arXiv subjects

Keiju Sono

Publications and source records attributed to Keiju Sono.

5 recordsLinked to original sources

arXiv2024

Zeros of Dirichlet $L$-functions on the critical line

In this paper, we estimate the proportion of zeros of Dirichlet $L$-functions on the critical line. Using Feng's mollifier and an asymptotic formula for the mean square of Dirichlet $L$-functions, we prove that averaged over primitive characters and conductors, at least 61.07 % of zeros of Dirichlet $L$-functions are on the critical line, and at least 60.44 % of zeros are simple and on the critical line. These results improve the work of Conrey, Iwaniec and Soundararajan.

math.NT↗

arXiv2024

An explicit lower bound for large gaps between some consecutive primes

Let $p_{n}$ denote the $n$th prime and for any fixed positive integer $k$ and $X\geq 2$, put \[ G_{k}(X):=\max _{p _{n+k}\leq X} \min \{ p_{n+1}-p_{n}, \ldots , p_{n+k}-p_{n+k-1} \}. \] Ford, Maynard and Tao proved that there exists an effective absolute constant $c_{LG}>0$ such that \[ G_{k}(X)\geq \frac{c_{LG}}{k^{2}}\frac{\log X \log \log X \log \log \log \log X}{\log \log \log X} \] holds for any sufficiently large $X$. The main purpose of this paper is to determine the constant $c_{LG}$ above. We see that $c_{LG}$ is determined by several factors related to analytic number theory, for example, the ratio of integrals of functions in the multidimensional sieve of Maynard, the distribution of primes in arithmetic progressions to large moduli, and the coefficient of upper bound sieve of Selberg. We prove that the above inequality is valid at least for $c_{LG}\approx 2.0\times 10^{-17}$.

math.NT↗

arXiv2022

Perfectly packing a square by squares of sidelength $f(n)^{-t}$

In this paper, we prove that for any $1/2<t<1$, there exists a positive integer $N_{0}$ depending on $t$ such that for any $n_{0}\geq N_{0}$, squares of sidelength $f(n)^{-t}$ for $n\geq n_{0}$ can be packed with disjoint interiors into a square of area $\sum_{n=n_{0}}^{\infty}f(n)^{-2t}$, if the function $f$ satisfies some suitable conditions. The main theorem (Theorem 1.1) is a generalization of Tao's theorem, which argued the case $f(n)=n$. As corollaries, we prove that there are such packings of squares when $f(n)$ represents the $n$th element of either an arithmetic progression or the set of prime numbers. In these cases, we give effective lower bounds for $N_{0}$ with respect to $t$. Furthermore, we consider the case that $f(n)$ represents the $n$th element of the set of twin primes and prove that squares of sidelength $f(n)^{-t}$ for $n\geq n_{0}$ can be packed with disjoint interiors into a slightly larger square than theoretically expected.

math.MG↗

arXiv2021

The second moment of Dirichlet twists of a $\textrm{GL}_{4}$ automorphic $L$-function

In this paper, we give an asymptotic formula for the second moment of Dirichlet twists of an automorphic $L$-function $L(s, π)$ on the critical line averaged over characters and conductors, where $π$ denotes an irreducible tempered cuspidal automorphic representation of $\textrm{GL}_{4}$ with unitary central character. We give some hybrid bound for the error term with respect to the size of conductors of Dirichlet characters and that of the automorphic representation.

math.NT↗

arXiv2018

Small gaps between the set of products of at most two primes

In this paper, we apply the method of Maynard and Tao to the set of products of two distinct primes (E2-numbers). We obtain several results on the distribution of E2-numbers and primes. Among others, the result of Goldston, Pintz, Yildirim and Graham on small gaps between m consecutive E2-numbers is improved.

math.NT↗