Starting from the classical integral representation of the $ζ(s)$ function introduced by Riemann in 1859, this paper reexamines its analytic symmetry structure. By performing a geometric decomposition of the integral representation, we demonstrate that on the critical line $\Re(s)=\frac{1}{2}$, the value of $ξ(s)$ corresponds strictly to the \textbf{real-part projection} of a specific analytic component. This discovery equivalently transforms the problem of complex zeros into a problem of \textbf{sign evolution} along the real axis.
Based on this geometric framework, we \textbf{construct} an analytic mechanism of \textbf{"Two-End Anchoring, Interval Counting"}: the global argument increment on the region boundary \textbf{anchors} the initial value of the phase function, while the geometric decomposition structure on the critical line \textbf{locks} its final value. This mechanism reveals an \textbf{intrinsic coherence} between global topological constraints and local sign oscillations. Unlike traditional methods that rely on asymptotic estimates (such as the Big $O$ error term), the analysis in this paper is \textbf{grounded in} exact identities. It unveils the \textbf{geometric determinism} underlying the zero-counting formula, offering a novel perspective for analytic number theory independent of asymptotic analysis.
A new interpretation of zeta functions is given for F1-schemes which do not satisfy Soulé's condition. Functional equations for reductive groups are computed and a new definition of zeta functions attached to more general counting functions is given which is based on regularization and puts on an equal footing F1-theory on the one hand and spectral theory of Laplace operators on manifolds on the other.
The aim of our present work here is to present few results in the theory of Mellin transforms using the method that S. Ramanujan used in proving his Master Theorem. Further applications of our results for some number-theoretic functions such as the prime counting function and the zeta function are established.
In this article, we show that $$ \left| N (T) - \frac{T}{ 2 π} \log \left( \frac{T}{2πe}\right) \right| \le 0.1038 \log T + 0.2573 \log\log T + 9.3675 $$ where $N(T)$ denotes the number of non-trivial zeros $ρ$, with $0<\Im(ρ) \le T$, of the Riemann zeta function. This improves the previous result of Trudgian for sufficiently large $T$. The improvement comes from the use of various subconvexity bounds and ideas from the work of Bennett $et$ $al.$ on counting zeros of Dirichlet $L$-functions.
Given a number field $K$ of degree $n_K$ and with absolute discriminant $d_K$, we obtain an explicit bound for the number $N_K(T)$ of non-trivial zeros (counted with multiplicity), with height at most $T$, of the Dedekind zeta function $ζ_K(s)$ of $K$. More precisely, we show that for $T \geq 1$, $$ \Big| N_K (T) - \frac{T}π \log \Big( d_K \Big( \frac{T}{2πe}\Big)^{n_K}\Big)\Big| \le 0.228 (\log d_K + n_K \log T) + 23.108 n_K + 4.520, $$ which improves previous results of Kadiri and Ng, and Trudgian. The improvement is based on ideas from the recent work of Bennett $et$ $al.$ on counting zeros…
In this paper we propose different notions of F_zeta-geometry, for zeta a root of unity, generalizing notions of F_1-geometry (geometry over the "field with one element") based on the behavior of the counting functions of points over finite fields, the Grothendieck class, and the notion of torification. We relate F_zeta-geometry to formal roots of Tate motives, and to functions in the Habiro ring, seen as counting functions of certain ind-varieties. We investigate the existence of F_zeta-structures in examples arising from general linear groups, matrix equations over finite fields, and some qu…
We prove a kind of integral expressions for finite multiple harmonic sums and multiple zeta-star values. Moreover, we introduce a class of multiple integrals, associated with some combinatorial data (called 2-labeled posets). This class includes both multiple zeta and zeta-star values of Euler-Zagier type, and also several other types of multiple zeta values. We show that these integrals can be used to obtain some relations among such zeta values quite transparently.
Nakasuji, Phuksuwan, and Yamasaki defined the Schur multiple zeta values and gave iterated integral expressions of the Schur multiple zeta values of the ribbon type. This paper generalizes their integral expressions to the ones of more general Schur multiple zeta values having constant entries on the diagonals. Furthermore, we also discuss the duality relations for Schur multiple zeta values obtained from the integral expressions.
We introduce an iterated integral version of (generalized) log-sine integrals (iterated log-sine integrals) and prove a relation between a multiple polylogarithm and iterated log-sine integrals. We also give a new method for obtaining relations among multiple zeta values, which uses iterated log-sine integrals, and give alternative proofs of several known results related to multiple zeta values and log-sine integrals.
