The formula for the number of domino tilings due to Kasteleyn and Temperley-Fisher is strikingly similar to Eisenstein's formula for the Legendre symbol. We study the connection between these two...

In this paper, we introduce formal sine functions whose coefficients are elements of a generalized harmonic algebra and investigate their properties corresponding to the classical addition formula...

We show that integer partitions, the fundamental building blocks in additive number theory, detect prime numbers in an unexpected way. Answering a question of Schneider, we show that the primes...

These are the notes for an undergraduate course at the University of Edinburgh, 2021-2023. Assuming basic knowledge of ring theory, group theory and linear algebra, the notes lay out the theory of...

In this paper, we introduce iterated integrals associated with colored rooted trees and give proofs for the shuffle relations for $\boldsymbol{p}$-adic finite and $t$-adic symmetric...

We give a simple proof of the fundamental theorem of Galois theory which provides a correspondence between the intermediate fields of a finite Galois extension and the subgroups of its Galois...

We introduce and study finite analogues of Euler's constant in the same setting as finite multiple zeta values. We define a couple of candidate values from the perspectives of a ``regularized...

Let $χ$ be a real primitive character to the modulus $D$. It is proved that $$ L(1,χ)\gg (\log D)^{-2022} $$ where the implied constant is absolute and effectively computable. In the...

In this paper, we consider a discrete version of iterated integrals by the naive (equally divided) Riemann sum. In particular, basic three formulas for usual iterated integrals are discritized....

A real number is called simply normal to base $b$ if its base-$b$ expansion has each digit appearing with average frequency tending to $1/b$. In this article, we discover a relation between the...

We present a new proof of the extended double shuffle relation for multiple zeta values which notably does not rely on the use of integrals. This proof is based on a formula recently obtained by...

We introduce finite multiple zeta values of general level and discuss the relationship between the non-zeroness of these values and regular or non-Wieferich primes. Because it's challenging to...

The $t$-adic symmetric multiple zeta value is a generalization of the symmetric multiple zeta value from the perspective of the Kaneko-Zagier conjecture. In this paper, we introduce a further...

Given any number field, we prove that there exist arbitrarily shaped constellations consisting of pairwise non-associate prime elements of the ring of integers. This result extends the celebrated...

We show that there exist infinite sets $A = \{a_1,a_2,\dots\}$ and $B = \{b_1,b_2,\dots\}$ of natural numbers such that $a_i+b_j$ is prime whenever $1 \leq i < j$.

Recently, Maesaka, Watanabe, and the third author discovered a phenomenon where the iterated integral expressions of multiple zeta values become discretized. In this paper, we extend their result...

In this paper, we will consider $(φ,Γ)$-modules over rings which are "combinations of discrete algebras and affinoid $\mathbb{Q}_p$-algebras", and prove basic results such as the...

Underlying the Riemann Hypothesis there is a question whose full answer still eludes us: what do the zeros of the Riemann zeta function really mean? As a step toward answering this, André Weil...

We indulge in what mathematicians call frivolous activities. In Arithmetic Billiards, a ball is bouncing around in a rectangle. In Parity Checkers we place checkers on a checkerboard under certain...

We discuss the evaluation of the average number of Goldbach representations for integers which are multiples of $q$ introduced by Granville. We improve an estimate given by Granville under the...

Inspired by Episode 3 of the Japanese manga "Seisu-tan" by Doom Kobayashi and Shin-ichiro Seki, we investigate the $k$-Göbel sequence $(g_{k,n})_n$ named after Fritz Göbel. Although the...

The Fourier coefficients $c_1(n)$ of the elliptic modular $j$-function are always even for $n \not\equiv 7 \pmod{8}$. In contrast, for $n \equiv 7 \pmod{8}$, it is conjectured that ``half" of the...

We prove a conjecture of K. Marton, widely known as the polynomial Freiman--Ruzsa conjecture, in characteristic $2$. The argument extends to odd characteristic, with details to follow in a...

In 2013 Bettin and Conrey have introduced a cotangent sum $c \colon \mathbb{Q}_{>0}\to \mathbb{R}$, which can be regarded as a variant of the Dedekind sum. They have discovered that the cotangent...

