We consider zeta functions: $Z(f ;P ;s)=\sum_{\m \in \N^{n}} f(m_1,..., m_n) P(m_1,..., m_n)^{-s/d}$ where $P \in \R [X_1,..., X_n]$ has degree $d$ and $f$ is a function arithmetic in origin, e.g. a multiplicative function. In this paper, I study the meromorphic continuation of such series beyond an a priori domain of absolute convergence when $f$ and $P$ satisfy properties one typically meets in applications. As a result, I prove an explicit asymptotic for a general class of lattice point problems subject to arithmetic constraints.
We construct the t-motivic mixed Carlitz zeta category over $\F_q(t)$ and show that it contains all the (mixed) t-motives with Carlitz-Thakur multi-zeta values as periods constructed by Anderson and Thakur. Our construction is canonical and our category is Tannakian and neutral and every object is equipped with a weight filtration whose graded pieces are Carlitz motives over $\F_q(t)$. For any finite separable extension L/\F_q(t) we show that existence of a similar category over $L$ is a consequence of a version of a conjecture of L. Taelman. Along the way we also prove the existence of the ca…
Recently, the author defined multiple Dedekind zeta values [5] associated to a number K field and a cone C. These objects are number theoretic analogues of multiple zeta values. In this paper we prove that every multiple Dedekind zeta value over any number field K is a period of a mixed Tate motive. Moreover, if K is a totally real number field, then we can choose a cone C so that every multiple Dedekind zeta associated to the pair (K;C) is unramified over the ring of algebraic integers in K. In [7], the author proves similar statements in the special case of a real quadratic fields for a part…
Two remarks related with the mixed joint universality for a polynomial Euler product and a periodic Hurwitz zeta-function with a transcendental parameter are given. One is the mixed joint functional independence, and the other is a generalized universality, which includes several periodic Hurwitz zeta-functions.
We analyse a collection of mixed moments of the Riemann zeta function and establish the validity of asymptotic formulae. Such examinations are performed both unconditionally and under the assumption of a weaker version of the $abc$ conjecture.
We present the most general at this moment results on the discrete mixed joint value-distribution and the universality property for the class of Matsumoto zeta-functions and periodic Hurwitz zeta-functions under certain linear independence condition on the relevant parameters, such as common differences of arithmetic progressions, prime numbers etc.
The property of zeta-functions on mixed joint universality in the Voronin's sense states that any two holomorphic functions can be approximated simultaneously with accuracy $ε>0$ by suitable vertical shifts of the pair consisting from the Riemann zeta- and Hurwitz zeta-functions. In [1], it was shown rather general result, i.e., an approximating pair was composed of the Matsumoto zeta-functions' class and the periodic Hurwitz zeta-function. In this paper, we prove that this set of shifts has a strict positive density for all but at most countably many $ε>0$. Also, we give the concluding remark…
We give a new type of mixed discrete joint universality properties, which is satisfied by a wide class of zeta-functions. We study the universality for a certain modification of a Matsumoto zeta-function and a periodic Hurwitz zeta-function under certain arithmetic condition on the common difference of arithmetical progression and the shifting parameter. Also we generalize a known discrete universality result in our former paper and the new one as well.
We analyse a collection of twisted mixed moments of the Riemann zeta function and establish the validity of asymptotic formulae comprising on some instances secondary terms of the shape $P(\log T) T^{C}$ for a suitable constant $C<1$ and a polynomial $P(x)$. Such examinations are performed both unconditionally and under the assumption of a weaker version of the $abc$-conjecture.
We study analytic properties of the pair consisting of a rather general form of zeta-function with an Euler product and a periodic Hurwitz zeta-function with a transcendental parameter. We first survey briefly previous results, and then investigate the mixed joint discrete value distribution and prove the mixed joint discrete universality theorem for these functions, in the case when common differences of relevant arithmetic progressions are not necessarily the same. Also some generalizations are given. For this purpose, certain arithmetic conditions on the common differences are necessary.
We prove a new case of mixed discrete joint universality theorem on approximation of certain target couple of analytic functions by the shifts of a pair consisting of the function belonging to wide class of Matsumoto zeta-functions and the periodic Hurwitz zeta-function. We work under the condition that the common difference of arithmetical progression satisfies a certain rational condition and the parameter is a transcendental number. The essential difference from the result in our previous article is that here we do not study the class of partial zeta-functions, but work with the class of th…
We prove a mixed joint discrete universality theorem for a Matsumoto zeta-function $φ(s)$ (belonging to the Steuding subclass) and a periodic Hurwitz zeta-function $ζ(s,α;{\mathfrak{B}})$. For this purpose, certain independence condition for the parameter $α$ and the minimal step of discrete shifts of these functions is assumed. This paper is a continuation of authors' works [12] and [arXiv:1601.03795].
