相关论文: The average order of the M\"{o}bius function for B…

For $p_1,...,p_n>0$, let $\mathbb E=\{z\in\mathbb C^n:\sum_{j=1}^n|z_j|^{2p_j}<1\}$ be a complex ellipsoid. We present effective formulas for the generalized M\"obius and Green functions $m_{\mathbb E}(A,\cdot)$, $g_{\mathbb E}(A,\cdot)$ in…

We show that the $L^1$ norm of an exponential sum of length $X$ and with coefficients equal to the Liouville or M\"{o}bius function is at least $\gg_{\varepsilon} X^{1/4 - \varepsilon}$ for any given $\varepsilon$. For the Liouville…

For any $m = 3 \left( 2n + 1 \right) with \ n \in \mathbb{N^*} ,$ the prime counting function $\pi(m) = 4 + \left \vert A_4(m) \right \vert + 2 \left \vert A_6(m) \right \vert $ where $A_6(m) $ and $ A_4(m) $ are the sets of Twin Primes and…

In this paper, we present a function-sharing criterion for the normality of meromorphic functions. Let $f$ be a meromorphic function in the unit disc $\mathbb{D}\subset \mathbb{C}$, $\psi_1$, $\psi_2$, and $\psi_3$ be three meromorphic…

This paper considers the query complexity of the functions in the family F_{N,M} of N-variable Boolean functions with onset size M, i.e., the number of inputs for which the function value is 1, where 1<= M <= 2^{N}/2 is assumed without loss…

Benjamini, Kalai and Schramm (2001) showed that weighted majority functions of $n$ independent unbiased bits are uniformly stable under noise: when each bit is flipped with probability $\epsilon$, the probability $p_\epsilon$ that the…

For a fixed odd prime $\ell$, we define a variant of the classical M\"{o}bius function on the poset of isomorphism classes of finite abelian $\ell$-groups, then we prove an analog of Hall's theorem on the vanishing of the M\"{o}bius…

We construct iteratively a sequence of numbers k_{n} and Beurling functions A_{n} converging pointwise to -1 in [0,1]. We prove results which seems to suggest that each A_{n} is equal to a well known approximating sequence of functions…

It is a well known result that for $\beta\in(1,\frac{1+\sqrt{5}}{2})$ and $x\in(0,\frac{1}{\beta-1})$ there exists uncountably many $(\epsilon_{i})_{i=1}^{\infty}\in {0,1}^{\mathbb{N}}$ such that…

We examine oscillations in a number of sums of arithmetic functions involving $\Omega(n)$, the total number of prime factors of $n$, and $\omega(n)$, the number of distinct prime factors of $n$. In particular, we examine oscillations in…

We consider the problem of computing a function of $n$ variables using noisy queries, where each query is incorrect with some fixed and known probability $p \in (0,1/2)$. Specifically, we consider the computation of the $\mathsf{OR}$…

Consider averages along the prime integers $ \mathbb P $ given by \begin{equation*} \mathcal{A}_N f (x) = N ^{-1} \sum_{ p \in \mathbb P \;:\; p\leq N} (\log p) f (x-p). \end{equation*} These averages satisfy a uniform scale-free $ \ell…

We prove that the average size of the squares of differences between consecutive primes less than $x$ is $O(x^{0.23+\varepsilon})$ for any fixed $\varepsilon>0$. This improves on a result of Peck, who gave bound $O(x^{0.25+\varepsilon})$ in…

This paper describes recent advances in the combinatorial method for computing $\pi(x)$, the number of primes $\leq x$. In particular, the memory usage has been reduced by a factor of $\log x$, and modifications for shared- and…

We examine correlations of the M\"obius function over $\mathbb{F}_q[t]$ with linear or quadratic phases, that is, averages of the form \begin{equation} \label{eq:average} \frac{1}{q^n}\sum_{\text{deg }f<n} \mu(f)\chi(Q(f)) \end{equation}…

The probability P(alpha, N) that search algorithms for random Satisfiability problems successfully find a solution is studied as a function of the ratio alpha of constraints per variable and the number N of variables. P is shown to be…

We establish quantitative bounds on the $U^k[N]$ Gowers norms of the M\"obius function $\mu$ and the von Mangoldt function $\Lambda$ for all $k$, with error terms of shape $O((\log\log N)^{-c})$. As a consequence, we obtain quantitative…

Research on the distribution of prime numbers has revealed a dual character: deterministic in definition yet exhibiting statistical behavior reminiscent of random processes. In this paper we show that it is possible to use an image-focused…

In Machine Learning, a benchmark refers to an ensemble of datasets associated with one or multiple metrics together with a way to aggregate different systems performances. They are instrumental in (i) assessing the progress of new methods…

Let $\lambda(n)$ and $\mu(n)$ denote the Liouville function and the M\"obius function, respectively. In this study, relationships between the values of $\lambda(n)$ and $\lambda(n+h)$ up to $n\leq10^8$ for $1\leq h\leq1,000$ are explored.…