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Multiplicative Functions in Short Intervals, with Applications

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Multiplicative Functions in Short Intervals, with Applications Feature Multiplicative Functions Functions in in Short Intervals, with With Applications Applications Kaisa Matomäki (University of Turku, Finland) Kaisa Matomäki (University of Turku, Finland) The understanding of the behaviour of multiplicative func- of prime factors. The result (1) is actually equivalent to the tions in short intervals has significantly improved during the prime number theorem, asserting that past decade. This has also led to several applications, in par- p X : p P lim |{ ≤ ∈ }| = 1, (2) ticular concerning correlations of multiplicative functions. X / →∞ X log X where P denotes the set of prime numbers. In this article, the 1 Introduction letter p will always denote a prime. Let us start by defining the key players. A function f : N C It will be very convenient for us to use o(1) and O(1) no- → = / is said to be multiplicative if f (mn) = f (m) f (n) whenever tations, so that A o(B) means that A B 0 for X = | | → →∞> gcd(m, n) = 1. We define the Liouville function λ: N and A O(B) means that A CB for some constant C 0 → | |≤ 1, 1 by λ(n):= ( 1)k when n has k prime factors (counted depending only on subscripts of O. In this notation (1) and (2) {− } − with multiplicity). For instance, λ(45) = λ(3 3 5) = ( 1)3 = can be written as · · − λ = 1. The function λ(n) is clearly multiplicative. (n) o(X) (3) − n X It is well known that the average value of λ(n) is 0, i.e. ≤ and 1 X X lim λ(n) = 0. (1) 1 = + o . (4) X X log X log X →∞ n X p X ≤ ≤ In other words, about half of the numbers have an odd number Before discussing the Liouville function further, let us define of prime factors and half of the numbers have an even number another important object: write ζ : C C for the Riemann → EMS Newsletter December 2020 39 Feature zeta function which is defined by ask how slowly H can tend to infinity with X so that we are 1 guaranteed to have ∞ 1 1 − ζ(s):= = 1 for s > 1, (5) ns − ps λ(n) = o(H), (9) n=1 p P ∈ X<n X+H where s denotes the real part of s. A series of the type ≤ s so that in the segment (X, X + H] roughly half of the numbers n N ann− with an C is called a Dirichlet series. The ζ- ∈ ∈ have an even and half of the numbers have an odd number of function can be analytically continued to the whole complex prime factors. plane apart from a simple pole at s = 1. The bound (8) together with the triangle inequality imme- It is easy to see that ζ(s) has no zeros with s > 1, diately implies (9) when and furthermore for s < 0 the only zeros are the “trivial C(log X)3/5 zeros” at negative even integers. The remaining zeros with H X exp . 0 s 1 are called the non-trivial zeros. One of the most ≥ −2(log log X)1/5 ≤ ≤ famous open problems in mathematics, the Riemann hypoth- However, one can show this for much shorter intervals. In esis, asserts that all these non-trivial zeros satisfy s = 1/2. 1972, Huxley proved prime number theorem in short inter- The zeros of the zeta function are closely related to the vals by showing that, for any ε>0, behaviour of the Liouville function. This relation stems from H H the fact that, for s > 1 1 = + o for H X7/12+ε. log X log X ≥ X<p X+H 1 1 ∞ λ(n)1n square-free ≤ = 1 = , (6) ζ(s) − ps ns Subsequently, in 1976 Motohashi [9] and Ramachandra [12] p P n=1 ∈ independently showed that Huxley’s ideas also work in the where 1n square-free denotes the characteristic function of the set case of the Liouville function, showing that, for any ε>0, of integers that are not divisible by a square of a prime. (9) holds for H X7/12+ . ≥ The function µ(n):= λ(n)1n square-free is called the Möbius The proofs of these results are based on zero-density es- function and its behaviour is very similar to that of the Liou- timates for the Riemann zeta function, i.e. estimates that give ville function. Consequently, the zeros of ζ(s) correspond to an upper bound for the number of zeta zeros in the rectangle s the poles of the Dirichlet series n N µ(n)n− that is closely ∈ s C: (s) [σ, 1] and (s) T related to the Liouville function. ∈ ∈ | |≤ One can show through (6) that (3) (and thus also the