Related papers: Arithmetic properties of the Ramanujan function

Let $\mathbb{A}=\mathbb{F}_{q}[T]$ be the polynomial ring over finite field $\mathbb{F}_{q}$, and $\mathbb{A}_{+}$ be the set of monic polynomials in $\mathbb{A}$. In this paper, we show that a large class of arithmetic functions in…

Ramanujan, in his famous first letter to Hardy, claimed a very precise estimate for the number of integers that can be written as a sum of two squares. Far less well-known is that he also made further claims of a similar nature for the…

After Landau, we define g(n) as the maximal order of a permutation of the symmetric group S(n) on n letters. We give several estimates of the largest prime divisor of g(n).

Let $b,n\in \mathbb{Z}$, $n\geq 1$, and ${\cal D}_1, \ldots, {\cal D}_{\tau(n)}$ be all positive divisors of $n$. For $1\leq l \leq \tau(n)$, define ${\cal C}_l:=\lbrace 1 \leqslant x\leqslant n \; : \; (x,n)={\cal D}_l\rbrace$. In this…

We prove that there are infinitely many $n$ such that $\omega(n+k) \ll \log k$ for all integers $k \ge 2$. This improves on a result of Tao-Ter\"{a}v\"{a}inen (2025), who has $O(k)$ in place of $O(\log k)$. As corollaries, we make progress…

Motivated by a question of V. Bergelson and F. K. Richter (2017), we obtain asymptotic formulas for the number of relatively prime tuples composed of positive integers $n\le N$ and integer parts of polynomials evaluated at $n$. The error…

Let P(n) denote the largest prime factor of $n \ge 2, P(1) = 1$, and let $$ \beta(n) = \sum_{p|n}p, \Beta(n) = \sum_{p^\alpha||n}\alpha p, \Beta_1(n) = \sum_{\p^\alpha||n}p^\alpha $$ denote "large" additive functions. A survey of results on…

Motivated by the recent work of Park on the analogue of the Ramanujan's function $k(\tau)=r(\tau)r^2(2\tau)$ for the Ramanujan's cubic continued fraction, where $r(\tau)$ is the Rogers-Ramanujan continued fraction, we use the methods of Lee…

The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that for all $x \geq R_n$ the interval $(x/2, x]$ contains at least $n$ primes. In this paper we undertake a study of the sequence $(\pi(R_n))_{n \in \mathbb{N}}$, which…

We explore the modularity of the continued fractions $I(\tau), J(\tau), T_1(\tau), T_2(\tau)$ and $U(\tau)=I(\tau)/J(\tau)$ of order $10$, where $I(\tau)$ and $J(\tau)$ are introduced by Rajkhowa and Saikia, which are special cases of…

In this paper, we consider the general divisor functions over Piatetski-Shapiro sequences. We can give some general results which contain some special divisor functions. Precisely, we extend the divisor problem over Piatetski-Shapiro…

We prove a remarkable formula of Ramanujan for the logarithmic derivative of the gamma function, which converges more rapidly than classical expansions, and which is stated without proof in Ramanujan's notebooks. The formula has a number of…

In this self-contained short note, we prove that {\it every arithmetic function} $F$ {\it has infinitely many Ramanujan coefficients} $G$ {\it giving an absolutely convergent Ramanujan expansion for $F$}. This is "coefficients'…

In this paper, we establish the irrationality of some open problems in mathematics based on using a recursive formula that generate the complete sequence of numbers. see [1] But before getting into that we begin with some Ramanujan notable…

In 1977, the first author observed a duality between the largest and smallest prime factors of integers, and established as a consequence some new results on the M\"obius function $\mu(n)$ using the Prime Number Theorem for Arithmetic…

We consider very general "random integers" and (attempt to) prove that many multiplicative and additive functions of such integers have limiting distributions. These integers include, for instance, the curvatures of Apollonian circle…

In this work we introduce interesting infinite series, related to Ramanujan-Soldner constant. Our method uses general properties of polynomials of binomial type and Lagrange inversion theorem. Also we study properties of the operator…

Let $P(m)$ denote the largest prime factor of an integer $m\geq 2$, and put $P(0)=P(1)=1$. For an integer $k\geq 2$, let $(F_{n}^{(k)})_{n\geq 2-k}$ be the $k-$generalized Fibonacci sequence which starts with $0,...,0,1$ ($k$ terms) and…

We derive an asymptotic formula for the divisor function $\tau(k)$ in an arithmetic progression $k\equiv a(\bmod \ q)$, uniformly for $q\leq X^{\Delta_{n,l}}$ with $(q,a)=1$. The parameter $\Delta_{n,l}$ is defined as $$…

Re presenting the traditional proof of Srinivasa Ramanujan's own favorite series for the reciprocal of $\pi$ :\begin{equation}\frac{1}{\pi} = \frac{\sqrt{8}}{9801} \sum_{n=0}^{+\infty} \frac{(4n)!}{(n!)^4} \frac{1103 + 26390n}{396^{4n}} \;…