Related papers: Triple Correlations of Multiplicative Functions

In this paper we consider the approximation of a function by its interpolating multilinear spline and the approximation of its derivatives by the derivatives of the corresponding spline. We derive formulas for the uniform approximation…

We introduce a general result relating "short averages" of a multiplicative function to "long averages" which are well understood. This result has several consequences. First, for the M\"obius function we show that there are cancellations…

We study the arithmetic function sopfr$(n)$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$. In particular we obtain the asymptotic formula $$ \sum_{n \leq x} \rm{sopfr}(n) \sim \frac{\pi^2}{12}…

Let $k$ be an integer greater than or equal $4$. We show that if a multiplicative function $f$ satisfies \[ f(x_1^2 + x_2^2 + \dots + x_k^2) = f(x_1)^2 + f(x_2)^2 + \dots + f(x_k)^2 \] for all positive integers $x_i$'s, then $f$ is the…

We consider a multiple arithmetical sum involving the Moebius function which despite its elementary appearance is in fact of a highly intriguing nature. We establish an asymptotic formula for the quadruple case that raises the first…

Let $f(n)$ denote a multiplicative function with range $\{-1,0,1\}$, and let $F(x) = \sum_{n\leq x} f(n)$. Then $F(x)/\sqrt{x} = a\sqrt{x} + b + E(x)$, where $a$ and $b$ are constants and $E(x)$ is an error term that either tends to $0$ in…

We present a class of multiplicative functions $f:\mathbb{N}\to\mathbb{C}$ with bounded partial sums. The novelty here is that our functions do not need to have modulus bounded by $1$. The key feature is that they pretend to be the constant…

In this paper we obtain a series of asymptotic formulae in the sum--product phenomena over the prime field $\mathbf{F}_p$. In the proofs we use usual incidence theorems in $\mathbf{F}_p$, as well as the growth result in ${\rm SL}_2…

We find the exact solutions of the $\Phi_{2}^{3}$ finite matrix model (Grosse-Wulkenhaar model). In the $\Phi_{2}^{3}$ finite matrix model, multipoint correlation functions are expressed as $G_{|a_{1}^{1}\ldots…

In this paper, we establish the following two congruences: \begin{gather*} \sum_{k=0}^{(p+1)/2}(3k-1)\frac{\left(-\frac{1}{2}\right)_k^2\left(\frac{1}{2}\right)_k4^k}{k!^3}\equiv…

We provide examples of multiplicative functions $f$ supported on the $k$-free integers such that at primes $f(p)=\pm 1$ and such that the partial sums of $f$ up to $x$ are $o(x^{1/k})$. Further, if we assume the Generalized Riemann…

We develop an approach to study character sums, weighted by a multiplicative function $f:\mathbb{F}_q[t]\to S^1$, of the form \begin{equation} \sum_{G\in \mathcal{M}_N}f(G)\chi(G)\xi(G), \end{equation} where $\chi$ is a Dirichlet character…

Let $ a_1(x)p_1(x)^n + \cdots + a_k(x)p_k(x)^n $ as well as $ b_1(x)q_1(x)^m + \cdots + b_l(x) q_l(x)^m $ be two polynomial power sums where the complex polynomials $ p_i(x) $ and $ q_j(x) $ are all non-constant. Then in the present paper…

Let $f: {\mathbb R}\to {\mathbb R}$ be a measurable function satisfying \begin{equation*} f(x+1)=f(x), \qquad \int_0^1 f(x)\, dx=0, \qquad \int_0^1 f^2(x)\, dx<\infty. \end{equation*} The asymptotic properties of series $\sum c_k f(kx)$…

Let $g_0,\dots,g_k: {\bf N} \to {\bf D}$ be $1$-bounded multiplicative functions, and let $h_0,\dots,h_k \in {\bf Z}$ be shifts. We consider correlation sequences $f: {\bf N} \to {\bf Z}$ of the form $$ f(a):= \widetilde{\lim}_{m \to…

Fix $k \ge 3$. If a multiplicative function $f$ satisfies \[ f(x_1+x_2+\dots+x_k) = f(x_1) + f(x_2) + \dots + f(x_k) \] for arbitrary positive triangular numbers $x_1, x_2, \dots, x_k$, then $f$ is the identity function. This extends Chung…

We establish an asymptotic formula for the logarithmic mean value of a 1-bounded multiplicative function that is sharp in many cases of interest. We derive from it a variety of applications, making progress on several old problems. As a…

We consider the asymptotic expansion of the functional series \[S_{\mu}^\pm(a;\lambda)=\sum_{n=0}^\infty \frac{(\pm 1)^n e^{-\lambda n}}{(n^2+a^2)^\mu}\] for $\lambda>0$ and $\mu\geq0$ as $|a|\to \infty$ in the sector $|\arg\,a|<\pi/2$. The…

Some identities are presented that generalize the formula x^3 = 3x floor(x floor(x)) - 3 floor(x) floor(x floor(x)) + floor(x)^3 + 3 frac(x) frac(x floor(x)) + frac(x)^3 to a representation of the product x_0x_1 ... x_{n-1}.

We study the existence of various sign and value patterns in sequences defined by multiplicative functions or related objects. For any set $A$ whose indicator function is 'approximately multiplicative' and uniformly distributed on short…