Related papers: Shifted convolution of divisor function $d_3$ and …

We obtain several asymptotic formulas for the sum of the divisor function $\tau(n)$ with $n \le x$ in an arithmetic progressions $n \equiv a \pmod q$ on average over $a$ from a set of several consecutive elements from set of reduced…

We obtain a new bound on the second moment of modified shifted convolutions of the generalized 3-fold divisor function, and show that, for applications, the modified version is sufficient.

Ramanujan sums have attracted significant attention in both mathematical and engineering disciplines due to their diverse applications. In this paper, we introduce an algebraic generalization of Ramanujan sums, derived through polynomial…

In this paper, we introduce the little $\mu$-function, which is obtained as a degenerate limit of the generalized $\mu$-function. We derive the little $\mu$-function as the image of the $q$-Borel summation of a divergent solution to the…

We study the average value of the divisor function $\tau(n)$ for $n\le x$ with $n \equiv a \bmod q$. The divisor function is known to be evenly distributed over arithmetic progressions for all $q$ that are a little smaller than $x^{2/3}$.…

In the sixth chapter of his notebooks Ramanujan introduced a method of summing divergent series which assigns to the series the value of the associated Euler-MacLaurin constant that arises by applying the Euler-MacLaurin summation formula…

In this paper, a generalization of Ramanujan's cubic transformation, in the form of an inequality, is proved for zero-balanced Gaussian hypergeometric function $F(a,b;a+b;x)$, $a,b>0$.

$R_{D^{(*)}}$ is the ratio of branching ratio $\overline{B} \rightarrow D^{(*)}\tau\overline{\nu}_{\tau}$ to $\overline{B} \rightarrow D^{(*)}l\overline{\nu}_{l}~(l=e,~\mu)$. There is a gap of 2$\sigma_{exp}$ or more between its…

We introduce and study a number of new spaces of ultradifferentiable functions and ultradistributions and we apply our results to the study of the convolution of ultradistributions. The spaces of convolutors…

For an arithmetical function $f$, its Ramanujan expansion is a series expansion in the form $f(n)=\sum\limits_{k=1}^{\infty}a(k) c_k(n)$ where $a(k)$ are complex numbers and $c_k(n):= \sum\limits_{\substack{m=1\\(m, k)=1}}^{k}e^{\frac{2\pi…

In this paper, we introduce a new generating function called $d$-polynomial for the dimensions of $\tau$-tilting modules over a given finite dimensional algebra. Firstly, we study basic properties of $d$-polynomials and show that it can be…

Let $K$ be a number field. This paper considers arithmetic functions over $K$, that are, complex valued functions on the set of nonzero integral ideals in $K$. Firstly we generalize some basic results on arithmetic functions. Next we define…

We use the weighted version of the arithmetic-mean-geometric-mean inequality to motivate new results about Zaremba's function, $z(n) = \sum_{d|n} \frac{\log d}{d}$. We investigate record-setting values for $z(n)$ and the related function…

The measurement of $R_D$ ($R_{D^*}$), the ratio of the branching fraction of $\overline{B} \to D \tau \bar{\nu}_\tau (\overline{B} \to D^* \tau \bar{\nu}_\tau)$ to that of $\overline{B} \to D l \bar{\nu}_l (\overline{B} \to D^* l…

A representation of divisor function $\tau(n)\equiv \sigma_{0}(n)$ by means of logarithmic residue of a function of complex variable is suggested. This representation may be useful theoretical instrument for further investigations of…

We give a simple proof and a multivariable generalization of an identity due to E. Alkan concerning a weighted average of the Ramanujan sums. We deduce identities for other weighted averages of the Ramanujan sums with weights concerning…

We investigate the relations between the K_{l3} and tau --> K pi nu_tau decays using the meson dominance approach. First, the experimental branching fractions (BF) for K^-_{e3} and K^0_{e3} are used to fix two normalization constants…

We define a new kind of classical digamma function, and establish its some fundamental identities. Then we apply the formulas obtained, and extend tools developed by Flajolet and Salvy to study more general Euler type sums. The main results…

In this note we explore the relationship between the operation of convolution of functions and the Eulerian integrals. This approach allow us to obtain some expressions for the convolution of a certain class of functions in terms of the…

We first study the mean value of certain restricted divisor sums involving the Chowla-Walum sums, improving in particular a recent estimate given by Iannucci. The aim of the second part of this work is the generalization of the previous…