We give a new and very concise proof of the existence of a holomorphic continuation for a large class of twisted multivariable zeta functions. To do this, we use a simple method of "decalage" that avoids using an integral representation of the zeta function. This allows us to derive explicit recurrence {\it relations} between the values at $T-$tuples of negative integers. This also extends some earlier results of several authors where the underlying polynomials were products of linear forms.
Motivated by recent work of Deitmar-Koyama-Kurokawa, Kurokawa-Ochiai, Connes-Consani, and the author, we define some multivariable deformed zeta functions of Hurwitz-Igusa type for a Noetherian $\F_1$-scheme $X$ in the sense of Connes-Consani. Our zeta functions generalize both the zeta functions studied by Deitmar-Koyama-Kurokawa, Kurokawa-Ochiai, and the log derivative of the modified Soulé type zeta function Connes-Consani. We give an explicit presentation for these zeta functions using the Hurwitz zeta functions, and so, we can derive its meromorphicity. When restricted to the log derivati…
We consider twisted zeta series of several variables associated to polynomials of several variables. Thanks to a totally new method (exchange lemma) we calculate the values at vectors formed of negative integers.After transformation of the formula we obtained, we make p-adic interpolation. ----- Nous considerons des series zetas multivariables associees a des polynomes de plusieurs variables. Grace a une methode totalement nouvelle (dite du lemme d'echange) nous calculons les valeurs aux T-uplets d'entiers negatifs. Apres transformation des formules ainsi obtenues nous faisons l'interpolation …
This article extends classical one variable results about Euler products defined by integral valued polynomial or analytic functions to several variables. We show there exists a meromorphic continuation up to a presumed natural boundary, and also give a criterion, a la Estermann-Dahlquist, for the existence of a meromorphic extension to $\C^n.$ Among applications we deduce analytic properties of height zeta functions for toric varieties over $\Q$ and group zeta functions.
We use multiple zeta functions to prove, under suitable assumptions, precise asymptotic formulas for the averages of multivariable multiplicative functions. As applications, we prove some conjectures on the average number of cyclic subgroups of the group ${\mathbb Z}_{m_1}\times\dots\times {\mathbb Z}_{m_n}$ and multivariable averages associated with the LCM function.
We conjecture average counting functions for prime $k$-tuples based on a gamma distribution hypothesis for prime powers. The conjecture is closely related to the Hardy-Littlewood conjecture for $k$-tuples but yields better estimates. Possessing average counting functions along with their corresponding exact counting functions allows to implicitly define pertinent $k$-tuple zeta functions. The $k$-tuple zeta functions in turn allow construction of $k$-tuple analogs of explicit formulae. If the zeros of the (implicitly defined) $k$-tuple zeta can be determined, the explicit formulae should yield…
Finite Euler product is known to be one of the classical zeta functions in number theory. In [1], [2] and [3], we have introduced some multivariable zeta functions and studied their definable probability distributions on R^d. They include functions which generate infinitely divisible, not infinitely divisible characteristic functions and not even to be characteristic functions. In this paper, we treat some multivariable finite Euler products and show how they behave in view of such properties.
Exact summatory functions that count the number of prime $k$-tuples up to some cut-off integer are presented. Related summatory $k$-tuple analogs of the first and second Chebyshev functions are then defined. Using a gamma distribution hypothesis for prime powers, associated average summatory functions are conjectured. With exact and average summatory functions in hand, pertinent $k$-tuple zeta functions can be identified, and Perron's formula allows the formulation of $k$-tuple analogs of explicit formulae. The $k$-tuple zeta functions are then used to make some inferences about $k$-tuple prim…
We consider a generalization of the Mahler measure of a multivariable polynomial $P$ as the integral of $\log^k|P|$ in the unit torus, as opposed to the classical definition with the integral of $\log|P|$. A zeta Mahler measure, involving the integral of $|P|^s$, is also considered. Specific examples are computed, yielding special values of zeta functions, Dirichlet $L$-functions, and polylogarithms.
We prove and conjecture several relations between multizeta values for $\mathbb{F}_q[t]$, focusing on zeta-like values, namely those whose ratio with the zeta value of the same weight is rational (or equivalently algebraic). In particular, we describe them conjecturally fully for $q=2$, or more generally for any $q$ for `even' weight (`eulerian' tuples). We provide some data in support of the guesses.
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 an asymptotic formula for the number of sublattices $Λ\subseteq \mathbb{Z}^d$ of index at most $X$ for which $\mathbb{Z}^d/Λ$ has rank at most $m$, answering a question of Nguyen and Shparlinski. We compare this result to recent work of Stanley and Wang on Smith Normal Forms of random integral matrices and discuss connections to the Cohen-Lenstra heuristics. Our arguments are based on Petrogradsky's formulas for the cotype zeta function of $\mathbb{Z}^d$, a multivariable generalization of the subgroup growth zeta function of $\mathbb{Z}^d$.
