arXiv subjects
Publications and source records attributed to Alexandru Zaharescu.
The sum $S(h,k):=\sum_{j=1}^{k-1}(-1)^{j+1+[hj/k]}$ appears in the modular transformation formulae of the classical theta function $\vartheta_3(z)$. The double sum $S(k) := \sum_{h=1}^{k-1}S(h,k)$ has a remarkable distribution of values. Although properties for $S(k)$ and a related sum can be established, several interesting conjectures are open.
Let $\mathcal{M}(n)$ be the subgroup of $GL(n,\mathbb{Z})$ generated by the particular involutions that are identical to the identity, except for a single line where $-1$ and $+1$ alternate. We study the properties of $\mathcal{M}(n)$, and then find several notable characteristics of the unions of trajectories obtained by iteratively applying a fixed sequence of such involutions to elements from $\mathbb{Z}^n$.
We consider the set ${\mathscr S}_Q$ of Farey fractions $d/b$ of order $Q$ with the property that there exists a matrix $\left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) \in \operatorname{SL}(2,{\mathbb Z})$ of trace at most $Q$, with positive entries and $a\ge \max\{ b,c\}$. For every $Q\ge 3$, the set ${\mathscr S}_Q \cup \{ 0\}$ is shown to define a unimodular partition of the interval $[0,1]$. We also prove that the elements of ${\mathscr S}_Q$ are asymptotically distributed with respect to the probability measure with density $(1/(1+x) -1/(2+x) )/\log (4/3) $ and that the sequence of sets $({\mathscr S}_Q)_Q$ has a limiting gap distribution as $Q\rightarrow \infty$.
Despite their simple definition as $\mathfrak{q}_p(b):=\frac{b^{p-1}-1}{p} \pmod p$, for $0\le b \le p^2-1$ and $\gcd(b,p)=1$, and their regular arrangement in a $p\times(p-1)$ Fermat quotient matrix $\mathtt{FQM}(p)$ of integers from $[0,p-1]$, Fermat quotients modulo $p$ are well known for their overall lack of regularity. Here, we discuss this contrasting effect by proving that, on the one hand, any line of the matrix behaves like an analogue of a randomly distributed sequence of numbers, and on the other hand, the spatial statistics of distances on regular $N$-patterns confirm the natural expectations.
Let $\mathfrak{p}_{\mathbb{P}_r}(n)$ denote the number of partitions of $n$ into $r$-full primes. We use the Hardy-Littlewood circle method to find the asymptotic of $\mathfrak{p}_{\mathbb{P}_r}(n)$ as $n \to \infty$. This extends previous results in the literature of partitions into primes. We also show an analogue result involving convolutions of von Mangoldt functions and the zeros of the Riemann zeta-function. To handle the resulting non-principal major arcs we introduce the definition of strange functions and pseudo-differentiability.
We provide an asymptotic formula for the average value of the sequence A351830: $a_{n} = |P_{n} - y^{2}_{n}|$ for $1 \leq n \leq x$, where $P_{n}$ is the $n$-th square pyramidal number and $y^{2}_{n}$ is the closest square to $P_{n}$. Moreover, we supply asymptotic formulas for the $k$-th moment of the same sequence, for any fixed natural number $k$.
Let $h$ be a fixed non-zero integer. For every $t\in \mathbb{R}_+$ and every prime $p$, consider the angles between rays from an observer located at the point $(-tJ_p^2,0)$ on the real axis towards the set of all integral solutions $(x,y)$ of the equation $y^{-1}-x^{-1}\equiv h \pmod{p}$ in the square $[-J_p,J_p]^2$, where $J_p=(p-1)/2$. We prove the existence of the limiting gap distribution for this set of angles as $p\rightarrow \infty$, providing explicit formulas for the corresponding density function, which turns out to be independent of $h$.
The iterated composition of two operators, both of which are involutions and translation invariant, partitions the set of lattice points in the plane into an infinite sequence of discrete parabolas. Each such parabola contains an associated stairway-like path connecting certain points on it, induced by the alternating application of the aforementioned operators. Any two lattice points in the plane can be connected by paths along the square grid composed of steps either on these stairways or towards taxicab neighbors. This leads to the notion of the parabolic-taxicab distance between two lattice points, obtained as the minimum number of steps of this kind needed to reach one point from the other. In this paper, we describe patterns generated by points on paths of bounded parabolic-taxicab length and provide a complete description of the balls centered at the origin. In particular, we prove an earlier conjecture on the area of these balls.
We examine exponential sums of the form $\sum_{n \le X} w(n) e^{2πiαn^k}$, for $k=1,2$, where $α$ satisfies a generalized Diophantine approximation and where $w$ are different arithmetic functions that might be multiplicative, additive, or neither. A strategy is shown on how to bound these sums for a wide class of functions $w$ belonging within the same ecosystem. Using this new technology we are able to improve current results on minor arcs that have recently appeared in the literature of the Hardy-Littlewood circle method. Lastly, we show how a bound on $\sum_{n \le X} |μ(n)| e^{2πiαn}$ can be used to study partitions asymptotics over squarefree parts and explain their connection to the zeros of the Riemann zeta-function.
For certain families of $L$-functions, we prove that if each $L$-function in the family has only real zeros in a fixed yet arbitrarily small neighborhood of $s=1$, then one may considerably improve upon the known results on Landau-Siegel zeros. Sarnak and the third author proved a similar result under much more restrictive hypotheses.
