Diophantine Analysis and Related Fields 2024
Scheduled Talks and Their Abstracts
Monday 4th March
13:3014:20 Shinichi Yasutomi (Toho University)
Title: Simultaneous convergent continued fraction algorithm with twoplaces

Abstract: Let p be a prime number and K be a eld with embeddings into R andQp . We propose some algorithms that generate continued fraction expansions converging in Qp and are expected to simultaneously converge in both R and Qp . Thisalgorithm produces nite continued fraction expansions for rational numbers. In thecase of p = 2 and if K is a quadratic eld, the continued fraction expansions generated by this algorithm converge in R, and they are eventually periodic or nite.
For an element α in K , let pn /qn denote the n-th convergent. There exist constants
u1 and u2 in R>0 with u1 + u2 = 2, and constants C1 and C2 in R>0 such that|α − pn /qn | < C1 /|qn |u1 and |α − pn /qn |2 < C2 /|qn |u2 . Here, | · |2 represents the2-adic distance. For prime numbers p > 2, we present numerical experiences andsome conjectures.
14:3015:20 Yu Yasufuku (Nihon University)
Title: Bounding GCD's of evaluations of two trivariate Polynomials
Abstract: Ru and Vojta devised an ecient way of applying Schmidt subspace theorem, via an algebraic-geometric invariant. This has already been applied to analyze
GCD's in various settings, but some subtle diculties arise for example when we tryto analyze the GCD of f (a, b, c) and g(a, b, c), where f and g are integer-coecientpolynomials of degrees 2 and 1 respectively and a, b, c are integers. We discuss thealgebro-geometric diculties in such cases, and compare what we can obtain fromRuVojta method with what is predicted from Vojta's conjecture.
15:4016:20 Yuta Suzuki (Rikkyo University)
Title: Some remarks on counting relatively prime pairs
Abstract: It is a well-known result that the density of relatively prime pairs withrespect to the set of pairs of integers is 6/π 2 . In this talk, we consider variationsof this result with the relatively prime pairs from a given sequence. A problem of
the classical method is that it stops working if the counting function of the given
sequence is of the size ≪ x1/2 . In this talk, we remark that we can overcome thisdiculty by a simple trick. As applications, we apply this trick to the [x/n]-sequence,a conjecture of BanksShparlinski on palindromic numbers and the Piatetski-Shapiro
sequences. This talk is based on joint work with Kota Saito, Wataru Takeda andYuuya Yoshida and joint work with Hirotaka Kobayashi and Ryota Umezawa.
16:3017:10 Wataru Takeda (Tokyo Universitu of Science)
Title: Multiplicative Titchmarsh divisor problem for the [x/n]-sequence
Abstract: The Titchmarsh divisor problem, a variant of the ordinal divisor problem,is the problem of giving the asymptotic behavior of the summatory function of thenumber of divisors of shifted primes. In this talk, I talk about certain "multiplicative" analogs of the Titchmarsh divisor problem. We consider two types of analogs
1and derive asymptotic formulas for such multiplicative Titchmarsh divisor problems
for some small arithmetic functions, and the Euler totient function with the vonMangoldt function. This work is a joint-work with Kota Saito, Yuta Suzuki, andYuuya Yoshida.
17:2018:00 Tomohiro Yamada (Osaka University)
Title: On Gauss-Kraïtchik formula for cyclotomic polynomials via sym-

metric function

Abstract: For any odd integer d ≥ 3, the d-th cyclotomic polynomial Φd (X) canbe represented in the way 4Φd (X) = Ψd (X)2 − DΞd (X)2 with D = (−1)(d−1)/2 d∑ d′∑d′d′ −n and Ξ (X) =d′ −nusing two polynomials Ψd (X) =dn=0 ad,n X
n=1 bd,n Xof integer coecients, which is proved by Gauss for prime cases and by Kraïtchikfor general cases. We obtain upper bounds for absolute values( of these coecients.
)√Gd,n + n − 1If d ≥ 3 is odd and squarefree, then |ad,n | + |bd,n | d < 2, wheren√
Gd,t = max{φ(f )/2 : f | d, 1 < f ≤ t} ∪ {|1 + d|/2}. As an application, we showthat if d ≥ 5 is odd and squarefree and x > 2G2d with Gd = Gd,φ(d)/4 , then1xΞd (x)√−<Ψd (x) 2x − µ(d)(2x − µ(d)) d((11−x)−GdGd−1−x).

