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We study the asymptotic behavior of the sequence $ \{\Omega (n) \}_{ n \in \mathbb {N} } $ from a dynamical point of view, where $ \Omega (n) $ denotes the number of prime factors of $ n $ counted with multiplicity. First, we show that for any non-atomic ergodic system $(X, \mathcal {B}, \mu , T)$, the operators $T^{\Omega (n)}: \mathcal {B} \to L^1(\mu )$ have the strong sweeping-out property. In particular, this implies that the pointwise ergodic theorem does not hold along $\Omega (n)$. Second, we show that the behaviors of $\Omega (n)$ captured by the prime number theorem and Erdős–Kac theorem are disjoint, in the sense that their dynamical correlations tend to zero.
For a positive integer $r\geq 2$, a natural number n is r-free if there is no prime p such that $p^r\mid n$. Asymptotic formulae for the distribution of r-free integers in the floor function set $S(x):=\{\lfloor x/ n \rfloor :1\leq n\leq x\}$ are derived. The first formula uses an estimate for elements of $S(x)$ belonging to arithmetic progressions. The other, more refined, formula makes use of an exponent pair and the Riemann hypothesis.
Let
$[t]$
be the integral part of the real number t and let
$\mathbb {1}_{{\mathbb P}}$
be the characteristic function of the primes. Denote by
$\pi _{\mathcal {S}}(x)$
the number of primes in the floor function set
$\mathcal {S}(x) := \{[{x}/{n}] : 1\leqslant n\leqslant x\}$
and by
$S_{\mathbb {1}_{{\mathbb P}}}(x)$
the number of primes in the sequence
$\{[{x}/{n}]\}_{n\geqslant 1}$
. Improving a result of Heyman [‘Primes in floor function sets’, Integers22 (2022), Article no. A59], we show
for
$x\to \infty $
, where
$C_{\mathbb {1}_{{\mathbb P}}} := \sum _{p} {1}/{p(p+1)}$
,
$c>0$
is a positive constant and
$\varepsilon $
is an arbitrarily small positive number.
For a fixed integer h, the standard orthogonality relations for Ramanujan sums
$c_r(n)$
give an asymptotic formula for the shifted convolution
$\sum _{n\le N} c_q(n)c_r(n+h)$
. We prove a generalised formula for affine convolutions
$\sum _{n\le N} c_q(n)c_r(kn+h)$
. This allows us to study affine convolutions
$\sum _{n\le N} f(n)g(kn+h)$
of arithmetical functions
$f,g$
admitting a suitable Ramanujan–Fourier expansion. As an application, we give a heuristic justification of the Hardy–Littlewood conjectural asymptotic formula for counting Sophie Germain primes.
We investigate the leading digit distribution of the kth largest prime factor of n (for each fixed $k=1,2,3,\dots $) as well as the sum of all prime factors of n. In each case, we find that the leading digits are distributed according to Benford’s law. Moreover, Benford behavior emerges simultaneously with equidistribution in arithmetic progressions uniformly to small moduli.
satisfying a familiar functional equation involving the gamma function $\Gamma (s)$. Two general identities are established. The first involves the modified Bessel function $K_{\mu }(z)$, and can be thought of as a ‘modular’ or ‘theta’ relation wherein modified Bessel functions, instead of exponential functions, appear. Appearing in the second identity are $K_{\mu }(z)$, the Bessel functions of imaginary argument $I_{\mu }(z)$, and ordinary hypergeometric functions ${_2F_1}(a,b;c;z)$. Although certain special cases appear in the literature, the general identities are new. The arithmetical functions appearing in the identities include Ramanujan’s arithmetical function $\tau (n)$, the number of representations of n as a sum of k squares $r_k(n)$, and primitive Dirichlet characters $\chi (n)$.
Let
$\mathcal {A}$
be the set of all integers of the form
$\gcd (n, F_n)$
, where n is a positive integer and
$F_n$
denotes the nth Fibonacci number. Leonetti and Sanna proved that
$\mathcal {A}$
has natural density equal to zero, and asked for a more precise upper bound. We prove that
for all sufficiently large x. In fact, we prove that a similar bound also holds when the sequence of Fibonacci numbers is replaced by a general nondegenerate Lucas sequence.
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.