In analogy with values of the classical Euler Gamma-function at rational numbers and the Riemann zeta-function at positive integers, we consider Thakur's geometric Gamma-function evaluated at rational arguments and Carlitz zeta-values at positive integers. We prove that, when considered together, all of the algebraic relations among these special values arise from the standard functional equations of the Gamma-function and from the Euler-Carlitz relations and Frobenius p-th power relations of the zeta-function.
An analysis of the zeta and gamma function is presented, using elementary functions like [] and {}, a general formula for the angle of zeta(1/2 + i*n) is found and the same for the gamma function.
Using three basic facts concerning Hurwitz zeta function,we give new natural proofs of the known results on Bernoulli polynomials,gamma function and also obtain Gauss' expression for Psi function at a rational point,all in a unified fashion.We also give a new proof of the relation between log gamma and derivative of the Hurwitz zeta function,including that of Stirling's expression for log gamma.
The following theorem is proven: Both real and imaginary parts of the function F(s) defined as F(s)=zeta(s)*Gamma(s/2)*pi**(-s/2)=xi(s)/(s*(s-1)), and whose zeroes exactly coincide with the non-trivial zeroes of the Riemann zeta-function, have infinitely many zeroes for any value of Re(s).
We consider the values at proper fractions of the arithmetic gamma function and the values at positive integers of the zeta function for F_q[theta] and provide complete algebraic independence results for them.
We study absolute zeta functions from the view point of a canonical normalization. We introduce the absolute Hurwitz zeta function for the normalization. In particular, we show that the theory of multiple gamma and sine functions gives good normalizations in cases related to the Kurokawa tensor product. In these cases, the functional equation of the absolute zeta function turns out to be equivalent to the simplicity of the associated non-classical multiple sine function of negative degree.
Using the Jackson integral, we obtain the $q$-integral analogue of the Raabe type formulas for Barnes multiple Hurwitz-Lerch zeta functions and Barnes and Vardi's multiple gamma functions. Our results generalize $q$-integral analogue of the Raabe type formulas for the Hurwitz zeta functions and log gamma functions in [N. Kurokawa, K. Mimachi, and M. Wakayama, Jackson's integral of the Hurwitz zeta function, Rend. Circ. Mat. Palermo (2) 56 (2007), no. 1, 43--56]. During the proof we also obtain a new formula on the relationship between the higher and lower orders Hurwitz zeta functions.
The special uniformity of zeta functions claims that pure non-abelian zeta functions coincide with group zeta functions associated to the special linear groups. Naturally associated are three aspects, namely, the analytic, arithmetic, and geometric aspects. In the first paper of this series, we expose intrinsic geometric structures of our zetas by counting semi-stable bundles on curves defined over finite fields in terms of their automorphism groups and global sections. We show that such a counting maybe read from Artin zetas which are abelian in nature. This paper also contains an appendix wr…
We derive several identities for the Hurwitz and Riemann zeta functions, the Gamma function, and Dirichlet $L$-functions. They involve a sequence of polynomials $α_k(s)$ whose study was initiated in an earlier paper. The expansions given here are practical and can be used for the high precision evaluation of these functions, and for deriving formulas for special values. We also present a summation formula and use it to generalize a formula of Hasse.
In 'The Lost Notebook and Other Unpublished Papers' of Ramanujan are present some manuscripts of Ramanujan in the handwriting of G. N. Watson which are 'copied from loose papers'. We present a proof of a beautiful formula of Ramanujan in one of these manuscripts, namely a transformation formula involving the Gamma function and Riemann Zeta function. This formula elegantly yields a modular relation. Later we also give another proof of the first equality of this transformation formula along the lines suggested by Andrew P. Guinand.
We show that coefficients in the Laurent series of Igusa Zeta functions are periods. This will be used in a subsequent paper (by P. Brosnan) to show that certain numbers occurring in study of Feynman amplitudes (upto gamma factors) are periods.
The purpose of this paper is to investigate coefficient matrices of functional equations of zeta functions associated with homogeneous cones, which are given explicitly in the previous paper, in detail. We prove that the coefficient matrix can be decomposed into variable-wise matrices regardless of the choice of homogeneous cones. Moreover, under a suitable condition, we show that the associated zeta functions admit a kind of completion forms.
As a generalization of the results [KW3],we study the functional equation of the higher Selberg zeta function for congruence subgroups. To obtain the gamma factor of this function, we introduce a higher Dirichlet $L$-function. Then we determine the gamma factor explicitly in terms of the Barnes triple gamma function and the higher Dirichlet $L$-function.
By a similar idea for the construction of Milnor's gamma functions, we introduce "higher depth determinants" of the Laplacian on a compact Riemann surface of genus greater than one. We prove that, as a generalization of the determinant expression of the Selberg zeta function, this higher depth determinant can be expressed as a product of multiple gamma functions and what we call a Milnor-Selberg zeta function. It is shown that the Milnor-Selberg zeta function admits an analytic continuation, a functional equation and, remarkably, has an Euler product.
