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T. Makoto Minamide

Publications and source records attributed to T. Makoto Minamide.

3 recordsLinked to original sources
arXiv2026

On a Turán's theorem for small primes

Denote by $ω(n)$ the number of distinct prime divisors of the natural number $n$. In 2007, Granville and Soundararajan gave a quite new method to compute the higher moments $\sum_{n\leq x}(ω(n)-\log\log x)^{k}$, for a wide range of integers $k\geq 2$. In this notes, we shall apply the method for $ω_{z}(n)$ which denotes the number of distinct prime divisors $\leq z$ of $n$. Especially, for odd integers $k\geq 3$, we lead asymptotic formulas for $\sum_{n\leq x}(ω_{z}(n)-\log\log z)^{k}$, under certain restrictions on $k$, $z$, and $x$. Also, we refer to a framework of the approach.

math.NT↗
arXiv2026

On a Turán's theorem for arithmetic progressions

Let $m\geq 1$ be a fixed integer, $a$ an integer satisfying $(a,m)=1$, and $z\geq 1$ a real parameter. Denote by $ω_{z}(n;m,a)$ the number of distinct prime divisors $p$ of $n$ satisfying $p\equiv a\, (m)$ and $p\leq z$. We study an asymptotic behaviour of $\sum_{n\leq x}\left(ω_{z}(n;m,a)-\frac{1}{φ(m)}\log\log z\right)^{k}$ as $x\to\infty$ for a wide range of positive integer $k\geq 2$, where $φ(\cdot)$ is the Euler function. Following a method of Granville and Soundararajan we lead an asymptotic formula for the above. Also, we investigate $\sum_{n\leq x}\left(ω(n;m,a)-\frac{1}{φ(m)}\log\log x\right)^{k}$, where $ω(n;m,a)$ denotes the number of distinct prime divisors $p$ of $n$ such that $p\equiv a\, (m)$.

math.NT↗
arXiv2024

On the sum of $Δ_{k}(n)$ in the Piltz divisor problem for $k=3$ and $k=4$

Let $Δ_{k}(x)$ be the error term in the classical asymptotic formula for the sum $\sum_{n\leq x}d_{k}(n)$, where $d_{k}(n)$ is the number of ways $n$ can be written as a product of $k$ factors. We study the analytic properties of the Dirichlet series $\sum_{n=1}^{\infty}Δ_{k}(n)n^{-s}$ and use Perron's formula to estimate the sums $\sum_{n\leq x}Δ_{3}(n)$ and $\sum_{n\leq x}Δ_{4}(n)$ for large $x>0$.

math.NT↗