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Yuta Suzuki

Publications and source records attributed to Yuta Suzuki.

51 records · Page 3Linked to original sources
arXiv2019

Jordan totient quotients

The Jordan totient $J_k(n)$ can be defined by $J_k(n)=n^k\prod_{p\mid n}(1-p^{-k})$. In this paper, we study the average behavior of fractions $P/Q$ of two products $P$ and $Q$ of Jordan totients, which we call Jordan totient quotients. To this end, we describe two general and ready-to-use methods that allow one to deal with a larger class of totient functions. The first one is elementary and the second one uses an advanced method due to Balakrishnan and Pétermann. As an application, we determine the average behavior of the Jordan totient quotient, the $k^{th}$ normalized derivative of the $n^{th}$ cyclotomic polynomial $Φ_n(z)$ at $z=1$, the second normalized derivative of the $n^{th}$ cyclotomic polynomial $Φ_n(z)$ at $z=-1$, and the average order of the Schwarzian derivative of $Φ_n(z)$ at $z=1$.

math.NT↗
arXiv2019

On the distribution of products of two primes

For a real parameter $r$, the RSA integers are integers which can be written as the product of two primes $pq$ with $p<q\leq rp$, which are named after the importance of products of two primes in the RSA-cryptography. Several authors obtained the asymptotic formulas of the number of the RSA integers. However, the previous results on the number of the RSA integers were valid only in a rather restricted range of the parameter $r$. Dummit, Granville, and Kisilevsky found some bias in the distribution of products of two primes with congruence conditions. Moree and the first author studied some similar bias in the RSA integers, but they proved that at least for fixed $r$, there is no such bias. In this paper, we provide an asymptotic formula for the number of the RSA integers available in wider ranges of $r$, and give some observations of the bias of the RSA integers, by interpolating the results of Dummit, Granville, and Kisilevsky and of Moree and the first author.

math.NT↗
arXiv2019

Linear independence results for certain sums of reciprocals of Fibonacci and Lucas numbers

The aim of this paper is to give linear independence results for the values of certain series. As an application, we derive arithmetical properties of the sums of reciprocals of Fibonacci and Lucas numbers associated with certain coprime sequences $\{n_\ell\}_{\ell\geq1}$. For example, the three numbers \[ 1,\qquad\sum_{p\text{:prime}}^{}\frac{1}{F_{p^2}},\qquad\sum_{p\text{:prime}}^{}\frac{1}{L_{p^2}} \] are linearly independent over $\mathbb{Q}(\sqrt{5})$, where $\{F_n\}$ and $\{L_n\}$ are the Fibonacci and Lucas numbers, respectively.

math.NT↗
arXiv2018

On the sum of a prime power and a power in short intervals

Let $R_{k,\ell}(N)$ be the representation function for the sum of the $k$-th power of a prime and the $\ell$-th power of a positive integer. Languasco and Zaccagnini (2017) proved an asymptotic formula for the average of $R_{1,2}(N)$ over short intervals $(X,X+H]$ of the length $H$ slightly shorter than $X^{\frac{1}{2}}$, which is shorter than the length $H=X^{\frac{1}{2}+ε}$ in the exceptional set estimates of Mikawa (1993) and of Perelli and Pintz (1995). In this paper, we prove that the same asymptotic formula for $R_{1,2}(N)$ holds for $H$ of the size $X^{0.337}$. Recently, Languasco and Zaccagnini (2018) extended their result to more general $(k,\ell)$. We also consider this general case, and as a corollary, we prove a conditional result of Languasco and Zaccagnini (2018) for the case $\ell=2$ unconditionally up to some small factors.

math.NT↗
arXiv2018

On error term estimates à la Walfisz for mean values of arithmetic functions

Walfisz (1963) proved the asymptotic formula \[ \sum_{n\le x}φ(n) = \frac{3}{π^2}x^2+O(x(\log x)^{\frac{2}{3}}(\log\log x)^{\frac{4}{3}}), \] which improved the error term estimate of Mertens (1874) and had been the best possible estimate for more than 50 years. Recently, H.-Q. Liu (2016) improved Walfisz's error term estimate to \[ \sum_{n\le x}φ(n) = \frac{3}{π^2}x^2+O(x(\log x)^{\frac{2}{3}}(\log\log x)^{\frac{1}{3}}). \] We generalize Liu's result to a certain class of arithmetic functions and improve the result of Balakrishnan and Pétermann (1996). To this end, we provide a refined version of Vinogradov's combinatorial decomposition available for a wider class of multiplicative functions.

math.NT↗
arXiv2017

On amicable tuples

For an integer $k\ge2$, a tuple of $k$ positive integers $(M_i)_{i=1}^{k}$ is called an amicable $k$-tuple if the equation \[ σ(M_1)=\cdots=σ(M_k)=M_1+\cdots+M_k \] holds. This is a generalization of amicable pairs. An amicable pair is a pair of distinct positive integers each of which is the sum of the proper divisors of the other. Gmelin (1917) conjectured that there is no relatively prime amicable pairs and Artjuhov (1975) and Borho (1974) proved that for any fixed positive integer $K$, there are only finitely many relatively prime amicable pairs $(M,N)$ with $ω(MN)=K$. Recently, Pollack (2015) obtained an upper bound \[ MN<(2K)^{2^{K^2}} \] for such amicable pairs. In this paper, we improve this upper bound to \[ MN<\frac{π^2}{6}2^{4^K-2\cdot 2^K} \] and generalize this bound to some class of general amicable tuples.

