## Periodic Reporting for period 4 - 1stProposal (An alternative development of analytic number theory and applications)

Reporting period: 2020-02-01 to 2020-07-31

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 aimed 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 were 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.

All of this emerged from the grant. Perhaps what was unexpected was how many other top researchers have been working on these ideas so there have been several enormous strides forward.

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 were 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.

All of this emerged from the grant. Perhaps what was unexpected was how many other top researchers have been working on these ideas so there have been several enormous strides forward.

I have successfully developed a group of researchers at UCL that are enthusiastically attacking the main questions in the field, four London-based postdoctoral Research Associates working with my group, as well as five doctoral students, and several masters level students and enthusiastic undergraduates.

Between us there are many papers and several of the key issues in the proposal have been resolved. I have been most pleased that the formulation of Halasz’s Theorem as well as the pretentious large sieve (with Harper and Soundararajan) has been finalized. We have determined new exciting on links with the circle method (with de la Breteche), and on the Selberg-Delange theorem (with Koukoulopoulos).

Between us there are many papers and several of the key issues in the proposal have been resolved. I have been most pleased that the formulation of Halasz’s Theorem as well as the pretentious large sieve (with Harper and Soundararajan) has been finalized. We have determined new exciting on links with the circle method (with de la Breteche), and on the Selberg-Delange theorem (with Koukoulopoulos).

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.

After that, Shao and I were able to fully resolve this issue, at least up to x^(1/2), showing that a BV theorem holds for F and its convolution converse g, if and only if they satisfy an SW property, anf f, supported now only on primes, also satisfies a BV property. This was a surprise to me but came out of our developments.

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.

After that, Shao and I were able to fully resolve this issue, at least up to x^(1/2), showing that a BV theorem holds for F and its convolution converse g, if and only if they satisfy an SW property, anf f, supported now only on primes, also satisfies a BV property. This was a surprise to me but came out of our developments.

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.