Analytic number theory in honor of helmut maier 19 birthday


Analytic Number Theory

A -derived multiplicative function will then be said to have an underived pole of order if it is the top order coefficient of a multiplicative function with a pole of order ; in terms of Dirichlet series, this roughly means that the Dirichlet series has a pole of order at. For instance, the singly derived multiplicative function has an underived pole of order , because it is the top order coefficient of , which has a pole of order ; similarly has an underived pole of order , being the top order coefficient of.

More generally, and have underived poles of order and respectively for any. By taking top order coefficients, we then see that the convolution of a -derived multiplicative function with underived pole of order and a -derived multiplicative function with underived pole of order is a -derived multiplicative function with underived pole of order. If there is no oscillation in the primes, the product of these functions will similarly have an underived pole of order , for instance has an underived pole of order. We then have the dimensional consistency property that in any of the standard identities involving derived multiplicative functions, all terms not only have the same derived order, but also the same underived pole order.

For instance, in 3 , 4 , 5 all terms have underived pole order with any Mobius function terms being counterbalanced by a matching term of or. This gives a second way to use dimensional analysis as a consistency check.

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For instance, any identity that involves a linear combination of and is suspect because the underived pole orders do not match being and respectively , even though the derived orders match both are. The results here are vaguely reminiscent of the recent progress on bounded gaps in the primes, but use different methods. About a decade ago, Ben Green and I showed that the primes contained arbitrarily long arithmetic progressions: given any , one could find a progression with consisting entirely of primes. In fact we showed the same statement was true if the primes were replaced by any subset of the primes of positive relative density.

Again, the same statement also applies if the primes were replaced by a subset of positive relative density. My previous result with Ben corresponds to the linear case. The result is still true if the primes are replaced by a subset of positive density , but unfortunately in our arguments we must then let depend on. However, in the linear case , we were able to make independent of although it is still somewhat large, of the order of. The polylogarithmic factor is somewhat necessary: using an upper bound sieve, one can easily construct a subset of the primes of density, say, , whose arithmetic progressions of length all obey the lower bound.

On the other hand, the prime tuples conjecture predicts that if one works with the actual primes rather than dense subsets of the primes, then one should have infinitely many length arithmetic progressions of bounded width for any fixed.


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The case of this is precisely the celebrated theorem of Yitang Zhang that was the focus of the recently concluded Polymath8 project here. The higher case is conjecturally true, but appears to be out of reach of known methods.

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Using the multidimensional Selberg sieve of Maynard, one can get primes inside an interval of length , but this is such a sparse set of primes that one would not expect to find even a progression of length three within such an interval. This correlation condition required one to control arbitrarily long correlations of , which was not compatible with a bounded value of particularly if one wanted to keep independent of. Conlon-Fox-Zhao did this for my original theorem with Ben; and in the current paper we apply the densification method to our previous argument to similarly remove the correlation condition.

This method does not fully eliminate the need to control arbitrarily long correlations, but allows most of the factors in such a long correlation to be bounded , rather than merely controlled by an unbounded weight such as. We believe though that this an artefact of our method, and one should be able to prove our theorem with an that is uniform in.

Here is a simple instance of the densification trick in action.

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Suppose that one wishes to establish an estimate of the form. Here I will be vague as to exactly what range the parameters are being averaged over. Suppose that the factor say has enough uniformity that one can already show a smallness bound. Since is bounded in magnitude by , we can bound the left-hand side of 1 as. The weight function will be normalised so that , so by the Cauchy-Schwarz inequality it suffices to show that. In view of this estimate, we now just need to show that. By setting to be the signum of , this is equivalent to. However, one can shift by and repeat the above arguments to achieve a similar densificiation of , at which point one has reduced 1 to 2.

Blog at WordPress. Ben Eastaugh and Chris Sternal-Johnson. Subscribe to feed. What's new Updates on my research and expository papers, discussion of open problems, and other maths-related topics. Monthly Archive. You are currently browsing the monthly archive for September Derived multiplicative functions 24 September, in expository , math. In this setting, we can add or multiply two arithmetic functions to obtain further arithmetic functions , and we can also form the Dirichlet convolution by the usual formula Regardless of what commutative ring is in used here, we observe that Dirichlet convolution is commutative, associative, and bilinear over.

An important class of arithmetic functions in analytic number theory are the multiplicative functions , that is to say the arithmetic functions such that and for all coprime. The specific multiplicative functions listed above are also related to each other by various important identities, for instance where is an arbitrary arithmetic function. More precisely: Definition 1 A derived multiplicative function is an arithmetic function that can be expressed as the formal derivative at the origin of a family of multiplicative functions parameterised by a formal parameter.