We study a relation between two refinements of the rank one abelian Gross-Stark conjecture: For a suitable abelian extension $H/F$ of number fields, a Gross-Stark unit is defined as a $p$-unit of $H$ satisfying some proporties. Let $τ\in \mathrm{Gal}(H/F)$. Yoshida and the author constructed the symbol $Y_p(τ)$ by using $p$-adic $\log$ multiple gamma functions, and conjectured that the $\log_p$ of a Gross-Stark unit can be expressed by $Y_p(τ)$. Dasgupta constructed the symbol $u_T(τ)$ by using the $p$-adic multiplicative integration, and conjectured that a Gross-Stark unit can be expressed by…
The higher rank Lefschetz formula for p-adic groups is used to prove rationality of a several-variable zeta function attached to the action of a p-adic group on its Bruhat-Tits building. By specializing to certain lines one gets one-variable zeta functions, which then can be related to geometrically defined zeta functions
Suppose that $ω_1,\ldots,ω_N$ are positive real numbers and $x$ is a complex number with positive real part. The multiple Barnes-Euler zeta function $ζ_{E,N}(s,x;\barω)$ with parameter vector $\barω=(ω_1,\ldots,ω_N)$ is defined as a deformation of the Barnes multiple zeta function as follows $$ ζ_{E,N}(s,x;\barω)=\sum_{t_1=0}^\infty\cdots\sum_{t_N=0}^\infty \frac{(-1)^{t_1+\cdots+t_N}}{(x+ω_1t_1+\cdots+ω_Nt_N)^s}. $$ In this paper, based on the fermionic $p$-adic integral, we define the $p$-adic analogue of multiple Barnes-Euler zeta function $ζ_{E,N}(s,x;\barω)$ which we denote by $ζ_{p,E,N}(…
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…
We provide several properties of the geometric polynomials discussed in earlier works of the authors. Further, the geometric polynomials are used to obtain a closed form evaluation of certain series involving Riemann's zeta function.
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…
Dating back to Euler, in classical analysis and number theory, the Hurwitz zeta function $$ ζ(z,q)=\sum_{n=0}^{\infty}\frac{1}{(n+q)^{z}}, $$ the Riemann zeta function $ζ(z)$, the generalized Stieltjes constants $γ_k(q)$, the Euler constant $γ$, Euler's gamma function $Γ(q)$ and the digamma function $ψ(q)$ have many close connections on their definitions and properties. There are also many integrals, series or infinite product representations of them along the history. In this note, we try to provide a parallel story for the alternating Hurwitz zeta function (also known as the Hurwitz-type Eul…
We prove that for a compact locally symmetric Riemannian space $M$ of rank 1 there exist infinitely many automorphic Tate motives $f$ such that the generalized Selberg zeta function $Z_{M(f)}(s)$ satisfies a simple functional equation in the sense that it has no gamma factors.
In one of his papers, using arguments about l-adic representations, Taniyama expresses the zeta function of an abelian variety over a number field as an infinite product of modified Artin L-functions. The latter can be further decomposed as products of modified Dedekind zeta functions. After recalling Taniyama's work, we give a simple geometric proof of the resulting product formula for abelian and more general group schemes.
We show that the motivic zeta functions of smooth, geometrically connected curves with no rational points are rational functions. This was previously known only for curves whose smooth projective models have a rational point on each connected component. In the course of the proof we study the class of a Severi-Brauer scheme over a general base in the Grothendieck ring of varieties.
In this paper, we give Hurwitz-Lerch zeta distributions with $0 < σ\ne 1$ by using the Gamma function. Moreover, we define Hurwitz-Lerch type of Euler-Zagier double zeta distributions not only in the region of absolute convergence but also the outside of the region of absolute convergence.
We have used the first 2600 nontrivial zeros gamma_l of the Riemann zeta function calculated with 1000 digits accuracy and developed them into the continued fractions. We calculated the geometrical means of the denominators of these continued fractions and for all cases we get values close to the Khinchin's constant, what suggests that gamma_l are irrational. Next we have calculated the n-th square roots of the denominators q_n of the convergents of the continued fractions obtaining values close to the Khinchin-Levy constant, again supporting the common believe that gamma_l are irrational.
We use a method, first developed for the Riemann zeta-function by Masser in ["Rational values of the Riemann zeta function", Journ. Num. Th. 131 (2011), 2037-2046], to prove a new zero estimate for polynomials in z and 1/Gamma(z). This allows us to prove that, for all n>=2, there exists an absolute effective positive constant C(n) such that, for all D>=3, there are at most C(n)log^2(D)/loglog(D) rational numbers z in [n-1,n] with denominator at most D and such that Gamma(z) is also rational with denominator at most D.
