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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.
For each positive integer n, let
$U(\mathbf {Z}/n\mathbf {Z})$
denote the group of units modulo n, which has order
$\phi (n)$
(Euler’s function) and exponent
$\lambda (n)$
(Carmichael’s function). The ratio
$\phi (n)/\lambda (n)$
is always an integer, and a prime p divides this ratio precisely when the (unique) Sylow p-subgroup of
$U(\mathbf {Z}/n\mathbf {Z})$
is noncyclic. Write W(n) for the number of such primes p. Banks, Luca, and Shparlinski showed that for certain constants
$C_1, C_2>0$
,
Under sufficiently strong assumptions about the first prime in an arithmetic progression, we prove that the number of Carmichael numbers up to
$X$
is
$\gg X^{1-R}$
, where
$R=(2+o(1))\log \log \log \log X/\text{log}\log \log X$
. This is close to Pomerance’s conjectured density of
$X^{1-R}$
with
$R=(1+o(1))\log \log \log X/\text{log}\log X$
.
We determine, up to multiplicative constants, the number of integers
$n\leq x$
that have a divisor in
$(y,2y]$
and no prime factor
$\leq w$
. Our estimate is uniform in
$x,y,w$
. We apply this to determine the order of the number of distinct integers in the
$N\times N$
multiplication table, which are free of prime factors
$\leq w$
, and the number of distinct fractions of the form
$(a_{1}a_{2})/(b_{1}b_{2})$
with
$1\leq a_{1}\leq b_{1}\leq N$
and
$1\leq a_{2}\leq b_{2}\leq N$
.
We determine the order of magnitude of
$\mathbb{E}|\sum _{n\leqslant x}f(n)|^{2q}$
, where
$f(n)$
is a Steinhaus or Rademacher random multiplicative function, and
$0\leqslant q\leqslant 1$
. In the Steinhaus case, this is equivalent to determining the order of
$\lim _{T\rightarrow \infty }\frac{1}{T}\int _{0}^{T}|\sum _{n\leqslant x}n^{-it}|^{2q}\,dt$
.
In particular, we find that
$\mathbb{E}|\sum _{n\leqslant x}f(n)|\asymp \sqrt{x}/(\log \log x)^{1/4}$
. This proves a conjecture of Helson that one should have better than squareroot cancellation in the first moment and disproves counter-conjectures of various other authors. We deduce some consequences for the distribution and large deviations of
$\sum _{n\leqslant x}f(n)$
.
The proofs develop a connection between
$\mathbb{E}|\sum _{n\leqslant x}f(n)|^{2q}$
and the
$q$
th moment of a critical, approximately Gaussian, multiplicative chaos and then establish the required estimates for that. We include some general introductory discussion about critical multiplicative chaos to help readers unfamiliar with that area.
Let
$K=\mathbb{Q}(\unicode[STIX]{x1D714})$
with
$\unicode[STIX]{x1D714}$
the root of a degree
$n$
monic irreducible polynomial
$f\in \mathbb{Z}[X]$
. We show that the degree
$n$
polynomial
$N(\sum _{i=1}^{n-k}x_{i}\unicode[STIX]{x1D714}^{i-1})$
in
$n-k$
variables takes the expected asymptotic number of prime values if
$n\geqslant 4k$
. In the special case
$K=\mathbb{Q}(\sqrt[n]{\unicode[STIX]{x1D703}})$
, we show that
$N(\sum _{i=1}^{n-k}x_{i}\sqrt[n]{\unicode[STIX]{x1D703}^{i-1}})$
takes infinitely many prime values, provided
$n\geqslant 22k/7$
.
Our proof relies on using suitable ‘Type I’ and ‘Type II’ estimates in Harman’s sieve, which are established in a similar overall manner to the previous work of Friedlander and Iwaniec on prime values of
$X^{2}+Y^{4}$
and of Heath-Brown on
$X^{3}+2Y^{3}$
. Our proof ultimately relies on employing explicit elementary estimates from the geometry of numbers and algebraic geometry to control the number of highly skewed lattices appearing in our final estimates.
We investigate the density of square-free values of polynomials with large coefficients over the rational function field 𝔽q[t]. Some interesting questions answered as special cases of our results include the density of square-free polynomials in short intervals, and an asymptotic for the number of representations of a large polynomial N as a sum of a k-th power of a small polynomial and a square-free polynomial.
We show that for all large enough x the interval [x, x + x1/2 log1.39x] contains numbers with a prime factor p > x18/19. Our work builds on the previous works of Heath–Brown and Jia (1998) and Jia and Liu (2000) concerning the same problem for the longer intervals [x, x + x1/2 + ϵ]. We also incorporate some ideas from Harman’s book Prime-detecting sieves (2007). The main new ingredient that we use is the iterative argument of Matomäki and Radziwiłł (2016) for bounding Dirichlet polynomial mean values, which is applied to obtain Type II information. This allows us to take shorter intervals than in the above-mentioned previous works. We have also had to develop ideas to avoid losing any powers of log x when applying Harman’s sieve method.
In this note we examine Littlewood’s proof of the prime number theorem. We show that this can be extended to provide an equivalence between the prime number theorem and the nonvanishing of Riemann’s zeta-function on the one-line. Our approach goes through the theory of almost periodic functions and is self-contained.
Given a positive integer n let ω (n) denote the number of distinct prime factors of n, and let a be fixed positive integer. Extending work of Kubilius, we develop a bivariate probabilistic model to study the joint distribution of the deterministic vectors (ω(n), ω(n + a)), with n ≤ x as x → ∞, where n and n + a belong to a subset of ℕ with suitable properties. We thus establish a quantitative version of a bivariate analogue of the Erdős–Kac theorem on proper subsets of ℕ.