Due to their singularities, multiple zeta functions behave sensitively at non-positive integer points. In this article, we focus on the asymptotic behavior at the origin $(0,\dots, 0)$ and unveil...

The object of this article is a family of $q$-series originating from Habiro's work on the Witten-Reshetikhin-Turaev invariants. The $q$-series usually make sense only when $q$ is a root of unity,...

We prove some weighted sum formulas for half multiple zeta values, half finite multiple zeta values, and half symmetric multiple zeta values. The key point of our proof is Dougall's identity for...

Recently, Maesaka, Seki and Watanabe discovered a surprising equality between multiple harmonic sums and certain Riemann sums which approximate the iterated integral expression of the multiple...

We sketch the construction of a derived enhancement of the reciprocity isomorphism of class field theory. Details will appear in a forthcoming joint paper of the authors with A. Raksit.

Recently, the author and Yamamoto invented a new proof of the duality for multiple zeta values. The technique is applicable in other series identities. In this article, we exhibit such proofs for...

In this paper we give a complete proof of the Brumer-Stark conjecture over $\mathbf{Z}$.

These are the notes of a course on Shimura varieties that I gave at the 2022 IHES summer school on the Langlands program. Lecture 1 gives an introduction to Shimura varieties over the complex...

In this paper, we introduce a new connector which generalizes the connector found by the third author and Yamamoto. The new connector gives a direct proof of the double Ohno relation recently...

We prove the irrationality of the classical Dirichlet L-value $L(2,χ_{-3})$. The argument applies a new kind of arithmetic holonomy bound to a well-known construction of Zagier. In fact our...

The duality relation of one-variable multiple polylogarithms was proved by Hirose, Iwaki, Sato and Tasaka by means of iterated integrals. In this paper, we give a new proof using the method of...

A prime number $p$ is said to be irregular if it divides the class number of the $p$-th cyclotomic field $\mathbb{Q}(ζ_{p}) = \mathbb{Q}(\mathbb{G}_m[p])$. In this paper, we study its elliptic...

We give an explicit formula for the well-known parity result for multiple zeta values as an application of the multitangent functions.

In this paper, we study the asymptotic behavior of the sum of twisted traces of self-dual or conjugate self-dual discrete automorphic representations of $\mathrm{GL}_n$ for the level aspect of...

Given a pure motive $M$ over $\mathbb{Q}$ with a multilinear algebraic structure $\mathsf{s}$ on $M$, and given a representation $V$ of the group respecting $\mathsf{s}$, we describe a functorial...

For $k\geq 0$, a $k$-generalized Markov number is an integer which appears in some positive integer solution to the $k$-generalized Markov equation $x^2 + y^2 + z^2 + k(yz + zx + xy) = (3 +...

Of the $(2H+1)^n$ monic integer polynomials $f(x)=x^n+a_1 x^{n-1}+\cdots+a_n$ with $\max\{|a_1|,\ldots,|a_n|\}\leq H$, how many have associated Galois group that is not the full symmetric group...

The refined Kaneko-Zagier conjecture claims that the algebras spanned by two kinds of "completed" finite multiple zeta values, called $\hat{A}$- and $\hat{S}$-MZVs, are isomorphic. Recently,...

The multiple gamma functions of BM (Barnes-Milnor) type and the $q$-multiple gamma functions have been studied independently. In this paper, we introduce a new generalization of the multiple gamma...

We study the Gauss and Jacobi sums from a viewpoint of motives. We exhibit isomorphisms between Chow motives arising from the Artin-Schreier curve and the Fermat varieties over a finite field,...

Suppose that $n$ is $0$ or $4$ modulo $6$. We show that there are infinitely many primes of the form $p^2 + nq^2$ with both $p$ and $q$ prime, and obtain an asymptotic for their number. In...

We classify the pairs of group morphisms $Γ\rightarrow {\rm Spin}(7)$ which are element conjugate but not globally conjugate. As an application, we study the case where $Γ$ is the Weil...

Let $K$ be a mixed characteristic complete discrete valuation field with perfect residue field of characteristic $p$. We construct a new linear category called the category of crystalline...