Let $N$ be a power of $2$ or $3$, and $μ_{N}$ the set of $N$-th roots of unity. We show that the ring of motivic periods of Mixed Tate motives over $\mathbb{Z}[μ_{N},\frac{1}{N}]$ is spanned by the motivic cyclotomic multiple zeta values of level $N$. This implies that the action of the motivic Galois group of mixed Tate motives over $\mathbb{Z}[μ_{N},\frac{1}{N}]$ on the motivic fundamental group of $\mathbb{G}_{m}-μ_{N}$ is faithful. This is a generalization of the known results for $N\in\{1,2,3,4,8\}$ by Deligne and Brown. We also discuss cyclotomic multiple zeta values of weight $2$ of oth…
We study a family of mixed Tate motives over $\mathbb{Z}$ whose periods are linear forms in the zeta values $ζ(n)$. They naturally include the Beukers-Rhin-Viola integrals for $ζ(2)$ and the Ball-Rivoal linear forms in odd zeta values. We give a general integral formula for the coefficients of the linear forms and a geometric interpretation of the vanishing of the coefficients of a given parity. The main underlying result is a geometric construction of a minimal ind-object in the category of mixed Tate motives over $\mathbb{Z}$ which contains all the non-trivial extensions between simple objec…
For each field k, we define an abelian category of rationally decomposed mixed motives with integer coefficients. When k is finite, we show that the category is Tannakian, and we prove formulas relating the behaviour of zeta functions near integers to certain Ext groups. This is the submitted version, with minor corrections and additions from the first version.
We define zeta-functions of weight lattices of compact connected semisimple Lie groups. If the group is simply-connected, these zeta-functions coincide with ordinary zeta-functions of root systems of associated Lie algebras. In this paper we consider the general connected (but not necessarily simply-connected) case, prove the explicit form of Witten's volume formulas for these zeta-functions, and further prove functional relations among them which include their volume formulas.
We study zeta-functions of weight lattices of compact connected semisimple Lie groups of type $A_3$. Actually we consider zeta-functions of SU(4), SO(6) and PU(4), and give some functional relations and new classes of evaluation formulas for them.
This paper draws connections between the double shuffle equations and structure of associators; universal mixed elliptic motives as defined by Hain and Matsumoto; and the Rankin-Selberg method for modular forms for $SL_2(\mathbb{Z})$. We write down explicit formulae for zeta elements $σ_{2n-1}$ (generators of the Tannaka Lie algebra of the category of mixed Tate motives over $\mathbb{Z}$) in depths up to four, give applications to the Broadhurst-Kreimer conjecture, and completely solve the double shuffle equations for multiple zeta values in depths two and three.
We set up a connection between the theory of spherical designs and the question of minima of Epstein's zeta function. More precisely, we prove that a Euclidean lattice, all layers of which hold a 4-design, achieves a local minimum of the Epstein's zeta function, at least at any real s>n/2. We deduce from this a new proof of Sarnak and Strömbergsson's theorem asserting that the root lattices D4 and E8, as well as the Leech lattice, achieve a strict local minimum of the Epstein's zeta function at any s>0. Furthermore, our criterion enables us to extend their theorem to all the so-called extremal…
We study the probability that a cycle of length k in the lattice [1, n]^s does not contain more lattice points than the k vertices of the cycle. Then we generalize this problem to other configurations induced by a given graph H, introducting the chromatic zeta fuction of a graph.
In this paper, we shall study Aéry-type series in which the central binomial coefficient appears as part of the summand. Let $b_n=4^n/\binom{2n}{n}$. Let $s_1,\dots,s_d$ be positive integers with $s_1\ge 2$. We consider the series \begin{align*} \sum_{n_1>\cdots>n_d>0} \frac{b_{n_1}}{n_1^{s_1}\cdots n_d^{s_d}} \end{align*} and the variants with some or all indices $n_j$ replaced by $2n_j\pm 1$ and some or all "$>$" replaced by "$\ge$", provided the series are defined. We can also replace $b_{n_1}$ by its square in the above series when $s_1\ge 3$. The main result is that all such series are $\…
We investigate the properties of the zeta-function of well-rounded sublattices of a fixed arithmetic lattice in the plane. In particular, we show that this function has abscissa of convergence at $s=1$ with a real pole of order 2, improving upon a recent result of S. Kuehnlein. We use this result to show that the number of well-rounded sublattices of a planar arithmetic lattice of index less or equal $N$ is $O(N \log N)$ as $N \to \infty$. To obtain these results, we produce a description of integral well-rounded sublattices of a fixed planar integral well-rounded lattice and investigate conve…
The theory of Ihara zeta functions is extended to non-compact arithmetic quotients of Bruhat-Tits trees. This new zeta function turns out to be a rational function, despite the infinite-dimensional setting. In general it has zeros and poles, in contrast to the compact case. The determinant formulas of Bass and Ihara hold true if one defines the determinant as limit of all finite principal minors. From this analysis, a prime geodesic theorem is derived, which, applied to special arithmetic groups, yields new asymptotic assertions on class numbers of orders in global fields.