We give three applications of this result. First, if y = x0(1) is not too small then we prove (in a quantitative way) that the y-truncated Möbius function μy has small binary autocorrelations. This gives a new proof of a result due to Daboussi and Sarkőzy. Second, if μ(n; u) :=e(uω(n)), where u ∈ ℝ then we show that μ(.; u) also has small binary autocorrelations whenever u = o(1) and
$u\sqrt {\mathop {\log }\nolimits_2 x} \to \infty$
, as x → ∞. These can be viewed as partial results in the direction of a conjecture of Chowla on binary correlations of the Möbius function.
Our final application is related to a problem of Erdős and Mirsky on the number of consecutive integers less than x with the same number of divisors. If
$y = x^{{1 \over \beta }}$
, where β = β(x) satisfies certain mild growth conditions, we prove a lower bound for the number of consecutive integers n ≤ x that have the same number of y-smooth divisors. Our bound matches the order of magnitude of the one conjectured for the original Erdős-Mirsky problem.
We define a natural topology on the collection of (equivalence classes up to scaling of) locally finite measures on a homogeneous space and prove that in this topology, pushforwards of certain infinite-volume orbits equidistribute in the ambient space. As an application of our results we prove an asymptotic formula for the number of integral points in a ball on some varieties as the radius goes to infinity.
Let
$p$
be a prime. If an integer
$g$
generates a subgroup of index
$t$
in
$(\mathbb{Z}/p\mathbb{Z})^{\ast },$
then we say that
$g$
is a
$t$
-near primitive root modulo
$p$
. We point out the easy result that each coprime residue class contains a subset of primes
$p$
of positive natural density which do not have
$g$
as a
$t$
-near primitive root and we prove a more difficult variant.
We prove that for every
$m\geq 0$
there exists an
$\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}(m)>0$
such that if
$0<\unicode[STIX]{x1D706}<\unicode[STIX]{x1D700}$
and
$x$
is sufficiently large in terms of
$m$
and
$\unicode[STIX]{x1D706}$
, then
The value of
$\unicode[STIX]{x1D700}(m)$
and the dependence of the implicit constant on
$\unicode[STIX]{x1D706}$
and
$m$
may be made explicit. This is an improvement of the author’s previous result. Moreover, we will show that a careful investigation of the proof, apart from some slight changes, can lead to analogous estimates when allowing the parameters
$m$
and
$\unicode[STIX]{x1D706}$
to vary as functions of
$x$
or replacing the set
$\mathbb{P}$
of all primes by primes belonging to certain specific subsets.
We construct a shifted version of the Turán sieve method developed by R. Murty and the second author and apply it to counting problems on tournaments. More precisely, we obtain upper bounds for the number of tournaments which contain a fixed number of restricted
$r$
-cycles. These are the first concrete results which count the number of cycles over “all tournaments”.
We discuss the generalizations of the concept of Chebyshev’s bias from two perspectives. First, we give a general framework for the study of prime number races and Chebyshev’s bias attached to general L-functions satisfying natural analytic hypotheses. This extends the cases previously considered by several authors and involving, among others, Dirichlet L-functions and Hasse–Weil L-functions of elliptic curves over Q. This also applies to new Chebyshev’s bias phenomena that were beyond the reach of the previously known cases. In addition, we weaken the required hypotheses such as GRH or linear independence properties of zeros of L-functions. In particular, we establish the existence of the logarithmic density of the set
$ \{x \ge 2:\sum\nolimits_{p \le x} {\lambda _f}(p) \ge 0\}$
for coefficients (λf(p)) of general L-functions conditionally on a much weaker hypothesis than was previously known.
We study the average error term in the usual approximation to the number of y-friable integers congruent to a modulo q, where a ≠ 0 is a fixed integer. We show that in the range exp{(log log x)5/3+ɛ} ⩽ y ⩽ x and on average over q ⩽ x/M with M → ∞ of moderate size, this average error term is asymptotic to −|a| Ψ(x/|a|, y)/2x. Previous results of this sort were obtained by the second author for reasonably dense sequences, however the sequence of y-friable integers studied in the current paper is thin, and required the use of different techniques, which are specific to friable integers.
We study the analogue of the Bombieri–Vinogradov theorem for
$\operatorname{SL}_{m}(\mathbb{Z})$
Hecke–Maass form
$F(z)$
. In particular, for
$\operatorname{SL}_{2}(\mathbb{Z})$
holomorphic or Maass Hecke eigenforms, symmetric-square lifts of holomorphic Hecke eigenforms on
$\operatorname{SL}_{2}(\mathbb{Z})$
, and
$\operatorname{SL}_{3}(\mathbb{Z})$
Maass Hecke eigenforms under the Ramanujan conjecture, the levels of distribution are all equal to
$1/2,$
which is as strong as the Bombieri–Vinogradov theorem. As an application, we study an automorphic version of Titchmarch’s divisor problem; namely for
$a\neq 0,$
where
$\unicode[STIX]{x1D70C}(n)$
are Fourier coefficients
$\unicode[STIX]{x1D706}_{f}(n)$
of a holomorphic Hecke eigenform
$f$
for
$\operatorname{SL}_{2}(\mathbb{Z})$
or Fourier coefficients
$A_{F}(n,1)$
of its symmetric-square lift
$F$
. Further, as a consequence, we get an asymptotic formula
where
$E_{1}(a)$
is a constant depending on
$a$
. Moreover, we also consider the asymptotic orthogonality of the Möbius function against the arithmetic function
$\unicode[STIX]{x1D70C}(n)d(n-a)$
.