• C G KARTHICK BABU

      Articles written in Proceedings – Mathematical Sciences

    • Primes in Beatty sequence

      C G KARTHICK BABU

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      For a polynomial $g(x)$ of deg $k \geq 2$ with integer coefficients and positive integer leading coefficient, we prove an upper bound for the least prime $p$ such that $g(p)$ is in non-homogeneous Beatty sequence $\{\lfloor \alpha n+ \beta\rfloor : n=1, 2, 3, \ldots\}$, where $\alpha , \beta\in R$ with $\alpha > 1$ is irrational and we prove an asymptotic formula for the number of primes $p$ such that $g(p) = \lfloor \alpha n+\beta\rfloor$. Next, we obtain an asymptotic formula for the number of primes $p$ of the form $p = \lfloor \alpha n+\beta \rfloor$ which also satisfies $p \equiv f (\mod d)$, where $f, d$ are integers with $1 \leq f$ < $d$ and $(f, d) = 1$.

    • Note on a problem of Ramanujan

      C G KARTHICK BABU USHA K SANGALE

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      For fixed positive real numbers $\omega , \omega '$, it is known that the number of lattice points $(u,v), u\ge 0, v \ge 0$ satisfying $0 \le u \omega + v\omega ' \le \eta $ is given by $\frac{1}{2}\big (\frac{\eta ^{2}}{\omega \omega ^{'}}+\frac{\eta }{\omega } +\frac{\eta }{\omega ^{'}}\big )+ O_{\varepsilon }(\eta ^{1-\frac{1}{\alpha _{0}} +\varepsilon })$, where $\alpha _0 \ge 1$ is a constant. In this paper, we explicitly compute $\alpha _0$ for certain values of $\omega /\omega '$. In particular, in Ramanujan's case (i.e., when $\omega = \log 2$ and $\omega ' = \log 3$), we show that $\alpha _0 = 2^{18}\log 3$ is admissible. This improves an earlier result of the paper (Ramachandra K, Sankaranarayanan A and Srinivas K, Hardy Ramanujan J. 19 (1996) 2--56), where it was shown that $\alpha _0 = 2^{40}\log 3$ holds.

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