Diophantine Analysis and Related Fields 2015
Scheduled Talks and Their Abstracts
Thursday 5 March
13:40–14:30
Takafumi Miyazaki (Nihon University)
Title: A polynomial-exponential Diophantine equation related to the sum of consecutive
k-th powers.
Abstract: Lucas’s square pyramid problem asks to solve the Diophantine equation
1 + 22 + 32 + · · · + x2 = y 2 .
It is known that the only non-trivial positive solution is (x, y) = (24, 70), that is,
1 + 22 + 32 + · · · + 242 = 702 .
Here, we consider the following general equation:
1 + 2k + 3k + · · · + xk = y n
in integers k, n, x, y with k ≥ 1, n ≥ 2, x ≥ 2, y ≥ 2. If the pair (k, n) is one of(1, 2), (3, 2), (3, 4), (5, 2),
then the problem is reduced to solve a Pellian equation, in particular, we ﬁnd inﬁnitely
many solutions (x, y). In 1956, Schäﬀer conjectured that the answer to Lucas’s problem
mentioned above is the only solution of our equation for which the pair (k, n) does notbelong to {(1, 2), (3, 2), (3, 4), (5, 2)}. This is still unsolved in spite of many studies inthe literature. In this talk, we solve our equation in a special case, and our result gives anaﬃrmative answer to Schäﬀer’s problem. This is a joint work with A. Bérczes, L. Hajdu
and I. Pink (University of Debrecen).
14:40–15:20
Tomohiro Ooto (Tsukuba University)
Title: Quadratic approximation of continued fractions with low complexity in Fq ((T −1 ))Abstract: We consider the Koksma’s exponent w2∗ in Fq ((T −1 )). We obtain a ﬁnitenessof the value of continued fractions with low complexity and some conditions taken by thefunction w2∗ . In particular, automatic or primitive morphic continued fractions satisfy theconditions.
15:40–16:30
Takumi Noda (Nihon University)
Title: Confluent hypergeometric type Ramanujan’s formulas
Abstract: Let ζ(s, x) be the Hurwitz zeta-function. Ramanujan’s formula (1916-17)
shows that ζ(s, x) is one generating function of the Riemann zeta-function, (the binomial
type power series):ζ(s, 1 + x) =∞∑Γ (s + m)

Γ (s) m!
m=0

ζ(s + m)(−x)m ,for |x| < 1 and s ∈ C\{−1, 0, 1, 2, · · · }. In this talk, we introduce some new zeta-functions
associated with conﬂuent hypergeometric functions as classical Dirichlet series. We willshow functional properties of these zeta-functions which lead to analogues of Ramanujan’s
formula.
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16:40–17:30
Yasutsugu Fujita (Nihon University)
Title: Bounds for Diophantine quintuples
Abstract: A set {a1 , . . . , am } of m positive integers is called a Diophantine m-tuple ifai aj + 1 is a perfect square for all i, j with i ̸= j. A folklore conjecture states that thereexists no Diophantine quintuple. It has been known that various kinds of Diophantine
triples or pairs cannot be extended to Diophantine quintuples, such as {1, 3, 8} by Baker
and Davenport or more generally {k − 1, k + 1} with k ≥ 2 an integer. In this talk wewill give some recent results on the Diophantine pairs {a, b} with a < b that cannot be
contained in Diophantine quintuples {a, b, c, d, e} with b < c < d < e, one of which is thepair {a, b} with b ≤ 3a (joint work with Mihai Cipu).

Friday 6 March
9:30–10:10
Jonathan Caalim (Tsukuba University)
Title: Rotational beta expansion: Ergodicity and Soficness
Abstract: Let 1 < β ∈ R and ζ ∈ C \ R with |ζ| = 1. Fix ξ, η1 , η2 ∈ C with η1 /η2 ̸∈ R.
Then X = {ξ + xη1 + yη2 | x ∈ [0, 1), y ∈ [0, 1)} is a fundamental domain of the lattice
L generated by η1 and η2 in C. We deﬁne a map T : X → X by T (z) = βζz − d whered = d(z) is the unique element in L such that βζz ∈ X + d. The above map gives atwo-dimensional generalization of the positive and negative β-transformations. In thistalk, we discuss the ergodicity and soﬁcness of T . This is a joint work with Prof. ShigekiAkiyama.
10:20–11:10
Yu Yasufuku(Nihon University)
Title: Logarithmic Kodaira dimension and Diophantine equations
Abstract: We describe a relationship between integral points on surfaces and certain
Diophantine equations. On one hand, we have the structure theorems based on log kodairadimension, studied by many Japanese mathematicians from the 70’s. On the other hand,
we have Diophantine tools, such as unit equations and Roth’s theorem. We use bothof these machineries to study Diophantine equations through log kodaira dimension. In
particular, we obtain what we believe to be new results on ﬁniteness of solutions to certain
Diophantine equations. If time permits, we will also discuss an application to arithmetic
dynamics in dimension 2. This is a joint work with Aaron Levin (Michigan State).
11:20–12:10
Masanori Katsurada (Keio Univ.)
Title: Asymptotic expansions for the Laplace-Mellin and Riemann-Liouville transformsof Lerch zeta-functions
Abstract: The Laplace-Mellin and Riemann-Liouville type transforms are ﬁrst introduced in a generic manner under certain natural settings. It is then shown that complete
asymptotic expansions exist for these transforms applied to Lerch zeta-functions, when
the pivotal parameter z in the transformations tends to both 0 and ∞ through the sector| arg z| ≤ π/2. Several applications of these results are also to be given.