Let $x$ be a complex number which has a positive real part, and $w_1,\ldots,w_N$ be positive rational numbers. We show that $w^s ζ_N (s, x \ |\ w_1,\ldots, w_N)$ can be expressed as a finite linear combination of the Hurwitz zeta functions over $\mathbb Q(x)$, where $ζ_N (s,x \ |\ w_1,\ldots, w_N)$ is the Barnes zeta function and $w$ is a positive rational number explicitly determined by $w_1,\ldots, w_N$. Furthermore, we give generalizations of Kummer's formula on the gamma function and Koyama-Kurokawa's formulae on the multiple gamma functions, and an explicit formula for the values at non-p…
The Ohno relation for multiple zeta values can be formulated as saying that a certain operator, defined for indices, is invariant under taking duals. In this paper, we generalize the Ohno relation to regularized multiple zeta values by showing that, although the suitably generalized operator is not invariant under taking duals, the relation between its values at an index and at its dual index can be written explicitly in terms of the gamma function.
As well known, the study of Riemanns zeta function ζ(s) involves the related entire function ξ(s). A close relative of ζ(s) is the alternating zeta function η(s). Similar to ζ(s), also η(s) has a corresponding entire function ξ_a (s). After establishing its definition and a related functional equation, formulas based on incomplete gamma functions are worked out, allowing to numerically evaluate ξ_a (s). Examples verifying the obtained formulas are included.
A variation of multiple $L$-values, which arises from the description of the special values of the spectral zeta function of the non-commutative harmonic oscillator, is introduced. In some special cases, we show that its generating function can be written in terms of the gamma functions. This result enables us to obtain explicit evaluations of them.
The paper deals with the analytic entire function Chi(s) closely related to Riemann Zeta Function Zeta(s). A formula is obtained for Chi(s) essentially within the so-called critical strip. This is achieved by applying Cauchy integral formula to an infinite loop encircling the critical strip. In the obtained formula a remarkable role is played by a special type of complex function known as Incomplete Gamma Function. Numerical examples verifying the obtained formula are included.
We compute in a direct (not algorithmic) way the zeta function of all supersingular curves of genus 2 over a finite field k, with many geometric automorphisms. We display these computations in an appendix where we select a family of representatives of all these curves up to geometric isomorphism and we exhibit equations and the zeta function of all their twists. As an application we obtain a direct computation of the cryptographic exponent of the Jacobians of these curves.
Let X be a complete, geometrically irreducible, singular, algebraic curve defined over a field of characteristic p big enough. Given a local ring O_{P,X} at a rational singular point P of X, we attached a universal zeta function which is a rational function and admits a functional equation if O_{P,X} is Gorenstein. This universal zeta function specializes to other known zeta functions and Poincare series attached to singular points of algebraic curves. In particular, for the local ring attached to a complex analytic function in two variables, our universal zeta function specializes to the gene…
In this paper, we study some Euler-Apéry-type series which involve central binomial coefficients and (generalized) harmonic numbers. In particular, we establish elegant explicit formulas of some series by iterated integrals and alternating multiple zeta values. Based on these formulas, we further show that some other series are reducible to ln(2), zeta values, and alternating multiple zeta values by considering the contour integrals related to gamma functions, polygamma functions and trigonometric functions. The evaluations of a large number of special Euler-Apéry-type series are presented as …
We compute integral moments of partial sums of the Riemann zeta function on the critical line and obtain an expression for the leading coefficient as a product of the standard arithmetic factor and a geometric factor. The geometric factor is equal to the volume of the convex polytope of substochastic matrices and is equal to the leading coefficient in the expression for moments of truncated characteristic polynomial of a random unitary matrix.
We introduce a new type of multiple zeta functions, which we call bilateral zeta functions, analogous to the Barnes zeta functions. The bilateral zeta function is a periodic function and shares certain basic properties of Barnes zeta function. Especially, we prove that the bilateral zeta function has a nice Fourier series expansion and the Barnes zeta function can be expressed as a finite sum of bilateral zeta functions. By these properties of the bilateral zeta functions, We obtain simple proofs of some formulas, for example the reflection formula for the multiple gamma function, the inversio…
We develop techniques for computing zeta functions associated with nilpotent groups, not necessarily associative algebras, and modules, as well as Igusa-type zeta functions. At the heart of our method lies an explicit convex-geometric formula for a class of $p$-adic integrals under non-degeneracy conditions with respect to associated Newton polytopes. Our techniques prove to be especially useful for the computation of topological zeta functions associated with algebras, resulting in the first systematic investigation of their properties.
