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Let $\mathbb {T}$ be the unit circle and ${\Gamma \backslash G}$ the $3$-dimensional Heisenberg nilmanifold. We consider the skew products on $\mathbb {T} \times {\Gamma \backslash G}$ and prove that the Möbius function is linearly disjoint from these skew products which improves the recent result of Huang, Liu and Wang [‘Möbius disjointness for skew products on a circle and a nilmanifold’, Discrete Contin. Dyn. Syst.41(8) (2021), 3531–3553].
We investigate uniform upper bounds for the number of powerful numbers in short intervals
$(x, x + y]$
. We obtain unconditional upper bounds
$O({y}/{\log y})$
and
$O(\kern1.3pt y^{11/12})$
for all powerful numbers and
$y^{1/2}$
-smooth powerful numbers, respectively. Conditional on the
$abc$
-conjecture, we prove the bound
$O({y}/{\log ^{1+\epsilon } y})$
for squarefull numbers and the bound
$O(\kern1.3pt y^{(2 + \epsilon )/k})$
for k-full numbers when
$k \ge 3$
. These bounds are related to Roth’s theorem on arithmetic progressions and the conjecture on the nonexistence of three consecutive squarefull numbers.
Given a large integer n, determining the relative size of each of its prime divisors as well as the spacings between these prime divisors has been the focus of several studies. Here, we examine the spacings between particular types of prime divisors of n, such as prime divisors in certain congruence classes of primes and various other subsets of the set of prime numbers.
Sarnak’s Möbius disjointness conjecture asserts that for any zero entropy dynamical system
$(X,T)$
,
$({1}/{N})\! \sum _{n=1}^{N}\! f(T^{n} x) \mu (n)= o(1)$
for every
$f\in \mathcal {C}(X)$
and every
$x\in X$
. We construct examples showing that this
$o(1)$
can go to zero arbitrarily slowly. In fact, our methods yield a more general result, where in lieu of
$\mu (n)$
, one can put any bounded sequence
$a_{n}$
such that the Cesàro mean of the corresponding sequence of absolute values does not tend to zero. Moreover, in our construction, the choice of x depends on the sequence
$a_{n}$
but
$(X,T)$
does not.
We establish the optimal order of Malliavin-type remainders in the asymptotic density approximation formula for Beurling generalized integers. Given
$\alpha \in (0,1]$
and
$c>0$
(with
$c\leq 1$
if
$\alpha =1$
), a generalized number system is constructed with Riemann prime counting function
$ \Pi (x)= \operatorname {\mathrm {Li}}(x)+ O(x\exp (-c \log ^{\alpha } x ) +\log _{2}x), $
and whose integer counting function satisfies the extremal oscillation estimate
$N(x)=\rho x + \Omega _{\pm }(x\exp (- c'(\log x\log _{2} x)^{\frac {\alpha }{\alpha +1}})$
for any
$c'>(c(\alpha +1))^{\frac {1}{\alpha +1}}$
, where
$\rho>0$
is its asymptotic density. In particular, this improves and extends upon the earlier work [Adv. Math. 370 (2020), Article 107240].
We prove that analogues of the Hardy–Littlewood generalised twin prime conjecture for almost primes hold on average. Our main theorem establishes an asymptotic formula for the number of integers
$n=p_1p_2 \leq X$
such that
$n+h$
is a product of exactly two primes which holds for almost all
$|h|\leq H$
with
$\log^{19+\varepsilon}X\leq H\leq X^{1-\varepsilon}$
, under a restriction on the size of one of the prime factors of n and
$n+h$
. Additionally, we consider correlations
$n,n+h$
where n is a prime and
$n+h$
has exactly two prime factors, establishing an asymptotic formula which holds for almost all
$|h| \leq H$
with
$X^{1/6+\varepsilon}\leq H\leq X^{1-\varepsilon}$
.
The Euler–Mascheroni constant
$\gamma =0.5772\ldots \!$
is the
$K={\mathbb Q}$
example of an Euler–Kronecker constant
$\gamma _K$
of a number field
$K.$
In this note, we consider the size of the
$\gamma _q=\gamma _{K_q}$
for cyclotomic fields
$K_q:={\mathbb Q}(\zeta _q).$
Assuming the Elliott–Halberstam Conjecture (EH), we prove uniformly in Q that
In other words, under EH, the
$\gamma _q /\!\log q$
in these ranges converge to the one point distribution at
$1$
. This theorem refines and extends a previous result of Ford, Luca and Moree for prime
$q.$
The proof of this result is a straightforward modification of earlier work of Fouvry under the assumption of EH.
We compute, via motivic wall-crossing, the generating function of virtual motives of the Quot scheme of points on
${\mathbb{A}}^3$
, generalising to higher rank a result of Behrend–Bryan–Szendrői. We show that this motivic partition function converges to a Gaussian distribution, extending a result of Morrison.
We prove that the Riemann hypothesis is equivalent to the condition
$\int _{2}^x\left (\pi (t)-\operatorname {\textrm {li}}(t)\right )\textrm {d}t<0$
for all
$x>2$
. Here,
$\pi (t)$
is the prime-counting function and
$\operatorname {\textrm {li}}(t)$
is the logarithmic integral. This makes explicit a claim of Pintz. Moreover, we prove an analogous result for the Chebyshev function
$\theta (t)$
and discuss the extent to which one can make related claims unconditionally.