For the multiple zeta function zeta2(s1,s2) of two variables,we obtain its integral representation(involving product of Hurwitz zeta functions) over the interval [1,infinity),with respect to second variable of Hurwitz zeta function and also obtain a good approximate to it as a function of s1 and s2,for s1>=1 and s2>1.In particular this approximate is explicitly computable,when s1,s2 differ by an even integer and is good,when s2 is large.We treat zeta3(s1,s2,s3) likewise.
In this paper, we prove the universality theorem for the iterated integrals of the logarithm of the Riemann zeta-function on some line parallel to the real axis.
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.
We investigate properties of zeta functions of polynomial rings and their quotients, generalizing and extending some classical results about Dedekind zeta functions of number fields. By an application of Delange's version of the Ikehara Tauberian Theorem, we are then able to determine the asymptotic order of the ideal counting function in such rings. As a result, we produce counting estimates on ideal lattices of bounded determinant coming from fixed number fields, as well as density estimates for any ideal lattices among all sublattices of $\mathbb Z^d$. We conclude with some more general spe…
We compute the Archimedean doubling zeta integrals which appear in the interpolation formulas for the p-adic L-functions of Siegel modular forms, and verify that they agree with the modified Archimedean Euler factors for p-adic interpolation conjectured by Coates--Perrin-Riou.
We consider a smooth counting function of the scaled zeros of the Riemann zeta function, around height T. We show that the first few moments tend to the Gaussian moments, with the exact number depending on the statistic considered.
Two integral representations of q-analogues of the Hurwitz zeta function are established. Each integral representation allows us to obtain an analytic continuation including also a full description of poles and special values at non-positive integers of the q-analogue of the Hurwitz zeta function, and to study the classical limit of this q-analogue. All the discussion developed here is entirely different from the previous work in [4]
Let F be a local non-archimedean field. We prove a formula relating orbital integrals in GL(n,F) (for the unit Hecke function) and the generating series counting ideals of a certain ring. Using this formula, we give an explicit estimate for such orbital integrals. We also derive an analogous formula for global fields, proving analytic properties of the Dedekind zeta function for orders in global fields.
We describe a decomposition algorithm for elliptic multiple zeta values, which amounts to the construction of an injective map $ψ$ from the algebra of elliptic multiple zeta values to a space of iterated Eisenstein integrals. We give many examples of this decomposition, and conclude with a short discussion about the image of $ψ$. It turns out that the failure of surjectivity of $ψ$ is in some sense governed by period polynomials of modular forms.
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 discuss a number of techniques for determining the Minkowski dimension of bounded subsets of some Euclidean space of any dimension, including: the box-counting dimension and equivalent definitions based on various box-counting functions; the similarity dimension via the Moran equation (at least in the case of self-similar sets); the order of the (box-)counting function; the classic result on compact subsets of the real line due to Besicovitch and Taylor, as adapted to the theory of fractal strings; and the abscissae of convergence of new classes of zeta functions. Specifically, we define bo…
This paper gives some results for the logarithm of the Riemann zeta-function and its iterated integrals. We obtain a certain explicit approximation formula for these functions. The formula has some applications, which are related with the value distribution of these functions and a relation between prime numbers and the distribution of zeros in short intervals.
We present many novel results in number theory, including a double series formula for the natural logarithm and a proof concerning the Hölder mean based on the functional equation for the Riemann zeta function. We find a harmonic mean analogue of Chebyshev's inequality for the prime counting function involving the Euler-Mascheroni constant. Furthermore, we define a function taking the Hölder mean of all positive integers up to a given number and investigate its asymptotic behavior, finding two different patterns which are separated by the harmonic mean. Additionally, we discuss the behavior of…
We present a new "integral=series" type identity of multiple zeta values, and show that this is equivalent in a suitable sense to the fundamental theorem of regularization. We conjecture that this identity is enough to describe all linear relations of multiple zeta values over Q. We also establish the regularization theorem for multiple zeta-star values, which too is equivalent to our new identity. A connection to Kawashima's relation is discussed as well.
In this paper,using methods of weight functions and techniques of real analysis, we provide a multidimensional Hilbert-type integral inequality with a homogeneous kernel of degree 0 as well as a best possible constant factor related to the Riemann zeta function. Some equivalent representations and certain reverses are obtained. Furthermore, we also consider operator expressions with the norm and some particular results.
