We show that the exponent of distribution of the ternary divisor function $d_{3}$ in arithmetic progressions to prime moduli is at least $1/2+1/46$, improving results of Friedlander–Iwaniec and Heath-Brown. Furthermore, when averaging over a fixed residue class, we prove that this exponent is increased to $1/2+1/34$.

We determine a lower gap property for the growth of an unbounded $\mathbb{Z}$-valued $k$-regular sequence. In particular, if $f:\mathbb{N}\to \mathbb{Z}$ is an unbounded $k$-regular sequence, we show that there is a constant $c>0$ such that $|f(n)|>c\log n$ infinitely often. We end our paper by answering a question of Borwein, Choi and Coons on the sums of completely multiplicative automatic functions.

Let $\nu _{f}(n)$ be the $n\mathrm{th}$ normalized Fourier coefficient of a Hecke–Maass cusp form $f$ for ${\rm SL }(2,\mathbb{Z})$ and let $\alpha $ be a real number. We prove strong oscillations of the argument of $\nu _{f}(n)\mu (n) \exp (2\pi i n \alpha )$ as $n$ takes consecutive integral values.

A natural number $n$ is called abundant if the sum of the proper divisors of $n$ exceeds $n$. For example, $12$ is abundant, since $1+ 2+ 3+ 4+ 6= 16$. In 1929, Bessel-Hagen asked whether or not the set of abundant numbers possesses an asymptotic density. In other words, if $A(x)$ denotes the count of abundant numbers belonging to the interval $[1, x] $, does $A(x)/ x$ tend to a limit? Four years later, Davenport answered Bessel-Hagen’s question in the affirmative. Calling this density $\Delta $, it is now known that $0. 24761\lt \Delta \lt 0. 24766$, so that just under one in four numbers are abundant. We show that $A(x)- \Delta x\lt x/ \mathrm{exp} (\mathop{(\log x)}\nolimits ^{1/ 3} )$ for all large $x$. We also study the behavior of the corresponding error term for the count of so-called $\alpha $-abundant numbers.

Let $\sigma (n)= {\mathop{\sum }\nolimits}_{d\mid n} d$ be the usual sum-of-divisors function. In 1933, Davenport showed that $n/ \sigma (n)$ possesses a continuous distribution function. In other words, the limit $D(u): = \lim _{x\rightarrow \infty }(1/ x){\mathop{\sum }\nolimits}_{n\leq x, n/ \sigma (n)\leq u} 1$ exists for all $u\in [0, 1] $ and varies continuously with $u$. We study the behaviour of the sums ${\mathop{\sum }\nolimits}_{n\leq x, n/ \sigma (n)\leq u} f(n)$ for certain complex-valued multiplicative functions $f$. Our results cover many of the more frequently encountered functions, including $\varphi (n)$, $\tau (n)$ and $\mu (n)$. They also apply to the representation function for sums of two squares, yielding the following analogue of Davenport’s result: for all $u\in [0, 1] $, the limit

$$\begin{eqnarray*}\tilde {D} (u): = \lim _{R\rightarrow \infty }\frac{1}{\pi R} \# \biggl\{ (x, y)\in { \mathbb{Z} }^{2} : 0\lt {x}^{2} + {y}^{2} \leq R\text{ and } \frac{{x}^{2} + {y}^{2} }{\sigma ({x}^{2} + {y}^{2} )} \leq u\biggr\}\end{eqnarray*}$$

$\tilde {D} (u)$

$[0, 1] $

Let $k\geq 2$ and $\Pi (n)= { \mathop{\prod }\nolimits}_{i= 1}^{k} ({a}_{i} n+ {b}_{i} )$ for some integers ${a}_{i} , {b}_{i} $ ($1\leq i\leq k$). Suppose that $\Pi (n)$ has no fixed prime divisors. Weighted sieves have shown for infinitely many integers $n$ that the number of prime factors $\Omega (\Pi (n))$ of $\Pi (n)$ is at most ${r}_{k} $, for some integer ${r}_{k} $ depending only on $k$. We use a new kind of weighted sieve to improve the possible values of ${r}_{k} $ when $k\geq 4$.

This paper establishes a version of the Chebotarev density theorem in which number fields are replaced by 3-manifolds.

We investigate sums of mixed powers involving two squares and three biquadrates. In particular, subject to the truth of the Generalised Riemann Hypothesis and the Elliott–Halberstam conjecture, we show that all large natural numbers $n$ with $8\nmid n$, $n\not\equiv 2~(\text{mod} ~3)$ and $n\not\equiv 14~(\text{mod} ~16)$ are the sum of two squares and three biquadrates.

Building on the concept of pretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.

Let $L(s, E)= {\mathop{\sum }\nolimits}_{n\geq 1} {a}_{n} {n}^{- s} $ be the $L$-series corresponding to an elliptic curve $E$ defined over $ \mathbb{Q} $ and $\mathbf{u} = \mathop{\{ {u}_{m} \} }\nolimits_{m\geq 0} $ be a nondegenerate binary recurrence sequence. We prove that if ${ \mathcal{M} }_{E} $ is the set of $n$ such that ${a}_{n} \not = 0$ and ${ \mathcal{N} }_{E} $ is the subset of $n\in { \mathcal{M} }_{E} $ such that $\vert {a}_{n} \vert = \vert {u}_{m} \vert $ holds with some integer $m\geq 0$, then ${ \mathcal{N} }_{E} $ is of density $0$ as a subset of ${ \mathcal{M} }_{E} $.

