## Periodic Reporting for period 1 - 1stProposal (An alternative development of analytic number theory and applications)

**Reporting period:**2015-08-01

**to**2017-01-31

## Summary of the context and overall objectives of the project

Analytic number theory is a fundamental area in pure mathematics. Research methods in the area have evolved primarily from Riemann’s 1859 monograph. This grant aims to develop the new alternative approach pioneered by Soundararajan at Stanford and the PI. This alternative approach has proved to be able to overcome several of the well-known limitations of Riemann’s technique, and we are enjoying a fertile period in which us, as well as some of the most exciting researchers around the world (many of whom trained with Soundararajan and/or me) are developing these ideas, and applying them to a host of previously unanswered important questions.

One way in which our branch of pure mathematics tends to be important for society is that many key applications of mathematics to computer science develop existing mathematical theories in which important conceptual work is in place to be built upon. Thus there are obvious and not so obvious applications of this work; the most obvious being to the analysis of the running time of algorithms, thus allowing computer designers to make informed choices in comparing what needs developing in their systems.

The main objectives of this grant are to gain a better understanding of the power of this theory, both by developing the core ideas, and by seeking new applications in neighbouring fields.

One way in which our branch of pure mathematics tends to be important for society is that many key applications of mathematics to computer science develop existing mathematical theories in which important conceptual work is in place to be built upon. Thus there are obvious and not so obvious applications of this work; the most obvious being to the analysis of the running time of algorithms, thus allowing computer designers to make informed choices in comparing what needs developing in their systems.

The main objectives of this grant are to gain a better understanding of the power of this theory, both by developing the core ideas, and by seeking new applications in neighbouring fields.

## Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

I have successfully developed a group of researchers at UCL that are enthusiastically attacking the main questions in the field. By next Autumn there will be four London-based postdoctoral Research Associates working with my group, as well as six doctoral students, and three or four masters level students and enthusiastic undergraduates. (Not all of these are funded by this grant.) Several of these, currently in residence, are working on deep and difficult aspects of the theory, particularly the two current Research Associates (Winston Heap, supported by the ERC grant, looking into Helson-like conjectures for L^1-norms of multiplicative functions; and Simon Myerson, supported by EPSRC, finding connections between his PhD thesis on an alternative approach to selecting major arcs in the circle method, and our alternative approach). My senior ERC-supported doctoral student, Oleksiy Klurman, has made startling progress on auto-correlation questions for multiplicative functions leading to several breakthroughs. My two new ERC-supported students Stephanie Chan and Nikoleta Kalaydzhieva have been doing background reading on the connection between elliptic curves and analytic number theory, and have started looking into the extraordinary very recent breakthroughs of Alexander Smith on congruent numbers, and on Goldfeld’s conjecture.

Between us we have worked on many of the questions raised in the grant proposal: On the formulation of Halasz’s Theorem as well as the pretentious large sieve (with Harper and Soundararajan), on links with the circle method (with de la Breteche and Soundararajan), and on the Selberg-Delange theorem (with Koukoulopoulos).

Some of my former students and postdocs have made substantial progress: Lamzouri (with Mangerel) on the spectrum of large character sums, and Koukoulopoulos on higher weight L-functions.

Several other projects will be mentioned in the next section.

Between us we have worked on many of the questions raised in the grant proposal: On the formulation of Halasz’s Theorem as well as the pretentious large sieve (with Harper and Soundararajan), on links with the circle method (with de la Breteche and Soundararajan), and on the Selberg-Delange theorem (with Koukoulopoulos).

Some of my former students and postdocs have made substantial progress: Lamzouri (with Mangerel) on the spectrum of large character sums, and Koukoulopoulos on higher weight L-functions.

Several other projects will be mentioned in the next section.

## Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

There have been several exciting developments by the research group funded by this grant:

Together with Fernando Shao at Oxford, we have understood why researchers had been unable, for the last 40+ years, to prove the analogy to the Bombieri-Vinogradov Theorem (BV) for all multiplicative functions, despite several well-known very flexible techniques that were felt to assure that any reasonable sequence would have the BV property. One issue, correlations to small conductor characters had been alluded to in the literature, and thus researchers assumed a “Siegel-Walfisz criteria” (SW) to circumvent this limitation. However we have found a new issue, now with large prime factors of the index of the sequence being counted, that also blocks the BV property. We were able to get around this in proving that smoothly supported multiplicative functions with the SW property also satisfy the BV property. This should appear in a top journal. Koukoulopoulos has suggested that we might (together) pursue an approach based on his converse theorem, instead of restricting to smoothly supported functions, which is a really interesting perspective, and well worth pursuing.

Going on from here, together with Sary Drappeau at Marseille (who was named as a postdoc in the grant proposal), we have extended the BV property well beyond the x^(1/2) barrier for a fixed modulus, on to x^(3/5), for smoothly supported functions. Using dispersion like this was proposed in our grant proposal but it has come much sooner and more easily than I expected, given that it is renowned to be such a tough technique to make work.