Thuesday 5th March
9:2010:10 István Pink (University of Debrecen)，Takafumi Miyazaki (GunmaUniversity)
Title: Number of solutions to a special type of unit equations in twounknowns III

Abstract: This talk is a continuation of the one given by the rst named speaker at
the previous DARF on March 3, 2023. The topic is the best possible general estimate
on the number of solutions to a special type of unit equations in two unknowns overthe rationals. It is conjectured that for any xed relatively prime positive integers
a, b and c all greater than 1 there is at most one solution to the equation ax + by = cz
in positive integers x, y and z , except for specic cases. In this talk we give a brief
introduction to the conjecture, and present our results with their proofs, which inparticular enables us to nd some new values of c being presumably innitely manysuch that for each such c the conjecture holds true except for only nitely many pairsof a and b. Most importantly we prove that if c = 13 then the equation has at mostone solution, except for (a, b) = (3, 10) or (10, 3). Our study can also conrm a largepart of Terai's conjecture involving the condition that all x, y and z are greater than1.
10:2011:10 Noriko Hirata-Kohno (Nihon University)
Title: Diophantine approximations and O-Minimality
Abstract: In the framework of the Pila-Wilkie theorem and O-minimality, Diophantine approximation has been recently applied by several authors. In this talk, we
explain how Diophantine approximation, in particular transcendence method applies to the counting question of points of low transcendence degree, under suitable
assumptions.
2This is a joint research with G. Binyamini (the Weizmann Institute) and M.
Kawashima (Nihon University).
11:2012:00 Makoto Kawashima (Nihon University)
Title: Gauss' continued fraction and irrationality
Abstract: In 1813, C. F. Gauss introduced a general continued fraction that represents the ratio of two Gauss hypergeometric functions 2 F1 by their three term
contiguous relations. While this continued fraction expansion is useful, explicitly
determining its numerator and denominator is not widely known. However, it isknown that these numerator and denominator are related to the Pade approximation of contiguous Gauss hypergeometric functions. In this lecture, we explain theexplicit construction of (n−1, n) and (n, n)-Pade approximations of contiguous Gausshypergeometric functions derived from the contiguous relation of Gauss's hypergeometric function. Additionally, we introduce a sharp irrationality measure of value ofthe ratios of contiguous Gauss hypergeometric functions at an algebraic number as
an application. This is a joint work with Akihito Ebisu.
14:0014:50 Yasutsugu Fujita (Nihon University)
Title: Bounding the size of D(±1)-tuples in imaginary quadratic rings
Abstract: Let R be a commutative ring with unity. For a nonzero integer n, a set{a1 , a2 , . . . , am } of m distinct elements in R \ {0} is called a D(n)-m-tuple in R if
ai aj + n is a square in R for any i and j with 1 ≤ i < j ≤ m.
In case R = Z and n ∈ {±1}, sharp bounds for m are known. More precisely,He, Togbé and Ziegler in 2019 settled the folklore conjecture asserting that there
exists no D(1)-quintuple in Z, while Bonciocat, Cipu and Mignotte in 2022 proved
the non-existence of D(−1)-quadruples in Z.
Let O be the ring of integers in an imaginary quadratic eld K . In 2021, notsharp, but absolute bounds for m in the cases where R = OK and n ∈ {±1} weregiven by Adºaga and Gupta, who proved the non-existence of D(1)-m-tuples in = OK
for m > 42 and of D(−1)-m-tuples in = OK for m > 36, respectively.
The aim of this talk is to improve both of the bounds above for m. This work is
a collaboration with Mihai Cipu.
15:0015:50 Taka-aki Tanaka (Keio University)
Title: On a certain function invariant under group action with remarkable