Let $ν_q(n)$ be the p-adic valuation of $n$. We show that the power series with coefficients $ν_q(n)$, respectively $ν_p(n)(\mathrm{ mod\;} k)$, are non-holonomic and not algebraic in characteristic 0. We find infinitely many rational numbers and infinitely many algebraic irrational numbers for which the values of these series are transcendental. We apply these results to some $p$-automatic sequences, one of them being the period-doubling sequence.
In light of Kim's conjecture on regular polytopes of dimension four, which is a generalization of Waring's problem, we establish asymptotic formulas for representing any sufficiently large integer as a sum of numbers in the form of those regular 4-polytopes. Moreover, we are able to obtain a more general result of the asymptotics for any degree-four polynomial $f$ satisfying $f(0)=0$ and $f(1)=1$.
The dynamical system generated by the iterated calculation of the high order gaps between neighboring terms of a sequence of natural numbers is remarkable and only incidentally characterized at the boundary by the notable Proth-Glibreath Conjecture for prime numbers. We introduce a natural extension of the original triangular arrangement, obtaining a growing hexagonal covering of the plane. This is just the base level of what further becomes an endless discrete helicoidal surface. % Although the repeated calculation of higher-order gaps causes the numbers that generate the helicoidal surface to decrease, there is no guarantee, and most often it does not even happen, that the levels of the helicoid have any regularity, at least at the bottom levels. However, we prove that there exists a large and nontrivial class of sequences with the property that their helicoids have all levels coinciding with their base levels. This class includes in particular many ultimately binary sequences with a special header. % For almost all of these sequences, we additionally show that although the patterns generated by them seem to fall somewhere between ordered and disordered, exhibiting fractal-like and random qualities at the same time, the distribution of zero and non-zero numbers at the base level has uniformity characteristics. Thus, we prove that a multitude of straight lines that traverse the patterns encounter zero and non-zero numbers in almost equal proportions.
Let $p$ be an odd prime, and let $ω$ be a primitive $p$th root of unity. In this paper, we introduce a metric on the cyclotomic field $K=\mathbb{Q}(ω)$. We prove that this metric has several remarkable properties, such as invariance under the action of the Galois group. Furthermore, we show that points in the ring of integers $\mathcal{O}_K$ behave in a highly uniform way under this metric. More specifically, we prove that for a certain hypercube in $\mathcal{O}_K$ centered at the origin, almost all pairs of points in the cube are almost equi-distanced from each other, when $p$ and $N$ are large enough. When suitably normalized, this distance is exactly $1/\sqrt{6}$.
For any odd prime $p$ and any integer $N\ge 0$, let $\mathcal{V}(p,N)$ be the set of vertices of the cyclotomic box $\mathscr{B} = \mathscr{B}(p,N)$ of edge size $2N$ and centered at the origin $O$ of the ring of integers $\mathbb{Z}[ω]$ of the cyclotomic field $\mathbb{Q}(ω)$, where $ω=\exp\big(\frac{2πi}{p}\big)$. Cyclotomic boxes represented as sets of points in the complex plane prove to have counter-intuitive super-regularity properties that are known to occur in high dimensional real hypercubes. Employing the naturally induced Euclidean-trace metric for distance measurement and letting the prime $p$ tend to infinity, we prove the following results. 1. Almost all triangles with vertices in $\mathcal{V}(p,N)$ are almost equilateral. 2. Almost all angles $\angle VOA$, where $V$ is in $\mathcal{V}(p,N)$, $O$ is the origin, which coincides with the center of $\mathscr{B}(p,N)$, and $A$ is fixed anywhere in $\mathscr{B}(p,N)$, are right angles. 3. Almost all pyramids with base on $\mathcal{V}(p,N)$ and the apex fixed anywhere in $\mathscr{B}(p,N)$ are super-regular, meaning that the base has all edges and diagonals almost equal and the lateral faces are nearly isosceles triangles, each nearly equal to the others.
The ternary Goldbach conjecture states that every odd number $m \geqslant 7$ can be written as the sum of three primes. We construct a set of primes $\mathbb{P}$ defined by an expanding system of admissible congruences such that almost all primes are not in $\mathbb{P}$ and still, the ternary Goldbach conjecture holds true with primes restricted to $\mathbb{P}$.
Given a half-integral weight holomorphic Kohnen newform $f$ on $Γ_0(4)$, we prove an asymptotic formula for large primes $p$ with power saving error term for \begin{equation*} \sideset{}{^*} \sum_{χ\hspace{-0.15cm} \pmod{p}} | L(1/2,f,χ) |^2. \end{equation*} Our result is unconditional, it does not rely on the Ramanujan--Petersson conjecture for the form $f$. This gives a very sharp Lindelöf on average result for Dirichlet series attached to Hecke eigenforms without an Euler product. The Lindelöf hypothesis for such series was originally conjectured by Hoffstein. There are two main inputs. The first is a careful spectral analysis of a highly unbalanced shifted convolution problem involving the Fourier coefficients of half-integral weight forms. The second input is a bound for sums of products of Salié sums in the Polya--Vinogradov range. Half--integrality is fully exploited to establish such an estimate. We use the closed form evaluation of the Salié sum to relate our problem to the sequence $αn^2 \pmod{1}$. Our treatment of this sequence is inspired by work of Rudnick--Sarnak and the second author on the local spacings of $αn^2$ modulo one.
We study a class of approximations to the Riemann zeta function introduced earlier by the second author on the basis of Euler product. This allows us to justify Euler Product Sieve for generation of prime numbers. Also we show that Bounded Riemann Hypothesis (stated in a paper by the fourth author) is equivalent to conjunction the Riemann Hypothesis + the simplicity of zeros. 16 pages.