Let
$g \geq 1$
be an integer and let
$A/\mathbb Q$
be an abelian variety that is isogenous over
$\mathbb Q$
to a product of g elliptic curves defined over
$\mathbb Q$
, pairwise non-isogenous over
$\overline {\mathbb Q}$
and each without complex multiplication. For an integer t and a positive real number x, denote by
$\pi _A(x, t)$
the number of primes
$p \leq x$
, of good reduction for A, for which the Frobenius trace
$a_{1, p}(A)$
associated to the reduction of A modulo p equals t. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions, we prove that
$\pi _A(x, 0) \ll _A x^{1 - \frac {1}{3 g+1 }}/(\operatorname {log} x)^{1 - \frac {2}{3 g+1}}$
and
$\pi _A(x, t) \ll _A x^{1 - \frac {1}{3 g + 2}}/(\operatorname {log} x)^{1 - \frac {2}{3 g + 2}}$
if
$t \neq 0$
. These bounds largely improve upon recent ones obtained for
$g = 2$
by Chen, Jones, and Serban, and may be viewed as generalizations to arbitrary g of the bounds obtained for
$g=1$
by Murty, Murty, and Saradha, combined with a refinement in the power of
$\operatorname {log} x$
by Zywina. Under the assumptions stated above, we also prove the existence of a density one set of primes p satisfying
$|a_{1, p}(A)|>p^{\frac {1}{3 g + 1} - \varepsilon }$
for any fixed
$\varepsilon>0$
.
Let $[t]$ be the integral part of the real number t. We study the distribution of the elements of the set $\mathcal {S}(x) := \{[{x}/{n}] : 1\leqslant n\leqslant x\}$ in the arithmetical progression $\{a+dq\}_{d\geqslant 0}$. We give an asymptotic formula
$$ \begin{align*} S(x; q, a) := \sum_{\substack{m\in \mathcal{S}(x)\\ m\equiv a \pmod q}} 1 = \frac{2\sqrt{x}}{q} + O((x/q)^{1/3}\log x), \end{align*} $$
which holds uniformly for $x\geqslant 3$, $1\leqslant q\leqslant x^{1/4}/(\log x)^{3/2}$ and $1\leqslant a\leqslant q$, where the implied constant is absolute. The special case $S(x; q, q)$ confirms a recent numerical test of Heyman [‘Cardinality of a floor function set’, Integers19 (2019), Article no. A67].
We prove uniform bounds for the Petersson norm of the cuspidal part of the theta series. This gives an improved asymptotic formula for the number of representations by a quadratic form. As an application, we show that every integer
$n \neq 0,4,7 \,(\textrm{mod}\ 8)$
is represented as
$n= x_1^2 + x_2^2 + x_3^3$
for integers
$x_1,x_2,x_3$
such that the product
$x_1x_2x_3$
has at most 72 prime divisors.
In 1946, Erdős and Niven proved that no two partial sums of the harmonic series can be equal. We present a generalisation of the Erdős–Niven theorem by showing that no two partial sums of the series
$\sum _{k=0}^\infty {1}/{(a+bk)}$
can be equal, where a and b are positive integers. The proof of our result uses analytic and p-adic methods.
Using a recent breakthrough of Smith [18], we improve the results of Fouvry and Klüners [4, 5] on the solubility of the negative Pell equation. Let
$\mathcal {D}$
denote the set of positive squarefree integers having no prime factors congruent to
$3$
modulo
$4$
. Stevenhagen [19] conjectured that the density of d in
$\mathcal {D}$
such that the negative Pell equation
$x^2-dy^2=-1$
is solvable with
$x, y \in \mathbb {Z}$
is
$58.1\%$
, to the nearest tenth of a percent. By studying the distribution of the
$8$
-rank of narrow class groups
$\operatorname {\mathrm {Cl}}^+(d)$
of
$\mathbb {Q}(\sqrt {d})$
, we prove that the infimum of this density is at least
$53.8\%$
.
There has been recent interest in a hybrid form of the celebrated conjectures of Hardy–Littlewood and of Chowla. We prove that for any $k,\ell \ge 1$ and distinct integers $h_2,\ldots ,h_k,a_1,\ldots ,a_\ell $, we have:
for all except $o(H)$ values of $h_1\leq H$, so long as $H\geq (\log X)^{\ell +\varepsilon }$. This improves on the range $H\ge (\log X)^{\psi (X)}$, $\psi (X)\to \infty $, obtained in previous work of the first author. Our results also generalise from the Möbius function $\mu $ to arbitrary (non-pretentious) multiplicative functions.
We investigate, for given positive integers a and b, the least positive integer
$c=c(a,b)$
such that the quotient
$\varphi (c!\kern-1.2pt)/\varphi (a!\kern-1.2pt)\varphi (b!\kern-1.2pt)$
is an integer. We derive results on the limit of
$c(a,b)/(a+b)$
as a and b tend to infinity and show that
$c(a,b)>a+b$
for all pairs of positive integers
$(a,b)$
, with the exception of a set of density zero.
We consider a concrete family of
$2$
-towers
$(\mathbb {Q}(x_n))_n$
of totally real algebraic numbers for which we prove that, for each
$n$
,
$\mathbb {Z}[x_n]$
is the ring of integers of
$\mathbb {Q}(x_n)$
if and only if the constant term of the minimal polynomial of
$x_n$
is square-free. We apply our characterization to produce new examples of monogenic number fields, which can be of arbitrary large degree under the ABC-Conjecture.
Using a method due to Rieger [‘Remark on a paper of Stux concerning squarefree numbers in non-linear sequences’, Pacific J. Math.78(1) (1978), 241–242], we prove that the Piatetski-Shapiro sequence defined by
$\{\lfloor n^c \rfloor : n\in \mathbb {N}\}$
contains infinitely many consecutive square-free integers whenever
$1<c<3/2$
.
We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over
${\mathbb{Q}}$
in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.