In this paper, we evaluate archimedean zeta integrals for automorphic $L$-functions on $GL_n \times GL_{n-1+\ell}$ and on $ SO_{2n+1} \times GL_{n+\ell}$, for $\ell=-1$, $0$, and $1$. In each of these cases, the zeta integrals in question may be expressed as Mellin transforms of products of class one Whittaker functions. Here, we obtain explicit expressions for these Mellin transforms in terms of Gamma functions and Barnes integrals.
In this paper we prove that there exists a Zariski dense open subset U defined over the rationals Q in the space of all one-variable rational functions with arbitrary k poles of prescribed orders, such that for every geometric point f in U(Qbar)$, the L-function of the exponential sum of f at a prime p has Newton polygon approaching the Hodge polygon as p approaches infinity. As an application to algebraic geometry, we prove that the p-adic Newton polygon of the zeta function of a p-cover of the projective line totally ramified at arbitrary k points of prescribed orders has an asymptotic gener…
We provide new representations for the finite parts at the poles and the derivative at zero of the Barnes zeta function in any dimension in the general case. These representations are in the forms of series and limits. We also give an integral representation for the finite parts at the poles. Similar results are derived for an associated function, which we term homogeneous Barnes zeta function. Our expressions immediately yield analogous representations for the logarithm of the Barnes Gamma function, including the particular case also known as multiple Gamma function.
We introduce and study "elliptic zeta values", a two-parameter deformation of the values of Riemann's zeta function at positive integers. They are essentially Taylor coefficients of the logarithm of the elliptic gamma function, and share the SL(3,Z) modular properties of this function. Elliptic zeta values at even integers are related to Eisenstein series and thus to sums of odd powers of divisors. The elliptic zeta values at odd integers can be expressed in terms of generating series of sums of even powers of divisors.
There are only aleph-zero rational numbers, while there are 2 to the power aleph-zero real numbers. Hence the probability that a randomly chosen real number would be rational is 0. Yet proving rigorously that any specific, natural, real constant, is irrational is usually very hard, witness that there are still no proofs of the irrationality of the Euler-Mascheroni constant, the Catalan constant, or Zeta(5). Inspired by Frits Beukers' elegant rendition of Apery's seminal proofs of the irrationality of Zeta(2) and Zeta(3), and heavily using algorithmic proof theory, we systematically searched fo…
In this paper we provide a new series representation for the values of Riemann zeta function at integer arguments, namely: $ ζ(m)=\sum_{n=1}^{\infty}\frac{m(-1)^{n-1}Γ(1-ω_{m}n)...Γ(1-ω_{m}^{m-1}n)}{n!n^m}$, where $n$ is an integer that lager than $1$ and $ω$ is the $m$-th root of unity. This series converges quite fast. It's derived by some technique of infinite partial fraction decomposition. With this technique we also establish other useful formulas related to gamma function.
I use the numerical values of the generalised Glaisher-Kinkelin-Bendersky (GKB) constants to give numerical values for the derivatives of the Hurwitz zeta function at negative integers, rather than the other way round. I point out that both Glaisher's numerical approach and Bendersky's recursion for the generalised gamma function were anticipated by Jeffery in 1862 who gave the value of the second constant as an example. I therefore propose that GKB become GKBJ.
In this article, we give a proof of the link between the zeta function of two families of hypergeometric curves and the zeta function of a family of quintics that was observed numerically by Candelas, de la Ossa, and Rodriguez Villegas. The method we use is based on formulas of Koblitz and various Gauss sums identities; it does not give any geometric information on the link.
We give a representation of the classical Riemann $ζ$-function in the half plane $\Re s>0$ in terms of a Mellin transform involving the real part of the dilogarithm function with an argument on the unit circle (associated Clausen $Gl_2$-function). We also derive corresponding representations involving the derivatives of the $Gl_2$-function. A generalized symmetrized Müntz-type formula is also derived. For a special choice of test functions it connects to our integral representation of the $ζ$-function, providing also a computation of a concrete Mellin transform. Certain formulae involving seri…
The analogue of the Bruhat-Tits building of a p-adic group in F1-geometry is a single apartment. In this setting, the trace formula gives rise to a several variable zeta function analogously to the p-adic case. The analogy carries on to the fact that the restriction to certain lines yield zeta functions which are defined in geometrical terms. Also, the classical formula of Ihara has an analogue in this setting.
Let $E$ be a quadratic semisimple extension of a local field $F$ of characteristic zero. We determine explicit relation between gamma factors for Asai representations of $R_{E/F}{\rm GL}_{2/E}$ defined by the Weil-Deligne representations and local zeta integrals. When $E = F\times F$, the results were due to Henniart and Jacquet. We completed the theory in this article based on explicit calculation.
In these notes, we explore possible stable properties for the zeta function of a geometric Zp-tower of curves over a finite field of characteristic p, in the spirit of Iwasawa theory. A number of fundamental questions and conjectures are proposed for those Zp towers coming from algebraic geometry.