This paper deals with a multiple version of zeta- and L-functions both in the complex case and in the p-adic case: [I] Our motivation in the complex case is to find suitable rigorous meaning of the values of multivariable multiple zeta-functions (MZFs) at non-positive integer points. (a) We reveal that MZFs turn to be entire on the whole space after taking the desingularization. Further we show that the desingularized function is given by a suitable finite linear combination of MZFs with some arguments shifted. It is also shown that specific combinations of Bernoulli numbers attain the special…
Let $p,x$ be real numbers, and $s$ be a complex number, with $\Re(s)>1-r$, $p\geq 1$, and $x+1>0$. The zeta function $Z^{\bfα}_p(s;x)$ is defined by $$ Z^{\bfα}_p(s;x) =\frac{1}{Γ(s)}\int^\infty_0 \frac{e^{-xt}} {e^t-1}\,Li_{\bfα}\left(\frac{1-e^{-t}}p\right) t^{s-1}\,dt, $$ where ${\bfα}=(α_1,\ldots,α_r)$ is a $r$-tuple positive integers, and $Li_{\bfα}(z)$ is the one-variable multiple polylogarithms. Since $Z^{\bfα}_1(s;0)=ξ(\bfα;s)$, we call this function as a generalized Arakawa-Kaneko zeta function. In this paper, we investigate the properties and values of $Z^{\bfα}_p(s;x)$ with differen…
Thakur (2010) showed that, for $r,$ $s\in \mathbb{N}$, a product of two Carlitz zeta values $ζ_A(r)$ and $ζ_A(s)$ can be expressed as an $\mathbb{F}_p$-linear combination of $ζ_A(r+s)$ and double zeta values of weight $r+s$. Such an expression is called shuffle relation by Thakur. Fixing $r,$ $s\in \mathbb{N}$, we construct a $t$-module $E'$. To determine whether an $(r+s)$-tuple $\mathfrak{C}$ in $\mathbb{F}_q(θ)^{r+s}$ gives a shuffle relation, we relate it to the $\mathbb{F}_q[t]$-torsion property of the point $\mathbf{v}_\mathfrak{C}\in E'(\mathbb{F}_q[θ])$ constructed with respect to the …
Let $k\in\mathbb{N}$ and let $f_1,\ldots,f_k$ belong to a Hardy field. We prove that under some natural conditions on the $k$-tuple $(f_1,\ldots,f_k)$ the density of the set $$ \big\{n\in \mathbb{N}: \text{gcd}(n,\lfloor f_1(n)\rfloor,\ldots,\lfloor f_k(n)\rfloor)=1\big\} $$ exists and equals $\frac{1}{ζ(k+1)}$, where $ζ$ is the Riemann zeta function.
For a pair of positive integers $n,k$ with $n\geq 2$, in this paper we prove that $$ \sum_{r=1}^k\sum_{|\bfα|=k}{k\choose\bfα} ζ(n\bfα)=ζ(n)^k =\sum^k_{r=1}\sum_{|\bfα|=k} {k\choose\bfα}(-1)^{k-r}ζ^\star(n\bfα), $$ where $\bfα=(α_1,α_2,\ldots,α_r)$ is a $r$-tuple of positive integers. Moreover, we give an application to combinatorics and get the following identity: $$ \sum^{2k}_{r=1}r!{2k\brace r}=\sum^k_{p=1}\sum^k_{q=1}{k\brace p}{k\brace q} p!q!D(p,q), $$ where ${k\brace p}$ is the Stirling numbers of the second kind and $D(p,q)$ is the Delannoy number.
Let $Λ\left(n\right)$ be the Von Mangoldt function, let \[ r_{G}\left(n\right)=\underset{{\scriptstyle m_{1}+m_{2}=n}}{\sum_{m_{1},m_{2}\leq n}}Λ\left(m_{1}\right)Λ\left(m_{2}\right), \] \[ r_{PT}\left(N,h\right)=\sum_{n=0}^{N}Λ\left(n\right)Λ\left(n+h\right),\,h\in\mathbb{N} \] be the counting function of the Goldbach numbers and the counting function of the prime tuples, respectively. Let $N>2$ be an integer. We will find the explicit formulae for the averages of $r_{G}\left(n\right)$ and $r_{PT}\left(N,h\right)$ in terms of elementary functions, the incomplete Beta function $B_{z}\left(a,b\…
This work deals with a new generalization of $r$-Stirling numbers using $l$-tuple of permutations and partitions called $(l,r)$-Stirling numbers of both kinds. We study various properties of these numbers using combinatorial interpretations and symmetric functions. Also, we give a limit representation of the multiple zeta function using $(l,r)$-Stirling of the first kind.
A difference equation analogue of the Knizhnik-Zamolodchikov equation is exhibited by developing a theory of the generating function $H(z)$ of Hurwitz polyzeta functions to parallel that of the polylogarithms. By emulating the role of the KZ equation as a connection on a suitable bundle, a difference equation version of the notion of connection is developed for which $H(z)$ is a flat section. Solving a family of difference equations satisfied by the Hurwitz polyzetas leads to the normalized multiple Bernoulli polynomials (NMBPs) as the counterpart to the Hurwitz polyzeta functions, at tuples o…
For each $(m+1)$-tuple ${\bf n}_m=(n_0,n_1,\ldots,n_m)$ of positive integers, the ${\bf n}_m$-derived zeta function $\widehatζ_{X,\mathbb F_q}^{\,({\bf n}_m)}(s)$ is defined for a curve $X$ over $\mathbb F_q$. This derived zeta function satisfies standard zeta properties. In particular, similar to the Artin Zeta function of $X/\mathbb F_q$, this ${\bf n}_m$-derived Zeta function of $X$ over $\mathbb F_q$ is a ratio of a degree $2g$ polynomial $P_{X,\mathbb F_q}^{({\bf n}_m)}$ in $T_{{\bf n}_m}=q^{-s\prod_{k=0}^mn_k}$ by $(1-T_{{\bf n}_m})(1-q_{{\bf n}_m}T_{{\bf n}_m})T_{{\bf n}_m}^{g-1}$ with …