math.NT↗
arXiv2017

Goldbach Representations in Arithmetic Progressions and zeros of Dirichlet L-functions

Assuming a conjecture on distinct zeros of Dirichlet L-functions we get asymptotic results on the average number of representations of an integer as the sum of two primes in arithmetic progression. On the other hand the existence of good error terms gives information on the the location of zeros of L-functions and possible Siegel zeros. Similar results are obtained for an integer in a congruence class expressed as the sum of two primes.

math.NT↗
arXiv2017

Real zeros of Hurwitz zeta-functions and their asymptotic behavior in the interval $(0,1)$

Let $0<a\leq1, s\in\mathbb{C}$, and $ζ(s,a)$ be the Hurwitz zeta-function. Recently, T.~Nakamura showed that $ζ(σ,a)$ does not vanish for any $0<σ<1$ if and only if $1/2\leq a \leq1$. In this paper, we show that $ζ(σ,a)$ has precisely one zero in the interval $(0,1)$ if $0<a<1/2$. Moreover, we reveal the asymptotic behavior of this unique zero with respect to $a$.

math.NT↗
arXiv2017

The Graph Ramsey Number $R(F_\ell,K_6)$

For a given pair of two graphs $(F,H)$, let $R(F,H)$ be the smallest positive integer $r$ such that for any graph $G$ of order $r$, either $G$ contains $F$ as a subgraph or the complement of $G$ contains $H$ as a subgraph. Baskoro, Broersma and Surahmat (2005) conjectured that \[ R(F_\ell,K_n)=2\ell(n-1)+1 \] for $\ell\ge n\ge3$, where $F_\ell$ is the join of $K_1$ and $\ell K_2$. In this paper, we prove that this conjecture is true for the case $n=6$.

math.CO↗
arXiv2016

On prime vs. prime power pairs

In this paper, we consider pairs of a prime and a prime power with a fixed difference. We prove an average result on the distribution of such pairs. This is a partial improvement of the result of Bauer (1998).

math.NT↗
arXiv2015

A remark on the conditional estimate for the sum of a prime and a square

Hardy and Littlewood conjectured that every sufficiently large integer is either a square or the sum of a prime and a square. Let $E(x)$ be the number of positive integers up to $x\ge4$ which does not satisfy this condition. We prove $E(x)\ll x^{1/2}(\log x)^A(\log\log x)^4$with $A=3/2$ under the Generalized Riemann Hypothesis. This is a small improvement of the previous remarks of Mikawa (1993) and Perelli-Zaccagnini (1995) which claims $A=4,3$ respectively.

math.NT↗
arXiv2015

A mean value of the representation function for the sum of two primes in arithmetic progressions

In this note, assuming a variant of the Generalized Riemann Hypothesis, which does not exclude the existence of real zeros, we prove an asymptotic formula for the mean value of the representation function for the sum of two primes in arithmetic progressions. This is an improvement of the result of F. Rüppel in 2009, and the generalization of the result of A. Languasco and A. Zaccagnini concerning the ordinary Goldbach problem in 2012.

math.NT↗
arXiv2013

Digital ultrasonically encoded (DUE) optical focusing into random media

Focusing light into opaque random or scattering media such as biological tissue is a much sought-after goal for biomedical applications such as photodynamic therapy, optical manipulation, and photostimulation. However, focusing with conventional lenses is restricted to one transport mean free path in scattering media, limiting both optical penetration depth and resolution. Focusing deeper is possible by using optical phase conjugation or wavefront shaping to compensate for the scattering. For practical applications, wavefront shaping offers the advantage of a robust optical system that is less sensitive to optical misalignment. Here, the phase of the incident light is spatially tailored using a phase-shifting array to pre-compensate for scattering. The challenge, then, is to determine the phase pattern which allows light to be optimally delivered to the target region. Optimization algorithms are typically employed for this purpose, with visible particles used as targets to generate feedback. However, using these particles is invasive, and light delivery is limited to fixed points. Here, we demonstrate a method for non-invasive and dynamic focusing, by using ultrasound encoding as a virtual guide star for feedback to an optimization algorithm. The light intensity at the acoustic focus was increased by an order of magnitude. This technique has broad biomedical applications, such as in optogenetics or photoactivation of drugs.

physics.optics↗
arXiv2012

The Mott State and Superconductivity in Face-Centred Cubic Structured Cs3C60: A 133Cs-Nuclear Magnetic Resonance Study under Pressure

Over the past 20 years, fullerides have been studied as the source of high-transition-temperature (Tc) superconductivity except for copper oxides. The recent finding of the Mott insulating state right beside superconductivity in Cs3C60 has suggested that magnetism helps raise Tc even in fullerides as in heavy-fermion compounds, high-Tc copper oxides, two-dimensional organic conductors, and iron pnictides. Namely, one tends to think that the link between Mott insulator and superconductivity takes place in fullerides, which can give rise to the mechanism beyond the Bardeen-Cooper-Schrieffer framework. However, the relationship between the Mott state and the superconductivity in Cs3C60 is still under debate. By nuclear magnetic resonance measurements under pressure, we find that the magnetism and superconductivity in Cs3C60 are competing orders. Different from previous reports, the phase separation of Cs3C60 crystals into the Mott and metallic states allows us to systematically study the evolution of the ground state under pressure. Our careful experiments have found that the prevention of a magnetic order is rather essential for the superconductivity in face-centred cubic Cs3C60, which presents a basic strategy for finding still higher Tc in this system.

cond-mat.supr-con↗