We present a result on short intervals about the moments of the free path length of the linear trajectory of a billiard in the unit square with small triangular pockets of size $\varepsilon$ removed at the corners, in which the trajectory ends in a specified corner pocket.

En posant $\mathbf{e_n}(t) =\{t / n\}$ et $\mathcal{H}: = L^2(0,+\infty; t^{-2}dt)$, nous dèmontrons $\frac{6}{5} + O(n^{-1}) \leq n^2 \cdot \mathrm{dist}_{\mathcal{H}}^2 \bigl(\mathbf{e_n}, \mathrm{Vect}(\mathbf{e_1}, \dots, \mathbf{e_{n-1}})\bigr) \leq \frac{3}{4} \log n + O(1). $

With $\mathbf{e_n}(t) =\{t / n\}$ and $\mathcal{H}: = L^2(0,+\infty; t^{-2}dt)$, we prove $\frac{6}{5} + O(n^{-1}) \leq n^2 \cdot \mathrm{dist}_{\mathcal{H}}^2 \bigl (\mathbf{e_n}, \mathrm{Vect}(\mathbf{e_1}, \dots, \mathbf{e_{n-1}})\bigr) \leq \frac{3}{4} \log n + O(1). $

Let $p$ be a sufficiently large prime and $\mathcal{A}$ be a sum-free subset of $\mathbb{Z} / p\mathbb{Z}$; improving on a previous result of V. F. Lev, we show that if $|\mathcal{A}|=\mathrm{card}(\mathcal{A}) \gt 0.324 p$, then $\mathcal{A}$ is contained in a dilation of the interval $[|\mathcal{A}|, p-|\mathcal{A}|]$ (mod. $p)$.

Let ${\mathcal{Q}}({\mathcal{T}})$ denote the set of integers which are composed of prime factors from a given set of primes ${\mathcal{T}}$ only. Suppose that $\mathcal{A}+\mathcal{B} \subseteq \mathcal{Q}'(\mathcal{T})$, where $\mathcal{Q}(\mathcal{T})$ and $\mathcal{Q}'(\mathcal{T})$ differ at finitely many elements only. Also assume that $\sum_{p \leq x, p \in \mathcal{T}} \frac{\log p}{p} = \tau \log x + O(1)$. We prove that $\mathcal{A}(N) \mathcal{B}(N) = O(N(\log N)^{2 \tau})$ holds. In the case $\tau \geq \frac{1}{2}$ we give an example where both $\mathcal{A}(N)$ and $\mathcal{B}(N)$ are of order of magnitude $\frac{N^\frac{1}{2}}{(\log N)^{\frac{1}{4}}}$, which shows that this is close to best possible.

In the present work we prove a new estimate for $\Delta_\nu:=\liminf_{n \to \infty} \frac{(p_{n+\nu}-p_n)}{\log p_n}$, where $p_n$ denotes the $n$th prime. Combining our recent method which led to $\Delta_1=0$ with Maier's matrix method, we show that $\Delta_\nu\leq e^{-\gamma}(\sqrt{\nu}-1)^2$. We also extend the result to primes in arithmetic perogressions where the modulus can tend slowly to infinity as a function of $p_n$.

Dans la mèthode des diffèrences divisèes pour majorer le nombre de points entiers au voisinage d'une courbe, Filaseta et Trifonov (1996) ont introduit un argument de divisibilitè qui s'applique lorsque le voisinage considèrè est très fin. Nous approfondissons cet argument et l'adaptons au cas plus gènèral où le voisinage n'est plus nècessairement aussi fin. Le problème se complique alors par l'apparition des arcs majeurs, dèjà rencontrès par les auteurs dans un article prècèdent (1995).

Filaseta and Trifonov (1996) introduced a divisibility argument into the divided differences method for bounding the number of integer points in a narrow strip close to a curve; their result is useful when the strip is very narrow. We sharpen their argument, and extend it to allow broader strips. An immediate complication is the possible appearance of major arcs, which were first encountered in our earlier paper (1995).

We obtain the precise asymptotics of the series $$\sum_{k=1}^\infty \frac {d_k}k \mathtt{P}(|S_k| \ge \varepsilon k)$$ as $\varepsilon \downarrow 0$ where $S_k$ are partial sums of independent identically distributed random variables attracted to a stable law of index $\alpha \gt 1$.

