The Fourier uniformity conjecture seeks to understand what multiplicative functions can have large Fourier coefficients on many short intervals. We will discuss recent progress on this problem and explain its connection with the distribution of prime numbers and with other central problems about the behaviour of multiplicative functions, such as the Chowla and Sarnak conjectures.
A conjecture of Chowla postulates that no L-function of Dirichlet characters over the rationals vanishes at s=1/2. Soundararajan has proved non-vanishing for a positive proportion of quadratic characters. Over function fields Li has discovered that Chowla's conjecture fails for infinitely many distinct quadratic characters. However, on the basis of the Katz-Sarnak heuristics, it is still widely believed that one should have non-vanishing for 100% of the characters in natural families (such as the family of quadratic characters). Works of Bui-Florea, David-Florea-Lalin, Ellenberg-Li-Shusterman, among others, provided evidence giving a positive proportion of non-vanishing in several such families. I will present an upcoming joint work with Peter Koymans and Mark Shusterman, where we prove that for each fixed q congruent to 3 modulo 4 one has 100% non-vanishing in the family of imaginary quadratic function fields.
The Chowla conjecture from 1965 predicts that all autocorrelations of the Liouville function vanish. In fact, after an adaptation, the Chowla conjecture was expected to hold for all aperiodic multiplicative functions with values in the unit disc (cf. Elliott’s conjecture from the 1990s). But in 2015, Matomäki, Radziwiłł and Tao gave a counterexample to Elliott’s conjecture by constructing aperiodic multiplicative functions (bounded by 1) for which (already) the Chowla conjecture of order 2 fails. During the talk I will explain however that the Chowla conjecture does not disappear entirely, in fact, for Matomäki, Radziwiłł, Tao’s functions it holds along a subsequence. The result is achieved by detecting certain dynamical systems (of algebraic origin), called Furstenberg systems, obtained by examining autocorrelations along subsequences. Moreover, the same technique disproves a recent conjecture by Frantzikinakis and Host concerning logarithmic autocorrelations of multiplicative functions.
called Furstenberg systems, obtained by examining autocorrelations along subsequences. Moreover, the same technique disproves a recent conjecture by Frantzikinakis and Host concerning logarithmic autocorrelations of multiplicative functions.
The talk is based on my joint work with Alex Gomilko and Thierry de la Rue.
I will discuss recent work with Harald Helfgott in which we establish roughly speaking that the graph connecting n to n ± p with p a prime dividing n is almost "locally Ramanujan". As a result we obtain improvements of results of Tao and Tao-Teravainen on logarithmic Chowla. I will discuss the main ideas in the proof and the connections with logarithmic Chowla.
Let λ be the Liouville function and P(x) any polynomial that is not a square. An open problem formulated by Chowla and others asks to show that the sequence λ(P(n)) changes sign infinitely often. We present a solution to this problem for new classes of polynomials P, including any product of linear factors or any product of quadratic factors of a certain type. The proofs also establish some nontrivial cancellation in Chowla and Elliott type correlation averages.
In 2016, Tao formulated a conjecture on the Gowers uniformity of the Möbius function in short intervals, which he showed to be equivalent to both the (logarithmic) Chowla and Sarnak conjectures. I will discuss work where we prove this conjecture for intervals of length X𝜺. I will then discuss applications to superpolynomial word complexity for the Liouville sequence and to a new averaged version of Chowla's conjecture.
The Liouville function is a multiplicative function that encodes important information related to distributional properties of the prime numbers. A conjecture of Chowla states that the values of the Liouville function fluctuate between plus and minus in such a random way, that all sign patterns of a given length appear with the same frequency. The Chowla conjecture remains largely open and in this talk we will see how ergodic theory combined with some feedback from number theory allows us to establish two variants of this conjecture. Key to our approach is an in-depth study of measure preserving systems that are naturally associated with the Liouville function. The talk is based on joint work with Bernard Host.
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