prime for given σ (1/2, 1] and T 2. ∈ ≥ number theorem (4)) is equivalent to the fact that the Riemann Using the multiplicativity of λ(n) in a crucial way, (9) was zeta function has no zeros with s = 1. The equivalence recently shown to hold for H X0.55+ε for any ε>0 by ≥ with the prime number theorem stems from the fact that, for Teräväinen and the author [7]. However, this result is still very s > 1, one has far from what is expected to be true – random models suggest that (9) holds in intervals much shorter than H = Xε. ζ ∞ log p (s) = , One can ask what about (9) in almost all short intervals − ζ pks k=1 p P ∈ rather than all (we say that a statement holds for almost all so the zeros of the zeta-function also correspond to the poles x X if the cardinality of the exceptional set is o(X)). In this ≤ of a Dirichlet series that is closely related to the characteristic case, the techniques based on the proof of Huxley’s prime function of the primes. number theorem get one down to H X1/6+ε for any ε>0; ≥ In general, the Liouville function is expected to behave whereas assuming the Riemann hypothesis, Gao has proved more or less randomly. In particular, we expect that it has so- it (in an unpublished work) for H (log X)A for certain fixed ≥ called square-root cancellation, i.e. one has, for all X 2, A > 0. ≥ 1/2+ε All the results discussed so far, with the exception of the λ(n) = O (X ) for any ε>0. (7) n X very recent work [7], have their counterparts for the primes. ≤ The conjecture (7) is in fact equivalent to the Riemann hy- Given these similarities and the so-called parity phenonenom, 1/2+ε 1 δ it is natural that until recently the problems for the primes and pothesis, and proving (7) even with X replaced by X − with a small fixed δ seems to be a distant dream which would for the Liouville function have been expected to be of equal ffi correspond to the Riemann zeta function having no zeros with di culty. real part 1 δ for some fixed δ>0. The best result currently However, in the recent years this expectation has turned ≥ − is that out to be wrong; in [2], Radziwiłł and the author made a breakthrough on understanding the Liouville function in short C(log X)3/5 λ(n) = O X exp (8) intervals by proving the following. − (log log X)1/5 n X ≤ Theorem 1. Let X H 2. Assume that H with for some absolute constant C > 0. This follows from the ≥ ≥ →∞ X . Then Vinogradov–Korobov zero-free region for the Riemann zeta →∞ λ(n) = o(H) (10) function that has been essentially unimproved for sixty years. x<n x+H ≤ for almost all x X. 2 Short intervals ≤ Note that H can go to infinity arbitrarily slowly here, e.g. A natural question is whether the average of the Liouville H = log log log log X, so this unconditionally improves upon function is still o(1) if taken over short segments; one can Gao’s result that was conditional on the Riemann hypothesis. 40 EMS Newsletter December 2020 Feature While our method fundamentally fails for the primes, it Theorem 6. Let α R and let X H 2. Assume that ∈ ≥ ≥ does work for more general multiplicative functions (see [2, H with X . Then →∞ →∞ Theorem 1]): λ(n)e(αn) = o(H) Theorem 2. Let X H 2. Let f : N [ 1, 1] be multi- x<n x+H ≥ ≥ → − ≤ plicative. Assume that H with X . Then for almost all x X. →∞ →∞ ≤ 1 1 f (n) f (n) = o(1) Theorem 5 follows from Theorem 6 through Fourier an- H − X x<n x+H n X alytic techniques. Note that the case α = 0 of Theorem 6 ≤ ≤ for almost all x X. corresponds to Theorem 1. Subsequently, Tao [14] used Theo- ≤ rem 6 alongside a novel entropy decrement argument to prove For many applications, it is helpful to also have a result a logarithmically averaged variant of Chowla’s conjecture for for complex-valued functions, and such an extension can be a fixed shift in the case k = 2: found in the recent pre-print [3], where we also extend our result in other directions, as we will explain below. Theorem 7. Let h 0. Then λ(n)λ(n + h) = o(log X). n 3 Applications n X ≤ Already in [2] we presented several applications of our gen- Both this and Theorems 5 and 6 have variants for much eral theorem such as more general multiplicative functions. Remarkably, Tao [13] was able to utilise the general version of Theorem 7 to prove ε> = ε Corollary 3.
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