We prove some conditions on the existence of natural boundaries of Dirichlet series. We show that generically the presumed boundary is the natural one. We also give an application of natural boundaries in determining asymptotic results.

Let $\mathcal{M}$ denote the class of multiplicative functions with values in the unit disk, and, for $x\geq 1$, $y\geq 1$, let $S(x,y)$ designate the set of $y$-friable positive integers not exceeding $x$. We provide, as $x$ and $y$ tend to infinity in prescribed ranges, upper bounds for exponential sums of the form $$E_f(x,y;\vartheta):=\sum_{n\in S(x,y)}f(n)\hbox{\rm e}^{2\pi i n\vartheta}$$ whenever $f\in \mathcal{M}$ and $\vartheta$ is a rational number with denominator not exceeding a fixed power of $\log x$.

Let $q$ be a power of an odd prime $p$ and let ${k}$ be a finite field with $q$ elements. Our main result is: If $q \notin \{3,9,5,13,17,25,29\},$ every polynomial $P\in{k}[t]$ of degree $\geq 269$ is a strict sum of 11 biquadrates. We first decompose $P$ as a strict mixed sum of biquadrates.

Let $M(x)$ be the summatory function of the Möbius function and $R(x)$ be the remainder term for the number of squarefree integers up to $x$. In this paper, we prove the explicit bounds $|M(x)|<x/4345$ for $x\ge 2160535$ and $|R(x)|\le 0.02767\sqrt x$ for $x\ge 438653$. These bounds are considerably better than preceding bounds of the same type and can be used to improve Schoenfeld type estimates.

For $d \leqq n^{1/8-\varepsilon}$, we determine in a large range of integers $N_1,\ldots,N_d$ the asymptotic number of partitions of $n$ with exactly $N_r$ parts congruent to r modulo $d$ for $1 \le r \le d$. In the second part of the paper we derive many results on the distributions of the parts in residue classes. In particular we obtain for $1 \leqq a < b \leqq d \leqq n^{1/8-\varepsilon}$, an asymptotic formula for the number of partitions of $n$ in which there are more parts $\equiv a (mod d)$ than parts $\equiv b (mod d)$.

We survey properties of the Ankeny-Onishi sieve function and establish inequalities for $j_\kappa(\kappa)$ and for $1-j_\kappa(u)$ for $u \to \infty$.

We give a new proof of a formula of Bloch for a special value of a certain Eisenstein series of weight one with an additive character

We apply recent bounds of the author for generalized Smirnov statistics to the distribution of integers whose prime factors satisfy certain systems of inequalities.

Let $A$ be a finite subset of an abelian group $G$. For every element $b_i$ of the sumset $2A = \{b_0, b_1, ...,b_{|2A|-1}\}$ we denote by $D_i = \{a-a': a, a'\in A; a + a' = b_i\}$ and $r_i = |\{(a,a'): a + a' = b_i; a, a' \in A \}|$. After an eventual reordering of $2A$, we may assume that $r_0 \geq r_1 \geq ... \geq r_{|2A|-1}.$ For every $1 \le s \le |2A|$ we define $R_s(A)=|D_0 \cup D_1 \cup ... \cup D_{s-1}|$ and $R_s(k) = \max \{R_s(A): A \subseteq G, |A| = k\}.$ Bourgain and Katz and Tao obtained an estimate of $R_s(k)$ assuming $s$ being of order $k$. In this note we find the {\it exact value } of $R_s(k)$ in cases $s = 1$, $s = 2$ and $s = 3$. The case $s = 3$ appeared to be not simple. The structure of {\it extremal sets} led us to sets isomorphic to planar sets having a rather unexpected form of a perfect hexagon. The proof suggests the way of dealing with the general case $s \ge 4$.

We prove an optimal `level of distribution' result sequences of integers of the type $X^2+Y^{2r}$.

We present three remarks on Goldbach's problem. First we suggest a refinement of Hardy and Littlewood's conjecture for the number of representations of $2n$ as the sum of two primes positing an estimate with a very small error term. Next we show that if a strong form of Goldbach's conjecture is true then every even integer is the sum of two primes from a rather sparse set of primes. Finally we show that an averaged strong form of Goldbach's conjecture is equivalent to the Generalized Riemann Hypothesis; as well as a similar equivalence to estimates for the number of ways of writing integers as the sum of $k$ primes.

In a given abelian group, let $A$ and $B$ be two finite subsets satisfying the small sumset condition $|A+B|\le K|A|$. We consider the problem of estimating how large $|A-B|$ can be in terms of $|A|$ and $K$ and the one of estimating the ratio $|X-B|/|X|$ when $X$ runs over all the non-empty subsets of $A$.

The Nyman-Beurling criterion states that the Riemann hypothesis is equivalent to the density in $L^2(0,+\infty;t^{-2} dt)$ of a certain space. We introduce an orthonormal family in $L^2(0,+\infty;t^{-2} dt)$, study the space generated by this family and reformulate the Nyman-Beurling criterion using this orthonormal basis. We then study three approximations that could lead to a proof of this criterion.

Let $N_+(X)$ denote the number of distinct real quadratic fields $\mathbb{Q}(\sqrt{d})$ with $d\leq X$ for which $3|h(\mathbb{Q}(\sqrt{d}))$. Define $N_-(X)$ similarly for $\mathbb{Q}(\sqrt{-d})$. It is shown that $N_+(X), N_-(X)\gg X^{9/10-\varepsilon}$ for any $\varepsilon>0$. This improves results of Byeon and Koh [2] and of Soundararajan [7], which had exponent $7/8-\varepsilon$.

Bombieri and Pila gave sharp estimates for the number of integer points $(m,n)$ on a given arc of a curve $y = F(x)$, enlarged by some size parameter $M$, for algebraic curves and for transcendental analytic curves. The transcendental case involves the maximum number of intersections of the given arc by algebraic curves of bounded degree. We obtain an analogous result for functions $F(x)$ of some class $C^k$ that satisfy certain differential inequalities that control the intersection number. We allow enlargement by different size parameters $M$ and $N$ in the $x$- and $y$-directions, and we also estimate integer points close to the curve, with $$\left|n - NF ( {m\over M} )| \leq \delta,$$ for $\delta$ sufficiently small in terms of $M$ and $N$. As an appendix we obtain a determinant mean value theorem which is a quantitative version of a linear independence theorem of Pólya.