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14:00–14:50
Hajime Kaneko(Tsukuba University)
Title: On the beta-expansions of algebraic numbers by a Pisot or Salem number beta
Abstract: Let β > 1 be a real number. Rényi introduced the notion of β-expansions ofreal numbers which is applicable to number theory and dynamical systems. In particular,
it is important to study the digits in the β-expansions of algebraic numbers in the casewhere β is a Pisot or Salem number. In this talk, we give new lower bounds for thenumber of nonzero digits.
15:00–15:40
Junya Iwaki (Gunma Univ.)
Title: Linear independence measures for the values of certain Mahler functions
Abstract: Let r ≥ 2 and m ≥ 1 be integers. Assume that an m-dimensional vector
F(z) = t (f1 (z), · · · , fm (z)) ∈ Q[[z]]m is a solution of a system of Mahler functional equations F(z r ) = A(z)F(z) with A(z) ∈ GL(m, Q(z)) such that f1 (z), · · · , fm (z) converge
on |z| < R with R > 0 and that 1, f1 (z), · · · , fm (z) are linearly independent over Q(z).
In this talk, we give an upper bound of the linear independence measure for the numbersk1, f1 (1/b), · · · , fm (1/b), where b > 1 is an integer with 1/b < R such that b−r are neither
zeros nor poles of det A(z) for all integers k ≥ 0. For example, let R(x) and B(x) bethe generating function of the Rudin-Shapiro sequence and that of Baum-Sweet sequence,respectively. Then we see that the linear independence measure of 1, R(1/b), (−1/b) is at
most 16, and that of 1, B(1/b), B(1/b2 ) is also at most 16. This is a joint work with Prof.
Masaaki Amou.
16:00–16:40
Akinari Goto (Keio University), Taka-aki Tanaka (Keio University)
Title: Algebraic independence of the values of Mahler functions in a certain case of
positive characteristic
Abstract: Mahler’s method gives algebraic independence of the values of functions satisfying functional equations such as (1) below, which has recently been studied over baseﬁeld K := Fq (θ) of characteristic p > 0. For example, Denis treated every period of
Carlitz’s exponential as a Mahler’s value and Pellarin gave, using Mahler’s method, completely diﬀerent proofs for the results of Papanikolas and of Chang and Yu for the valuesof its inverse function, Carlitz’s logarithm. We consider diﬀerent type of functional equations in positive characteristic than those treated in the previous works mentioned above.
For an integer d ≥ 2, deﬁnef (x, z) :=∞∑kxk z d ,k=0which satisﬁes the functional equation
f (x, z) = xf (x, z d ) + z.

(1)

The main result of this talk is the following:
Theorem. Assume that the characteristic p does not divide d and that nonzero elements
a1 , . . . , am of K alg. , the algebraic closure of K, are distinct. Then for any α ∈ K alg. with
0 < |α|v < 1, the values∂jf (ai , α) (1 ≤ i ≤ m, 0 ≤ j ≤ p − 1)
∂xj

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are algebraically independent over K. Hence, for the function F (x) :=infinite set of the values{ (j)}
F (a) 0 ≤ j ≤ p − 1, a ∈ K alg. \{0}∑∞k=0 αdk k

x , theis algebraically independent over K.
16:50-17:40
Makoto Kawashima (Osaka University)
Title: Linear independence of values of certain p-adic functions
Abstract: In this talk, we study two types of criteria of linear independence of p-adic
numbers. First, we introduce a criterion of linear independence of p-adic numbers due toPierre Bel. Then, by using the above result, we axiomatize two types of criteria of linear
independence of special values of certain p-adic functions. Finally, we show the linear
independence of special values of p-adic Lerch function by using this axiom.

Saturday 7 March
9:30-10:20
Hirofumi Nagoshi (Gunma University)
Title: The existence of zeros of non-primitive L-functions for SL(2, Z) in the strip
1/2 < ℜs < 1
Abstract: Let f (z) be a holomorphic cusp form of even positive integral weight for
SL(2, Z). In this talk, we will show that if f (z) is not a Hecke eigenform then theassociated L-function L(s, f ), which satisﬁes a functional equation, has inﬁnitely many
zeros in the strip 1/2 < ℜs < 1, where the critical line of L(s, f ) is normalized to beℜs = 1/2. Actually, a stronger assertion holds. In the proof, Kronecker’s approximation
theorem in the theory of Diophantine analysis and other things are used. A similar result
was obtained by Voronin for certain Epstein zeta functions. However, we cannot adapt
his proof straightforward to our case. Another similar result was obtained by Kaczorowski
and Kulas for certain L-functions of degree 1 in the extended Selberg class, from a variantof the so-called “joint universality” property of Dirichlet L-functions. It should be notedthat our proof as well as Voronin’s proof does not use such an universality property.
We would also like to discuss some results concerning the independence of L-functions
L(s, fj ) for distinct primitive forms fj (z), since these results are closely related to theabove one.
10:40-11:30
Takao Komatsu (Wuhan university)
Title: Associate and restricted Stirling numbers and their applications
Abstract: We deﬁne the associate Stirling numbers of the ﬁrst kind and of the second kind, and the restricted Stirling numbers of the ﬁrst kind and of the second kind,as generalizations of the classical Stirling numbers of the ﬁrst kind and of the secondkind. By studying their arithmetical and combinatorial properties, we give several theirapplications.