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More generally, for any , a -derived multiplicative function is an arithmetic function that can be expressed as the formal derivative at the origin of a family of multiplicative functions parameterised by formal parameters. There are Leibniz rules similar to 2 but they are harder to state; for instance, a doubly derived multiplicative function comes with singly derived multiplicative functions and a multiplicative function such that for all coprime.

For instance, the Leibniz rule for any arithmetic functions can be viewed as the top order term in in the ring with one infinitesimal , and then we see that the Leibniz rule is a special case or a derivative of 1 , since is completely multiplicative. Similarly, the formulae are top order terms of and the variant formula is the top order term of which can then be deduced from the previous identities by noting that the completely multiplicative function inverts multiplicatively, and also noting that annihilates.

Jaffe, Arthur M. Limited documentation for this scholar.

School of Mathematics explained that this can happen if a scholar is invited by Faculty. This circumstance may apply in this instance. Information about students of Lenstra's, who were invited to come to the Institute to join him, accompany his materials. The scholars mentioned are: Oliver Schirokauer, Everett W.

Howe, Bart de Smit, , and Carl F. Further documentation for some of these individual scholars can be found in this series. Jean-Marc Deshouillers. Files lack documentation for fall visit to the Institute. School of Mathematics found nothing further at the time these files were transferred to the Archives Center.

These documents also pertain to Pierre Colmez. Sethi, Savdeep S. Other appointments to School of Natural Science only.


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  8. Files lacking documentation for all visits to the Institute. Documents for Woods include two pages of information about the Second Meeting of the Mid-Atlantic Logic Seminar December , , which was held on the Institute campus. Speakers listed include: M. Makkai, S. Thomas, H. Woodin, J. Schmerl, S. Haran, C. Videla, and S. The Staff and Member directories would also provide information that could be useful, including for the years when the registries lack this information residential addresses, which are most often on the Institute campus. In some instances, the funding source is a Professor of the School of Mathematics to whom the listed scholar is an Assistant.

    The Faculty member's name is given in full or indicated with initials.

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    Pomerance, Carl

    Documents for most years additionally include the following: whether or not that scholar was an American Mathematical Society nominee, campus arrival date sometimes departure date, too , telephone numbers, and residential address. These digital scans of the original paper documents in this same subseries were created by the School of Mathematics, but the quality is not very good. Since many of the documents are handwritten, better scanning won't improve keyword search capability. Data collected to meet requirements of the National Science Foundation NSF , which provides financial support to visiting mathematicians.

    Roughly half of each year's cadre in the School of Mathematics are postdocs i. Only those scholars who were affiliated with the School of Mathematics for more than one consecutive academic year would be tracked for the full five years. Scholars of the School for a single academic year were tracked only for an additional three years after they left the Institute.

    These records include tracking of scholars with the School for the academic years through and the data was collected annually by staff of the School of Mathematics, who contacted each scholar in order to request an update regarding academic or professional affiliation which is the only parameter provided in these documents. These documents gather together data points of interest about Members and Visitors for the academic years through that are useful reference both within the School and more broadly, including for the Institute's annual report.

    The data fields are: full name, IAS year, position and term, gender and age, most recent institutional affiliation, citizenship, funds, and field of research. The "Position" field provides the Institute's terminology for that scholars' affiliation, e. Chiefly digital images of index cards for visiting scholars starting at the founding of the School. Contains detailed information, especially in the earliest years, such as the exact date of a scholar's departure.

    For a limited number of individuals, there are also casual portrait photographs similar to those featured in the School's headshot photo albums. The physical records from which this digital version was created are still in the active records of the School of Mathematics.

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    As reported by the office of origin, there were many individuals contributing to this digitization project, and quality control was not always possible, so comparison to the originals should be pursued if something appears to be missing. Three files fall outside the pattern of alphabetical ordering: Dotties Cards, new scanned, and Groups.

    These are described more fully in the Arrangement note. Most of this material is foldered alphabetically with a separate folder for each letter and a separate file for each person. Three files diverge from this pattern: Dotties Cards, new scanned, and Groups. The first of these are digital images of the Rolodex cards maintained by Dorothea Phares, who was on the staff of the School of Mathematics from , and features only contact information for scholars, which may be available from other sources.

    The other two non-alpha files are work in progress and the contents were most often incorporated into the main, larger digital files. In the case of "new scanned," some files had not been incorporated into the more complete files. A further copy was made to crop the pages, combine the files, and make the document searchable. When the physical photograph albums through were transferred, the Archives Center staff digitized those albums. Scans of physical photo albums featuring headshots of visiting scholars and also, sometimes, Faculty and Staff.

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