the AIM Preprint Series and on Arxiv. This 36 page paper is the revision of our original manuscript from February. The results presented here are all from before the two papers uploaded in May and June. I hope the first four sections of the paper will give the non-expert reader the main ideas in our method. At that point a reader can pretty much choose whether to look into the details of the proofs from either this paper or the two other papers. There is already a "second generation" approach of Soundararajan which reduces everything to an optimization problem, and reveals some limitations in the method.
Here is the Science magazine article from May on our work.
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June, 2005: The paper Small Gaps Between Primes and Almost Primes (joint with S. W; Graham, J. Pintz, and C. Y. Yildirim) is now on the AIM Preprint Series and on Arxiv. This 49 page paper gives an alternative proof of the our new results on small gaps between primes using sieve methods. This includes an improved conditional result that assuming the Elliott-Halberstam conjecture there will be infinitely often primes within 16 of each other.One noteworthy feature of the proofs here is that the main terms are all evaluated without using zero-free regions. The paper also gives applications to almost prime numbers, where we obtain unconditionally results for almost prime numbers which we can only prove conditionally for primes.
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May, 2005: The paper Small Gaps Between Primes Exist (joint with Y. Motohashi, J. Pintz, and C. Y. Yildirim) is now on the AIM Preprint Series and on Arxiv. This 8 page paper gives a complete proof that there are infinitely often two primes within any small multiple of the average spacing. It also contains the conditional result that any improvement in the level of distribution of primes in arithmetic progressions beyond the Bombieri-Vinogradov level 1/2 will imply infinitely often the existence of bounded gaps between primes, and in particular the Elliott-Halberstam conjecture implies gaps of less than or equal to 20 infinitely often (this has been improved to 16).. This paper gives a very short proof of the main results proved by Goldston-Pintz-Yildirim in a manuscript that is currently being rewritten into two or more papers. The method can be viewed as in some sense repairing the mistaken approximation from two years ago, but it is more accurate to view it as a weighted Selberg sieve on almost-prime tuples.
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April, 2005: The paper Small Gaps Between Primes I (joint with C. Y. Yildirim) is now on the AIM Preprint Series and on Arxiv. This paper is what came out of our work two years ago once we removed the mistaken approximation. (However see above May 2005.) This paper is probably most notable because it contains the lemma used by Green and Tao in their great work The primes contain arbitrarily long aritmetic progressions. Related to Gaps I is Note on Twin Primes. This note shows that extending the range of validity of the formulas in Gaps I would have significant applications but is probably very difficult.
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February, 2005: Here are some pictures from September 2005. The first pictures are from the 60th Birthday Celebration for Hugh Montgomery in Ann Arbor. After these are pictures from the workshop on the Riemann zeta-function at Oberwolfach. It rained almost the whole week, but on the free afternoon it stopped long enough to walk down to the town of Oberwolfach, and most of the pictures were taken then. Finally I spent 5 minutes shooting pictures in the lounge one night.
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February, 2005: In 2003 I became famous for a few weeks. Here is an article, My 30 Minutes of Fame, which I wrote for the Spring 2004 Math Department Newsletter (on page two) concerning my brush with fame. Shortly after this was written I found how to partly fix the idea that I claimed in the article totally failed. Never give up.
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January, 2005: Here are two papers that are in the AIM Preprint Series and on the Arxiv. Notes on Pair Correlation of Zeros of the Zeta Function is based on some lectures I gave during a workshop on Random Matrices and Analytic Number Theore at the Newton Institute in Cambridge, England in April 2004. All of the lectures notes for the workshop will soon be published in a book by Cambridge University Press. Higher Correlations of Divisor Sums II with Cem Yildirim obtains the same triple correlations obtained in Higher Correlation of Short Divisor Sums I: Triple Correlations but for a refined divisor sum which arises naturally in the Selberg Sieve. But this paper should make clear that what is natural for a binary problem i.e. squaring, is not so natural for 3-tuples i.e. cubing. The advantaqge however over the earlier paper is that some error terms are smaller and we obtain on GRH new results on primes in short intervals. We are able in this way to obtain assuming GRH both positive and negative oscillations for the difference between the number of primes in the interval [x,x+h], h=x^c, 0<c<1/7, and the expected number, where previously this could only be shown for the absolute value of this difference. .
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January, 2005: Here are some 2003-2004 pictures. 2003-2004 Goldston Family Pictures
Shota, Momoko (cousin), Hiromi (cousin), Aiko, Makoto. August, 2004 in Portland Oregon