It is shown explicitly how the sign of Hardy's function $Z(t)$ depends on the parity of the zero-counting function $N(T)$. Two existing definitions of this function are analyzed, and some related problems are discussed.
We refine a recent heuristic developed by Keating and the second author. Our improvement leads to a new integral expression for the conjectured asymptotic formula for shifted moments of the Riemann zeta-function. This expression is analogous to a formula, recently discovered by Brad Rodgers and Kannan Soundararajan, for moments of characteristic polynomials of random matrices from the unitary group.
The Legendre type relation for the counting function of ordinary twin primes is reworked in terms of the inverse of the Riemann zeta function. Its analysis sheds light on the distribution of the zeros of the Riemann zeta function in the critical strip and their links to primes and the twin prime problem.
We show that integrals involving log-tangent function, with respect to certain square-integrable functions on $(0, π/2)$, can be evaluated by some series involving the harmonic number. Then we use this result to establish many closed forms relating to the Riemann zeta function at odd positive integers. In addition, we show that the log-tangent integral with respect to the Hurwitz zeta function defines a meromorphic function and that its values depend on the Dirichlet series $ζ_h(s) :=\sum_{n = 1}^\infty h_n n^{-s}$, where $h_n = \sum_{k=1}^n(2k-1)^{-1}$.
Inspired by a famous formula of Ramanujan for odd zeta values, we prove an analogous formula involving the Hurwitz zeta function. We introduce a new integral kernel related to the Hurwitz zeta function, generalizing the integral kernel associated to Ramanujan's identity. We also derive several infinite families of identities analogous to Ramanujan's formula.
In this paper we extend the Zeta function regularization technique, which gives a meaningful solution to divergent power series, in order to assign finite values to divergent integral of certain transcendental functions $f(x)$. The functions are assumed to be analytical and hence they have a convergent Maclaurin series with infinity radius of convergence. Using Maclaurin series and binomial expansion, we equivalently convert the divergent integral to an infinite series in terms of Riemann zeta function. It is shown that the infinite series is convergent under a certain condition and consequent…
We construct an analytic approach to evaluate odd Euler sums, multiple zeta value $ζ(3,2,\ldots,2)$ and multiple $t$-value $t\left(3,2,\ldots,2\right)$. Moreover, we also conjecture a closed expression for multiple $t$-value $t\left(2,\ldots,2,1\right)$.
We have established novel integral representations of the Riemann zeta-function and Dirichlet eta-function based on powers of trigonometric functions and digamma function, and then use these representations to find close forms of Laurent series expansions of these same powers of trigonometric functions and digamma function. The so obtained series can be used to find numerous summation rules for certain values of the Riemann zeta and related functions and numbers, such as e.g. Bernoulli and Euler numbers.
By using the method of iterated integral representations of series, we establish some explicit relationships between multiple zeta values and Integrals of logarithmic functions. As applications of these relations, we show that multiple zeta values of the form \[ζ( {\bar 1,{\left\{ 1 \right\}_{m - 1}},\bar 1,{\left\{ 1 \right\}_{k - 1}}} ),\ (k,m\in \mathbb{N})\] for $m=1$ or $k=1$, and \[ζ( {\bar 1,{\left\{ 1 \right\}_{m - 1}},p,{\left\{ 1 \right\}_{k - 1}}}),\ (k,m\in\mathbb{N})\] for $p=1$ and $2$, satisfy certain recurrence relations which allow us to write them in terms of zeta values, pol…
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 iterated integrals of $\logζ(s)$ on certain vertical and horizontal lines. Here, the function $ζ(s)$ is the Riemann zeta-function. It is a well known open problem whether or not the values of the Riemann zeta-function on the critical line are dense in the complex plane. In this paper, we give a result for the denseness of the values of the iterated integrals on the horizontal lines. By using this result, we obtain the denseness of the values of $\int_{0}^{t} \log ζ(1/2 + it')dt'$ under the Riemann Hypothesis. Moreover, we show that, for any $m\geq 2$, the denseness of the values of…
The double sum sum_(s >= 1) sum_p 1/(p^s log p^s) = 2.00666645... over the inverse of the product of prime powers p^s and their logarithms, is computed to 24 decimal digits. The sum covers all primes p and all integer exponents s>=1. The calculational strategy is adopted from Cohen's work which basically looks at the fraction as the underivative of the Prime Zeta Function, and then evaluates the integral by numerical methods.
