Related papers: Notes de lecture de l'article "Partial sums of the…
We show that, for the M\"obius function $\mu(n)$, we have $$ \sum_{x < n\leq x+x^{\theta}}\mu(n)=o(x^{\theta}) $$ for any $\theta>0.55$. This improves on a result of Ramachandra from 1976, which is valid for $\theta>7/12$. Ramachandra's…
A multiplicative function $f$ is said to be resembling the M\"{o}bius function if $f$ is supported on the square-free integers, and $f(p)=\pm 1$ for each prime $p$. We prove $O$- and $\Omega$-results for the summatory function $\sum_{n\leq…
We present an elementary problem on analytic polynomials with coefficients $\pm 1$ or in $\{0,\pm 1 \}$ which implies Riemann hypothesis. It is turns out that this problem is a particular case of the weak form of a flat polynomials problem…
In this paper, we study the sum of additive characters over finite fields, with a focus on those of specified \(\mathbb{F}_q\)-Order. We establish a general formula for these character sums, providing an additive analogue to classical…
Inspired by the work of Bourgain and Garaev (2013), we provide new bounds for certain weighted bilinear Kloosterman sums in polynomial rings over a finite field. As an application, we build upon and extend some results of Sawin and…
Unary theta functions have played a significant role in the theory of holomorphic modular forms and modular $L$-functions. A partial theta functions is defined analogously, but the sum is over part of the integer lattice. Such sums fail to…
We provide some upper bounds for the Mertens function ($M(n)$: the cumulative sum of the M$\ddot{\mathrm{o}}$bius function) by an approach of statistical mechanics, in which the M$\ddot{\mathrm{o}}$bius function is taken as a particular…
The motivation of the note is to obtain a H\"{o}rmander-type $L^2$ estimate for $\bar\partial$ equation. The feature of the new estimate is that the constant is independent of the weight function. Moreover, our estimate can be used for…
In this short research note, we aim to establish an interesting extension of a summation due to Ramanujan.The result is derived with the help of an extension of Gauss's summation theorem available in the literature.
In this paper we derive some identities and inequalities on the M\"obius mu function. Our main tool is phi functions for intervals of positive integers and their unions.
This paper reviews and extends some recent results on the multivariate fractional Brownian motion (mfBm) and its increment process. A characterization of the mfBm through its covariance function is obtained. Similarly, the correlation and…
In their recent work, the authors (2016) have combined classical ideas of A. G. Postnikov (1956) and N. M. Korobov (1974) to derive improved bounds on short character sums for certain nonprincipal characters with powerful moduli. In the…
We study the growth rate of the summatory function of the M\"obius function in the context of an algebraic curve over a finite field. Our work shows a strong resemblance to its number field counterpart, which was proved by Ng in 2004. We…
Let $F$ be a number field, $k$ a positive integer. In this paper, we define the Mobius and Liouville functions of order $k$ in $F$. We give a formula about the partial sums of them by using elementary number theory and complex analysis.…
We revisit several hybrid multiplicative-to-additive type functions from a recent preprint article. These functions, $g(n)$ with Dirichlet generating function (DGF) $\zeta(s)^{-1} (1+P(s))^{-1}$ for $\Re(s) > 1$ where $P(s) = \sum_p p^{-s}$…
This note proposes an improved estimate of the coefficient t(n) of the discriminant modular form using elementary method. It improves a well known estimate of the tau function t(n) by Deligne.
This paper is aimed to prove a quantitative estimate (in terms of the modulus of continuity) for the convergence in the nonlinear version of Korovkin's theorem for sequences of weakly nonlinear and monotone operators defined on spaces of…
We study the double sum $S_\varepsilon(X)$$=$$\sum_{\substack{d,e\le X}}\frac{\mu(d)\mu(e)}{[d,e]^{1+\varepsilon}}$, which converges even in the case $\varepsilon=0$, where $\mu$ denotes the M\"obius function and $[d,e]$ is the least common…
We obtain new bounds for short sums of isotypic trace functions associated to some sheaf modulo prime $p$ of bounded conductor, twisted by the Mobius function and also by the generalised divisor function. These trace functions include…
This paper studies the M\"obius function and related questions about the finiteness of the poset of submodules of semisimple and general modules. We show how to calculate the M\"obius function for semisimple modules based on endomorphism…