Soundararajan, Harper and I have published a paper on the basic alternative theory in function fields. This is much easier than the number field version, but has some if its own twists and turns; in effect, because it has a discrete spectrum alongside the continuous spectrum inherited from the number field case. My student Klurman has found some other phenomena unique to the function field case as I explain below.

My doctoral student, Oleksiy Klurman, has made some startling progress on some important questions. One, which appeared in the proposal, is to better understand the Gowers U^k-norms of multiplicative functions, using techniques from analytic number theory, rather than the (more difficult) ergodic theory approach of Frantzikinakis and Host, and of Matthieson. This allows Oleksiy to much better understand the value of the U^k-norm. He is therefore able to prove the bounds on the U^k-norm of the Mobius function, which is central to the work of Green, Tao and Ziegler on prime k-tuplets. These bounds are not, as yet, good enough to reprove the Green-Tao-Ziegler theorem but there is hope. If he/we can achieve this it would be a massive win for classical-type techniques in the subject, especially as it would be a so much simpler approach, avoiding many of the very deep ideas that had been needed by GTZ.

More generally, Klurman has been able to get a strong grip on auto-correlations of multiplicative functions, proving the Erdos-Coons-Tao conjecture which has led to many exciting consequences inside the area and out. He has also developed the “alternative theory” for function fields well beyond where Harper, Soundararajan and I left it, finding a new type of pseudo-character obstruction. This work will be central to any further developments in this direction.

I finished the preprint with Koukoulopoulos and Maynard in which we sought to better understand moments of partial sums of the Mobius function, both smoothed and unsmoothed. This has led to extraordinary new questions about integrating (in very high dimension), products of many zeta-functions. The eventual paper is difficult, 87 pages long, but has led to lots of interest by other researchers.

Together with Fernando Shao at Oxford, we have understood why researchers had been unable, for the last 40+ years, to prove the analogy to the Bombieri-Vinogradov Theorem (BV) for all multiplicative functions, despite several well-known very flexible techniques that were felt to assure that any reasonable sequence would have the BV property. One issue, correlations to small conductor characters had been alluded to in the literature, and thus researchers assumed a “Siegel-Walfisz criteria” (SW) to circumvent this limitation. However we have found a new issue, now with large prime factors of the index of the sequence being counted, that also blocks the BV property. We were able to get around this in proving that smoothly supported multiplicative functions with the SW property also satisfy the BV property. This should appear in a top journal. Koukoulopoulos has suggested that we might (together) pursue an approach based on his converse theorem, instead of restricting to smoothly supported functions, which is a really interesting perspective, and well worth pursuing.

Going on from here, together with Sary Drappeau at Marseille (who was named as a postdoc in the grant proposal), we have extended the BV property well beyond the x^(1/2) barrier for a fixed modulus, on to x^(3/5), for smoothly supported functions. Using dispersion like this was proposed in our grant proposal but it has come much sooner and more easily than I expected, given that it is renowned to be such a tough technique to make work.

Soundararajan, Harper and I have published a paper on the basic alternative theory in function fields. This is much easier than the number field version, but has some if its own twists and turns; in effect, because it has a discrete spectrum alongside the continuous spectrum inherited from the number field case. My student Klurman has found some other phenomena unique to the function field case as I explain below.

My doctoral student, Oleksiy Klurman, has made some startling progress on some important questions. One, which appeared in the proposal, is to better understand the Gowers U^k-norms of multiplicative functions, using techniques from analytic number theory, rather than the (more difficult) ergodic theory approach of Frantzikinakis and Host, and of Matthieson. This allows Oleksiy to much better understand the value of the U^k-norm. He is therefore able to prove the bounds on the U^k-norm of the Mobius function, which is central to the work of Green, Tao and Ziegler on prime k-tuplets. These bounds are not, as yet, good enough to reprove the Green-Tao-Ziegler theorem but there is hope. If he/we can achieve this it would be a massive win for classical-type techniques in the subject, especially as it would be a so much simpler approach, avoiding many of the very deep ideas that had been needed by GTZ.

More generally, Klurman has been able to get a strong grip on auto-correlations of multiplicative functions, proving the Erdos-Coons-Tao conjecture which has led to many exciting consequences inside the area and out. He has also developed the “alternative theory” for function fields well beyond where Harper, Soundararajan and I left it, finding a new type of pseudo-character obstruction. This work will be central to any further developments in this direction.

I finished the preprint with Koukoulopoulos and Maynard in which we sought to better understand moments of partial sums of the Mobius function, both smoothed and unsmoothed. This has led to extraordinary new questions about integrating (in very high dimension), products of many zeta-functions. The eventual paper is difficult, 87 pages long, but has led to lots of interest by other researchers.