algebraic independence properties

Abstract: Using specic linear recurrences of positive integers, we construct functionsof three complex variables which are invariant under the action of the dihedral groups.
For such a function, we consider the map dened on the set of orbits of the actionof the dihedral group on its domain of denition and the restriction of the map to
the set of algebraic points. Then we show that the restriction of the map takes
algebraically independent values at any distinct orbits.
16:1016:50 Shintaro Murakami (Hirosaki University)
Title: Linear independence of certain innite series with monomial orders
Abstract:(1957) proved that for any integers i ≥ 1 and b, j ≥ 2 the number∑ P. Erd®s−inj is transcendental or an algebraic number with degree at least j ;αi,j := ∞bn=1
however, not much is known about algebraic relations among the numbers αi,j (i =1, 2, . . . , j = 2, 3, . . . ). Recently, V. Kumar (2019) gave linear independence results
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for the numbers αi,j under some conditions, and shortly thereafter, the speaker andY. Tachiya (2023) have removed the conditions and proved that for any integer b(|b| ≥ 2) the numbers 1, αi,j (i = 1, 2, . . . , j = 2, 3, . . . ) are linearly independent overQ.
In this talk, we give linear independence results for Lambert series and the above
numbers αi,j . More precisely, we show linear independence over Q of the numbers

1,∞∑ai,j (n)
n=1

binj

(i = 1, 2, . . . , j = 2, 3, . . . ),

∞
∑1

n=1

bknℓ − 1(k, ℓ = 1, 2, . . . ),where {ai,j (n)}n≥1 are sequences of integers with suitable non-zero and growth conditions. This is a joint work with Yohei Tachiya (Hirosaki University).
17:0017:40 Motoya Teranishi (Kyoto University)
Title: On potential density of integral points on the complement of somesubvarieties in the projective space

Abstract: In this talk, we will introduce our density results for integral points on
the complement of a closed subvariety in the n-dimensional projective space over anumber eld. For instance, we consider a subvariety whose components consist of
n-1 hyperplanes plus one quadric hyperpsurface in general position, or four planes ingeneral position plus a nite number of concurrent straight lines in the 3-dimensional
projective space. In these cases, under some conditions on their intersection, we showthat the integral points on the complements are potentially dense.

Wedsday 6th March
9:2010:10 Takumi Noda (Nihon University)
Title: On the generating Dirichlet series of the Riemann zeta-function
Abstract: Let ζ(s) be the Riemann zeta-function and ζ(s, x) be the Hurwitz zetafunction. It is well known that ζ(s, x) is the binomial type generating function of
ζ(s):∞∑Γ (s + m)
ζ(s + m)(−x)mζ(s, 1 + x) =Γ (s) m!
m=0for |x| < 1 and s ∈ C \ {1}, which gives a number of formula involving the Riemannzeta-function.
∞∑In this talk, we start with the classical Goldbach's formula{ζ(k) − 1} = 1, andk=2

discuss the generating Dirichlet series of the Riemann zeta-function.
10:3011:10 Kota Saito (University of Tsukuba)
Title: The simple normality of the fractional powers of two and the Rie-mann zeta function

Abstract: A real number x is called a simply normal number to base b if the base-b
expansion of x has each digit 0, 1, . . . , b−1 appearing with average frequency tending
to 1/b. We do not determine
the simple normality of many non-articial numbers√
such as π , e, log 2, 2, etc. In this talk, we report discovering a relation between
the frequency that the digit 1 appears in the binary expansion of 2p/q and a meanvalue of the Riemann zeta function on arithmetic progressions. This is joint work4

with Yuya Kanado (Nagoya University).
11:2012:00 Hajime Kaneko (University of Tsukuba)
Title: Marko and Lagrange spectra on one-sided shift spaces with cylin-

der orders

Abstract: Various analogies of the Marko-Lagrange spectrum have been investigated by many mathematicians. Multiplicative versions of the Marko-Lagrange
spectrum are related to the fractional parts of linear recurrences. Dubickas investigated the minimal limit points of certain multiplicative Marko-Lagrange spectrum
in terms of symbolic dynamical systems and certain substitution.
In this talk, we study an analogy of the Marko-Lagrange spectrum for one-sided
shift spaces with cylinder orders. As our main results, we determine the discrete partsand minimal limit points in terms of S -adic sequences, where S is an innite set ofsubstitutions. This is a joint work with Wolfgang Steiner.

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