Liouville type theorems for solutions of semilinear equations on non-compact Riemannian manifolds

It is proved that the Liouville function associated with the semilinear equation $\Delta u -g(x,u)=0$ is identical to zero if and only if there is only a trivial bounded solution of the semilinear equation on non-compact Riemannian manifolds. This result generalizes the corresponding result of S.A. Korolkov for the case of the stationary Schrödinger equation $ \Delta u-q (x) u = 0$. The concept of the capacity of a compact set associated with the stationary Schrödinger equation is also introduced and it is proved that if the capacity of any compact set is equal to zero, then the Liouville function is identically zero.

Extremal behaviour of ± 1-valued completely multiplicative functions in function fields

We investigate the classical Pólya and Turán conjectures in the context of rational function fields over finite fields 𝔽q. Related to these two conjectures we investigate the sign of truncations of Dirichlet L-functions at point s=1 corresponding to quadratic characters over 𝔽q[t], prove a variant of a theorem of Landau for arbitrary sets of monic, irreducible polynomials over 𝔽q[t] and calculate the mean value of certain variants of the Liouville function over 𝔽q[t].

Sarnak’s conjecture for sequences of almost quadratic word growth

We prove the logarithmic Sarnak conjecture for sequences of subquadratic word growth. In particular, we show that the Liouville function has at least quadratically many sign patterns. We deduce the main theorem from a variant which bounds the correlations between multiplicative functions and sequences with subquadratically many words which occur with positive logarithmic density. This allows us to actually prove that our multiplicative functions do not locally correlate with sequences of subquadratic word growth. We also prove a conditional result which shows that if the ( $\kappa -1$ )-Fourier uniformity conjecture holds then the Liouville function does not correlate with sequences with $O(n^{t-\varepsilon })$ many words of length n where $t = \kappa (\kappa +1)/2$ . We prove a variant of the $1$ -Fourier uniformity conjecture where the frequencies are restricted to any set of box dimension less than $1$ .

UNEXPECTED AVERAGE VALUES OF GENERALIZED VON MANGOLDT FUNCTIONS IN RESIDUE CLASSES

In order to study integers with few prime factors, the average of $\unicode[STIX]{x1D6EC}_{k}=\unicode[STIX]{x1D707}\ast \log ^{k}$ has been a central object of research. One of the more important cases, $k=2$ , was considered by Selberg [‘An elementary proof of the prime-number theorem’, Ann. of Math. (2)50 (1949), 305–313]. For $k\geq 2$ , it was studied by Bombieri [‘The asymptotic sieve’, Rend. Accad. Naz. XL (5)1(2) (1975/76), 243–269; (1977)] and later by Friedlander and Iwaniec [‘On Bombieri’s asymptotic sieve’, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4)5(4) (1978), 719–756], as an application of the asymptotic sieve. Let $\unicode[STIX]{x1D6EC}_{j,k}:=\unicode[STIX]{x1D707}_{j}\ast \log ^{k}$ , where $\unicode[STIX]{x1D707}_{j}$ denotes the Liouville function for $(j+1)$ -free integers, and $0$ otherwise. In this paper we evaluate the average value of $\unicode[STIX]{x1D6EC}_{j,k}$ in a residue class $n\equiv a\text{ mod }q$ , $(a,q)=1$ , uniformly on $q$ . When $j\geq 2$ , we find that the average value in a residue class differs by a constant factor from the expected value. Moreover, an explicit formula of Weil type for $\unicode[STIX]{x1D6EC}_{k}(n)$ involving the zeros of the Riemann zeta function is derived for an arbitrary compactly supported ${\mathcal{C}}^{2}$ function.

On the Pseudorandomness of the Liouville Function of Polynomials over a Finite Field

We study several pseudorandom properties of the Liouville function and the Möbius function of polynomials over a finite field. More precisely, we obtain bounds on their balancedness as well as their well-distribution measure, correlation measure, and linear complexity profile.

THE LOGARITHMICALLY AVERAGED CHOWLA AND ELLIOTT CONJECTURES FOR TWO-POINT CORRELATIONS

Let $\unicode[STIX]{x1D706}$ denote the Liouville function. The Chowla conjecture, in the two-point correlation case, asserts that $$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}\unicode[STIX]{x1D706}(a_{1}n+b_{1})\unicode[STIX]{x1D706}(a_{2}n+b_{2})=o(x)\end{eqnarray}$$ as $x\rightarrow \infty$, for any fixed natural numbers $a_{1},a_{2}$ and nonnegative integer $b_{1},b_{2}$ with $a_{1}b_{2}-a_{2}b_{1}\neq 0$. In this paper we establish the logarithmically averaged version $$\begin{eqnarray}\mathop{\sum }_{x/\unicode[STIX]{x1D714}(x)<n\leqslant x}\frac{\unicode[STIX]{x1D706}(a_{1}n+b_{1})\unicode[STIX]{x1D706}(a_{2}n+b_{2})}{n}=o(\log \unicode[STIX]{x1D714}(x))\end{eqnarray}$$ of the Chowla conjecture as $x\rightarrow \infty$, where $1\leqslant \unicode[STIX]{x1D714}(x)\leqslant x$ is an arbitrary function of $x$ that goes to infinity as $x\rightarrow \infty$, thus breaking the ‘parity barrier’ for this problem. Our main tools are the multiplicativity of the Liouville function at small primes, a recent result of Matomäki, Radziwiłł, and the author on the averages of modulated multiplicative functions in short intervals, concentration of measure inequalities, the Hardy–Littlewood circle method combined with a restriction theorem for the primes, and a novel ‘entropy decrement argument’. Most of these ingredients are also available (in principle, at least) for the higher order correlations, with the main missing ingredient being the need to control short sums of multiplicative functions modulated by local nilsequences. Our arguments also extend to more general bounded multiplicative functions than the Liouville function $\unicode[STIX]{x1D706}$, leading to a logarithmically averaged version of the Elliott conjecture in the two-point case. In a subsequent paper we will use this version of the Elliott conjecture to affirmatively settle the Erdős discrepancy problem.