In this paper, we first reveal an intrinsic relation between non-abelian zeta functions and Epstein zeta functions for algebraic number fields. Then, we expose a fundamental relation between stability of lattices and distance to cusps. Next, using these two relations, we explicitly express rank two zeta functions in terms of the well-known Dedekind zeta functions. Finally, based on such an expression, we show that all zeros of rank two non-abelian zeta functions are entirely sitting on the critical line whose real part equals to 1/2. This is an integrated part of our Geo-Arithmetic Program.
Let $k$ be a totally real number field. For every odd $n\geq 3$, we construct a Dedekind zeta motive in the category $\MT(k)$ of mixed Tate motives over $k$. By directly calculating its Hodge realisation, we prove that its period is a rational multiple of $π^{n[k:\Q]}ζ^*_k(1-n)$, where $ζ^*_k(1-n)$ denotes the special value of the Dedekind zeta function of $k$. We deduce that the group $\Ext^1_{\MT(k)} (\Q(0),\Q(n))$ is generated by the cohomology of a quadric relative to hyperplanes. This proves a surjectivity result for certain motivic complexes for $k$ that have been conjectured to calculat…
The symmetric group $\mathfrak S_{n+1}$ of degree $n+1$ admits an $n$-dimensional irreducible $\mathbf Q \mathfrak S_n$-module $V$ corresponding to the hook partition $(2,1^{n-1})$. By the work of Craig and Plesken we know that there are $σ(n+1)$ many isomorphism classes of $\mathbf Z \mathfrak S_{n+1}$-lattices which are rationally equivalent to $V$, where $σ$ denotes the divisor counting function. In the present paper we explicitly compute the Solomon zeta function of these lattices. As an application we obtain the Solomon zeta function of the $\mathbf Z \mathfrak S_{n+1}$-lattice defined by…
For these two decades, the Arakawa-Kaneko zeta function has been studied actively. Recently Kaneko and Tsumura constructed its variants from the viewpoint of poly-Bernoulli numbers. In this paper, we generalize their zeta functions of Arakawa-Kaneko type to those with indices in which positive and negative integers are mixed. We show that values of these functions at positive integers can be expressed in terms of the multiple Hurwitz zeta star values.
If K_0 is the fraction field of the Witt vectors over an algebraically closed field k of characteristic p, we calculate upper bounds on the conductor of higher ramification for (the Galois closure of) extensions K_0(zeta_{p^r}, sqrt[p^r]{a})/K_0, where a is in K_0(zeta_{p^r}). Here zeta_{p^r} is a primitive p^r-th root of unity. In certain cases, including when a is in K_0 and p=2, we calculate the conductor exactly. These calculations can be used to determine the discriminants of various extensions of Q obtained by adjoining roots of unity and radicals.
We consider the prehomogeneous vector space of pairs of ternary quadratic forms. For the lattice of pairs of integral ternary quadratic forms and its dual lattice, there are six zeta functions associated with the the prehomogeneous vector space. We present a conjecture which states that there are simple relations among the six zeta functions. We prove that the coefficients coincide on fundamental discriminants.
Zeta functions of periodic cubical lattices are explicitly derived by computing all the eigenvalues of the adjacency operators and their characteristic polynomials. We introduce cyclotomic-like polynomials to give factorization of the zeta function in terms of them and count the number of orbits of the Galois action associated with each cyclotomic-like polynomial to obtain its further factorization. We also give a necessary and sufficient condition for such a polynomial to be irreducible and discuss its irreducibility from this point of view.
We investigate the location of zeros and poles of a dynamical zeta function arizing in a class of lattice spin models introduced in the 60-ties by M. Kac. The transfer operator method allows us to prove the xistence of infinitely nontrivial zeros of this function on the real line. For certain parameter values there exist also infinitely many trivial equally spaced zeros on a line parallel to the imaginary axis. Hence also for this kind of dynamical zeta function some kind of Riemann hypothesis seems to hold.