In this paper, the first part of a larger work, we prove the spectral decomposition of $$ \int_{-\infty}^\infty|\zeta(\sigma+it)|^4g(t) dt \qquad(\tfrac{1}{2} < \sigma < 1 {\rm {fixed}}), $$ where $g(t)$ is a suitable weight function of fast decay. This is used to obtain estimates and omega results for the function \begin{align*} E_2(T,\sigma) &: =\int_0^T|\zeta(\sigma+it)|^4 dt - {\zeta^4(2\sigma)\over\zeta(4\sigma)}T -{\frac{T}{3-4\sigma}}{\left(\frac{T}{2\pi} \right)}^{2-4\sigma}{\zeta^4(2-2\sigma)\over\zeta(4-4\sigma)}\cr & \quad- T^{2-2\sigma}(a_0(\sigma) + a_1(\sigma)\log T + a_2(\sigma)\log^2T), \end{align*} the error term in the asymptotic formula for the fourth moment of $|\zeta(\sigma+it)|$.

Roughly speaking, we prove that the Hecke $L$-functions associated with the cusp forms of the Hecke groups $\mathsf{G}(\lambda)$ belong to the extended Selberg class, and for $\lambda \leq 2$ we characterize the Hecke $L$-functions belonging to the Selberg class.

We give an improved lower bound for $\max_{|T-t|\leq H} |\zeta(\tfrac{1}{2} + it)|$ when $2 \leq \alpha H \leq \log\log T - c$, $1 \leq \alpha \lt \pi$. Our theorem slightly refines the result in [11]. We also prove a theorem about an upper bound for the multiplicities of zeros of $\zeta(s)$ conditionally, assuming some lower bound for $\max_{|s - s_1| \leq \Delta} |\zeta(s)|$.

In a recent paper A. Sankaranarayanan and the author using a novel method prove a special case of a recent result of G. Bachmann on exponential sums with multiplicative coefficients. Here we apply this method to the case in hich the exponential sum is extended over smooth numbers only.

We characterize some non-negative multiplicative functions $f(n)$ such that $\lim_{x\rightarrow +\infty}\frac{1}{x}\sum_{{1\leq n\leq x }\atop {n\in A }} f(n)$ exists and is positive, but there exists a subset $A(f)$ of $N$ of density $1$ such that $\lim_{x\rightarrow +\infty }\frac{1}{x}\sum_{{1\leq n\leq x }\atop {n\in A(f)}} f(n)=0$. An application to the case of the Ramanujan $\tau$-function is provided.

We assume the Riemann Hypothesis (RH). It is classical that there is an absolute constant $C > 1$ such that $\psi(x)-x$ changes sign in every interval $[x, Cx]$ for $x\ge 1$. We prove that $\psi(x)-x$ changes sign in $[x,19x]$ for all $x\ge 1$, and also that for $x\ge x_0$, $\psi(x)-x$ changes sign in the interval $[x,Cx]$ where $C = 2.02$.

We discuss to what extent Euclid's elementary proof of the infinitude of primes can be modified so as to show infinitude of primes in arithmetic progressions (Dirichlet's theorem). Murty had shown earlier that such proofs can exist if and only if the residue class (mod $k$) has order 1 or 2. After reviewing this work, we consider generalizations of this question to algebraic number fields.

Cycle lengths of polynomial maps in one variable in rings of integers of cubic fields of negative discriminants are determined

The length $L(P)$ of a polynomial $P$ is the sum of the absolute values of the coefficients. For $P\in\mathbb{R}[x]$ the properties of $l(P)$ are studied, where $l(P)$ is the infimum of $L(PG)$ for $G$ running through monic polynomials over $\mathbb{R}$.

K. Yu [16] had generalized Mahler's classification to several variables. Here we will deal with generalizations of both the Mahler and Koksma classification. We will discuss the relationship between these classifications, formulate a metric theorem, and pose a number of open problems.

Let $\varphi$ denote Euler's totient function. A classical result of Schoenberg asserts that $G(t):=\mathrm{dens}\{n \ge 1 : \varphi(n)/n \le t\}$ is well-defined for every $t\in[0,1]$ and recent results of the second author show that the local behaviour of $G$ around any given $t$ may essentially be described in terms of the variations around $t = 1$. We provide, as $\varepsilon\to 0+$, an asymptotic expansion of $G(1-\varepsilon)$ according to negative powers of $\log(1/\varepsilon)$, together with an evaluation of the coefficients and an explicit bound for the remainder.

In 1873 Hermite proved the transcendence of the basis $e$ of the natural logarithm. In the more than 140 years since this paper was published, several quite different proofs were given (see Transcendental Numbers by N.I. Fel'dman and Yu.V. Nesterenko, Springer 1998). One of the best known versions goes back to Hurwitz, who used divisibility properties to show that a certain linear form doesn't vanish. Recently I generalized this method (see [20]), using some results on higher congruences going back to Dedekind and Hasse. We regained a former result of Carlson on linear independence of certain values of the function $$G(x) = \sum_{n=0}^\infty ~ \frac{x^n}{Q(1)\cdots Q(n)},$$ $Q$ a certain polynomial. Here I change the method used in [20] (see Chap. 5) to give a new proof of Galochkin's measure for the $\mathbb{Q}$--linear independence of these values.