Geometric/arithmetic theta correspondences provide correspondences between automorphic forms and cohomology classes/algebraic cycles on Shimura varieties. We give an introduction focusing on the...

We specify a Turing machine $T_{\text{Mordell}}$ with the following properties. 1. On input $(K,C/K)$, with $K/\mathbb{Q}$ a number field and $C/K$ a smooth projective hyperbolic curve, if...

Let $K$ be a number field and $S$ a set of primes of $K$. We write $K_S/K$ for the maximal extension of $K$ unramified outside $S$ and $G_{K,S}$ for its Galois group. In this paper, we prove the...

Let $F$ be a totally real field of degree $n$ and $p$ an odd prime. We prove the $p$-part of the integral Gross--Stark conjecture for the Brumer--Stark $p$-units living in CM abelian extensions of...

We study the universal family of odd hyperelliptic curves of genus $g \geq 1$ over $\mathbb{Q}$. We relate the heights of $\mathbb{Q}$-points of Jacobians of curves in this family to the reduction...

Let $f$ be a normalized newform of weight 2 on $Γ_0(N)$ whose coefficients lie in $\mathbb{Q}$ and let $χ_M$ be a primitive quadratic Dirichlet character with conductor $M$. In this paper,...

It is well known that the set of origami constructible numbers is larger than the classical straight-edge and compass constructible numbers. However, the Huzita-Justin-Hatori origami constructible...

For several evaluations of special values and several relations known only in $\mathcal{A}_n$-multiple zeta values or $\mathcal{S}_n$-multiple zeta values, we prove that they are uniformly valid...

Shnirel'man's inequality and Shnirel'man's basis theorem are fundamental results about sums of sets of positive integers in additive number theory. It is proved that these results are inherently...

In this work we resolve several conjectures stated in the On-Line Encyclopedia of Integer sequences.

In this paper, we present results on the uniqueness of the real zeros of the Hurwitz zeta function in given intervals. The uniqueness in question, if the zeros exist, has already been proved for...

It is proven that there are infinitely prime pairs whose difference is no greater than 20.

In this paper we study the twisted Shintani zeta function over number fields.

We relate two different solutions of a Mahler equation; one solution is only defined at certain roots of unity, while the other is an analytic function inside the unit disk.

The Kodaira dimension of Shimura varieties has been studied by many people. Kondo and Gritsenko-Hulek-Sankaran studied the singularities of orthogonal Shimura varieties related to the moduli...

$p$-adic Hodge Theory is one of the most powerful tools in modern Arithmetic Geometry. In this survey, we will review $p$-adic Hodge Theory for algebraic varieties, present current developments in...

Faltings proved that there are finitely many abelian varieties of genus $g$ over a number field $K$, with good reduction outside a finite set of primes $S$. Fixing one of these abelian varieties...

The aim of this paper is to give a full detail of the proof given by Harder of a theorem on the denominator of the Eisenstein class for $\mathrm{SL}_2(\mathbb{Z})$ and to show that the theorem has...

In 1998, Borwein, Bradley, Broadhurst and Lisoněk posed two families of conjectural identities among multiple zeta values, later generalized by Charlton using his alternating block notation....

It is proven that there are infinitely prime pairs whose difference is no greater than 84. In proving that result, a new sieve weight is constructed. The new idea is to construct the largest sieve...

Let $\ell$ and $p$ be distinct primes, and let $Γ$ be an abelian pro-$p$-group. We study the structure of the algebra $Λ:=\mathbb{Z}_\ell[[Γ]]$ and of $Λ$-modules. In the...

In a recent article a breakthrough was made in quantum computer science which established the existence of a phenomenon commonly known as $\mathsf{MIP}^\ast = \mathsf{RE}$. We show that the...

We prove new bounds for how often Dirichlet polynomials can take large values. This gives improved estimates for a Dirichlet polynomial of length $N$ taking values of size close to $N^{3/4}$,...

We survey recent developments on generalizing the Gross--Zagier formula to high dimensional Shimura varieties, with an emphasis on the Arithmetic Gan--Gross--Prasad conjecture and the relative...