We prove that when $(a, m)= 1$ and $a$ is a quadratic residue $\hspace{0.167em} \mathrm{mod} \hspace{0.167em} m$, there are infinitely many Carmichael numbers in the arithmetic progression $a\hspace{0.167em} \mathrm{mod} \hspace{0.167em} m$. Indeed the number of them up to $x$ is at least ${x}^{1/ 5} $ when $x$ is large enough (depending on $m$).

Let $K$ be a number field of degree $n$, and let $d_K$ be its discriminant. Then, under the Artin conjecture, the generalized Riemann hypothesis and a certain zero-density hypothesis, we show that the upper and lower bounds of the logarithmic derivatives of Artin $L$-functions attached to $K$ at $s=1$ are $\log \log |d_K|$ and $-(n-1) \log \log |d_K|$, respectively. Unconditionally, we show that there are infinitely many number fields with the extreme logarithmic derivatives; they are families of number fields whose Galois closures have the Galois group $C_n$ for $n=2,3,4,6$, $D_n$ for $n=3,4,5$, $S_4$ or $A_5$.

For any positive integer $n$, let $f(n)$ denote the number of solutions to the Diophantine equation

$$\begin{eqnarray*}\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\end{eqnarray*}$$

$x, y, z$

$f(n)\gt 0$

$n\geq 2$

$f(n)$

$f(p)$

$n$

$p$

$$\begin{eqnarray*}N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\ll \displaystyle \sum _{p\leq N}f(p)\ll N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\log \log N.\end{eqnarray*}$$

Two arithmetic functions $f$ and $g$ form a Möbius pair if $f(n)= {\mathop{\sum }\nolimits}_{d\mid n} g(d)$ for all natural numbers $n$. In that case, $g$ can be expressed in terms of $f$ by the familiar Möbius inversion formula of elementary number theory. In a previous paper, the first-named author showed that if the members $f$ and $g$ of a Möbius pair are both finitely supported, then both functions vanish identically. Here we prove two significantly stronger versions of this uncertainty principle. A corollary is that in a nonzero Möbius pair, one cannot have both ${\mathop{\sum }\nolimits}_{f(n)\not = 0} 1/ n\lt \infty $ and ${\mathop{\sum }\nolimits}_{g(n)\not = 0} 1/ n\lt \infty $.

We study the distribution of the size of Selmer groups arising from a 2-isogeny and its dual 2-isogeny for quadratic twists of elliptic curves with a non-trivial $2$-torsion point over $\mathbb {Q}$. This complements the work [Xiong and Zaharescu, Distribution of Selmer groups of quadratic twists of a family of elliptic curves. Adv. Math.219 (2008), 523–553] which studied the same subject for elliptic curves with full 2-torsions over $\mathbb {Q}$ and generalizes [Feng and Xiong, On Selmer groups and Tate–Shafarevich groups for elliptic curves $y^2=x^3-n^3$. Mathematika 58 (2012), 236–274.] for the special elliptic curves $y^2=x^3-n^3$. It is shown that the 2-ranks of these groups all follow the same distribution and in particular, the mean value is $\sqrt {\frac {1}{2}\log \log X}$ for square-free positive integers $n \le X$ as $X \to \infty $.

Given an elliptic curve E and a positive integer N, we consider the problem of counting the number of primes p for which the reduction of E modulo p possesses exactly N points over 𝔽p. On average (over a family of elliptic curves), we show bounds that are significantly better than what is trivially obtained by the Hasse bound. Under some additional hypotheses, including a conjecture concerning the short-interval distribution of primes in arithmetic progressions, we obtain an asymptotic formula for the average.

In [Gorodnik and Nevo, Counting lattice points, J. Reine Angew. Math. 663 (2012), 127–176] an effective solution of the lattice point counting problem in general domains in semisimple S-algebraic groups and affine symmetric varieties was established. The method relies on the mean ergodic theorem for the action of G on G/Γ, and implies uniformity in counting over families of lattice subgroups admitting a uniform spectral gap. In the present paper we extend some methods developed in [Nevo and Sarnak, Prime and almost prime integral points on principal homogeneous spaces, Acta Math. 205 (2010), 361–402] and use them to establish several useful consequences of this property, including:

1. (1) effective upper bounds on lifting for solutions of congruences in affine homogeneous varieties;

2. (2) effective upper bounds on the number of integral points on general subvarieties of semisimple group varieties;

3. (3) effective lower bounds on the number of almost prime points on symmetric varieties;

4. (4) effective upper bounds on almost prime solutions of congruences in homogeneous varieties.

We study the differentiability of the limiting distribution function associated to the normalized Euler function defined on the shifted primes.

For a class of multiplicative integer-valued functions $f$ the distribution of the sequence $f(n)$ in restricted residue classes modulo $N$ is studied. We consider a property weaker than weak uniform distribution and study it for polynomial-like multiplicative functions, in particular for $\varphi (n)$ and $\sigma (n)$.