We introduce certain lattice sums associated with hyperplane arrangements, which are (multiple) sums running over integers, and can be regarded as generalizations of certain linear combinations of zeta-functions of root systems. We also introduce generating functions of special values of those lattice sums, and study their properties by virtue of the theory of convex polytopes. Consequently we evaluate special values of those lattice sums, especially certain special values of zeta-functions of root systems and their affine analogues. In some special cases it is possible to treat sums running o…
We introduce the zeta function of the prehomogenous vector space of binary cubic forms, twisted by the real analytic Eisenstein series. We prove the meromorphic continuation of this zeta function and identify its poles and their residues. We also identify the poles and residues of the zeta function when restricted to irreducible binary cubic forms. This zeta function can be used to prove the equidistribution of the lattice shape of cubic rings.
In the mirror symmetry of Calabi-Yau threefolds, the instanton expansion of the prepotential has a constant term that is a rational multiple of $ζ(3)/(2 πi)^3$, the motivic origin of which has been carefully studied in the author's paper with M. Kim. The Gamma conjecture claims that higher zeta values in fact appear in the theory of Calabi-Yau $n$-folds for $n \geq 4$. Moreover, a natural question is whether they also have a motivic origin. In this paper, we will study the limit mixed Hodge structure (MHS) of the Fermat pencil of Calabi-Yau $n$-folds at the large complex structure limit, which…
We define a generalisation of the completed Riemann zeta function in several complex variables. It satisfies a functional equation, shuffle product identities, and has simple poles along finitely many hyperplanes, with a recursive structure on its residues. The special case of two variables can be written as a partial Mellin transform of a real analytic Eisenstein series, which enables us to relate its values at pairs of positive even points to periods of (simple extensions of symmetric powers of the cohomology of) the CM elliptic curve corresponding to the Gaussian integers. In general, the t…
Given an elliptic curve $E$ defined over $\mathbb{C}$, let $E^{\times}$ be an open subset of $E$ obtained by removing a point. In this paper, we show that the $i$-th Betti number of the unordered configuration space $\mathrm{Conf}^{n}(E^{\times})$ of $n$ points on $E^{\times}$ appears as a coefficient of an explicit rational function in two variables. We also compute its Hodge numbers as coefficients of another explicit rational function in four variables. Our result is interesting because these rational functions resemble the generating function of the $\mathbb{F}_{q}$-point counts of $\mathr…
In this paper we show how to use elementary methods to prove that the volume of Sl_k R / Sl_k Z is zeta(2) * zeta(3) * ... * zeta(k) / k. Using a version of reduction theory presented in this paper, we can compute the volumes of certain unbounded regions in Euclidean space by counting lattice points and then appeal to the machinery of Dirichlet series to get estimates of the growth rate of the number of lattice points appearing in the region as the lattice spacing decreases. We also present a proof of the closely related result that the Tamagawa number of Sl_k Q is 1 that is somewhat simpler a…
We study the interplay between recurrences for zeta related functions at integer values, `Minor Corner Lattice' Toeplitz determinants and integer composition based sums. Our investigations touch on functional identities due to Ramanujan and Grosswald, the transcendence of the zeta function at odd integer values, the Li Criterion for the Riemann Hypothesis and pseudo-characteristic polynomials for zeta related functions. We begin with a recent result for ζ(2s) and some seemingly new Bernoulli relations, which we use to obtain a generalised Ramanujan polynomial and properties thereof.
In this paper, new probability bounds are derived for algebraic lattice codes. This is done by using the Dedekind zeta functions of the algebraic number fields involved in the lattice constructions. In particular, it is shown how to upper bound the error performance of a finite constellation on a Rayleigh fading channel and the probability of an eavesdropper's correct decision in a wiretap channel. As a byproduct, an estimate of the number of elements with a certain algebraic norm within a finite hyper-cube is derived. While this type of estimates have been, to some extent, considered in algeb…
We study universality properties of the Epstein zeta function $E_n(L,s)$ for lattices $L$ of large dimension $n$ and suitable regions of complex numbers $s$. Our main result is that, as $n\to\infty$, $E_n(L,s)$ is universal in the right half of the critical strip as $L$ varies over all $n$-dimensional lattices $L$. The proof uses an approximation result for Dirichlet polynomials together with a recent result on the distribution of lengths of lattice vectors in a random lattice of large dimension and a strong uniform estimate for the error term in the generalized circle problem. Using the same …