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.
In this paper we give two idelic representations of the multiple zeta values - one using iterated integrals over the finite ideles and the other using iterated integrals over the idele class group. Each of the representations leads to a shuffle relation. Thus, we recover in a unified way the two types of shuffle relations of multiple zeta values via the iterated integrals over finite ideles and via iterated integrals over the idele class group.
Mordell-Tornheim multiple zeta-functions, their integral analogues, and relations among multiple polylogarithms
Graduate School of Mathematics, Nagoya University（名古屋大学大学院数学研究科）；Center for General Education, Aichi Institute of Technology（爱知工业大学通识教育中心）；Division of Mathematics, Chiba Institute of Technology（千叶工业大学数学系）；Harish-Chandra Research Institute（哈里什-钱德拉研究所）
We categorify the Hecke L-functions of $\mathrm{GL}(1)$ by replacing the L-functions with "modules of zeta integrals". These modules of zeta integrals are generated by the classical L-function. This approach allows us to categorify questions regarding L-functions, as well as make their construction more canonical by avoiding the GCD procedure usually used to define them. Let $F$ be a number field, and let $\mathfrak{X}(F)$ be the space of Hecke characters on $F$. We define a ring $\mathscr{S}$ of holomorphic functions on $\mathfrak{X}(F)$ and an $\mathscr{S}$-module $\mathscr{L}$ of zeta integ…
Several results are obtained concerning multiplicities of zeros of the Riemann zeta-function $ζ(s)$. They include upper bounds for multiplicities, showing that zeros with large multiplicities have to lie to the left of the line $σ= 1$. A zero-density counting function involving multiplicities is also discussed.
A Unified Approach to Formal Arakawa-Kaneko Type Zeta Values via Iterated Integrals
College of Teacher Education, Quzhou University（衢州学院教师教育学院）；School of Mathematics and Statistics, Anhui Normal University（安徽师范大学数学与统计学院）；Department of Mathematics, The Bishop’s School（圣公会中学数学系）
It is known that multiple zeta values can be written in terms of certain iterated log-sine integrals. Conversely, we evaluate iterated log-sine integrals in terms of multiple polylogarithms and multiple zeta values in this paper. We also suggest some conjectures on multiple zeta values, multiple Clausen values, multiple Glaisher values and iterated log-sine integrals.
This paper is a continuation of our recent paper with the same title, arXiv:0806.1596v1 [math.NT], where a number of integral equalities involving integrals of the logarithm of the Riemann zeta-function were introduced and it was shown that some of them are equivalent to the Riemann hypothesis. A few new equalities of this type are established; contrary to the preceding paper the focus now is on integrals involving the argument of the Riemann zeta-function (imaginary part of logarithm) rather than the logarithm of its module (real part of logarithm). Preliminary results of the numerical resear…
The functional equation for nonarchimedean Rankin-Selberg local Euler factors was proved by Jacquet, Piatetski-Shapiro, and Shalika in 1983. In this expository note we translate the original proof into the purely functorial language of parabolic induction-restriction of Bernstein-Zelevinsky. This new language gives a clearer presentation of the ideas, and works over arbitrary fields with characteristic not equal to the residue characteristic.
We give a survey of Denef's rationality theorem on $p$-adic integrals, its uniform in $p$ versions, the relevant model theory, and a number of applications to counting subgroups of finitely generated nilpotent groups and conjugacy classes in congruence quotients of Chevalley groups over rings of integers of local fields. We then state results on analytic properties of Euler products of such $p$-adic integrals over all $p$, and an application to counting conjugacy classes in congruence quotients of certain algebraic groups over the rationals. We then briefly discuss zeta functions arising from …
In this paper we consider iterated integrals on $\mathbb{P}^{1}\setminus\{0,1,\infty,z\}$ and define a class of $\mathbb{Q}$-linear relations among them, which arises from the differential structure of the iterated integrals with respect to $z$. We then define a new class of $\mathbb{Q}$-linear relations among the multiple zeta values by taking their limits of $z\rightarrow1$, which we call \emph{confluence relations} (i.e., the relations obtained by the confluence of two punctured points). One of the significance of the confluence relations is that it gives a rich family and seems to exhaust …
