2,946 results on '"*ZETA functions"'
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2. Absolute zeta functions arising from ceiling and floor Puiseux polynomials.
- Author
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Hirakawa, Yoshinosuke and Tomita, Takuki
- Abstract
For the ℤ-lift Xℤ of a monoid scheme X of finite type, Deitmar
et al. calculated its absolute zeta function by interpolating #Xℤ(픽q) for all prime powers q using the Fourier expansion. This absolute zeta function coincides with the absolute zeta function of a certain polynomial. In this paper, we characterize the polynomial as a ceiling polynomial of the sequence (#Xℤ(픽q))q, which we introduce independently. Extending this idea, we introduce a certain pair of absolute zeta functions of a separated scheme X of finite type over ℚ by means of a pair of Puiseux polynomials which estimate “#X(픽pm)” for sufficiently large p. We call them the ceiling and floor Puiseux polynomials of X. In particular, if X is an elliptic curve, then our absolute zeta functions of X do not depend on its isogeny class. [ABSTRACT FROM AUTHOR]- Published
- 2024
- Full Text
- View/download PDF
3. Spectral asymptotics of elliptic operators on manifolds.
- Author
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Avramidi, Ivan G.
- Subjects
- *
DIFFERENTIAL invariants , *ELLIPTIC operators , *PARTIAL differential operators , *QUANTUM field theory , *MATHEMATICAL physics , *DIFFERENTIAL operators , *ZETA functions - Abstract
The study of spectral properties of natural geometric elliptic partial differential operators acting on smooth sections of vector bundles over Riemannian manifolds is a central theme in global analysis, differential geometry and mathematical physics. Instead of studying the spectrum of a differential operator L directly one usually studies its spectral functions, that is, spectral traces of some functions of the operator, such as the spectral zeta function ζ (s) = Tr L − s and the heat trace Θ (t) = Tr exp (− t L). The kernel U (t ; x , x ′) of the heat semigroup exp (− t L) , called the heat kernel, plays a major role in quantum field theory and quantum gravity, index theorems, non-commutative geometry, integrable systems and financial mathematics. We review some recent progress in the study of spectral asymptotics. We study more general spectral functions, such as Tr f (t L) , that we call quantum heat traces. Also, we define new invariants of differential operators that depend not only on the eigenvalues but also on the eigenfunctions, and, therefore, contain much more information about the geometry of the manifold. Furthermore, we study some new invariants, such as Tr exp (− t L +) exp (− s L −) , that contain relative spectral information of two differential operators. Finally, we show how the convolution of the semigroups of two different operators can be computed by using purely algebraic methods. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
4. D-brane Masses at Special Fibres of Hypergeometric Families of Calabi–Yau Threefolds, Modular Forms, and Periods.
- Author
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Bönisch, Kilian, Klemm, Albrecht, Scheidegger, Emanuel, and Zagier, Don
- Subjects
- *
MODULAR forms , *QUINTIC equations , *ZETA functions , *FINITE fields , *MIRROR symmetry , *D-branes , *EIGENVALUES - Abstract
We consider the fourteen families W of Calabi–Yau threefolds with one complex structure parameter and Picard–Fuchs equation of hypergeometric type, like the mirror of the quintic in P 4 . Mirror symmetry identifies the masses of even-dimensional D-branes of the mirror Calabi–Yau M with four periods of the holomorphic (3, 0)-form over a symplectic basis of H 3 (W , Z) . It was discovered by Chad Schoen that the singular fiber at the conifold of the quintic gives rise to a Hecke eigenform of weight four under Γ 0 (25) , whose Hecke eigenvalues are determined by the Hasse–Weil zeta function which can be obtained by counting points of that fiber over finite fields. Similar features are known for the thirteen other cases. In two cases we further find special regular points, so called rank two attractor points, where the Hasse–Weil zeta function gives rise to modular forms of weight four and two. We numerically identify entries of the period matrix at these special fibers as periods and quasiperiods of the associated modular forms. In one case we prove this by constructing a correspondence between the conifold fiber and a Kuga–Sato variety. We also comment on simpler applications to local Calabi–Yau threefolds. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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5. Heat coefficients for magnetic Laplacians on the complex projective space Pn(ℂ).
- Author
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Ahbli, K., Hafoud, A., and Mouayn, Z.
- Subjects
- *
BERNOULLI numbers , *ZETA functions , *BERNOULLI polynomials , *TRACE formulas , *THETA functions , *PROJECTIVE spaces - Abstract
We denote by $ \Delta _\nu $ Δ ν the Fubini-Study Laplacian perturbed by a uniform magnetic field whose strength is proportional to ν. When acting on bounded functions on the complex projective n-space, this operator has a discrete spectrum consisting on eigenvalues $ \beta _m, \ m\in \mathbb {Z}_+ $ β m , m ∈ Z + . For the corresponding eigenspaces, we give a new proof for their reproducing kernels by using Zaremba's expansion directly. These kernels are then used to obtain an integral representation for the heat kernel of $ \Delta _\nu $ Δ ν . Using a suitable polynomial decomposition of the multiplicity of each $ \beta _m $ β m , we write down a trace formula for the heat operator associated with $ \Delta _\nu $ Δ ν in terms of Jacobi's theta functions and their higher order derivatives. Doing so enables us to establish the asymptotics of this trace as $ t\searrow 0^+ $ t ↘ 0 + by giving the corresponding heat coefficients in terms of Bernoulli numbers and polynomials. The obtained results can be exploited in the analysis of the spectral zeta function associated with $ \Delta _\nu $ Δ ν . [ABSTRACT FROM AUTHOR]
- Published
- 2024
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6. $\mu ^*$ -ZARISKI PAIRS OF SURFACE SINGULARITIES.
- Author
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EYRAL, CHRISTOPHE and OKA, MUTSUO
- Subjects
- *
HOMOGENEOUS polynomials , *COMPLEX variables , *ZETA functions , *TOPOLOGY - Abstract
Let $f_0$ and $f_1$ be two homogeneous polynomials of degree d in three complex variables $z_1,z_2,z_3$. We show that the Lê–Yomdin surface singularities defined by $g_0:=f_0+z_i^{d+m}$ and $g_1:=f_1+z_i^{d+m}$ have the same abstract topology, the same monodromy zeta-function, the same $\mu ^*$ -invariant, but lie in distinct path-connected components of the $\mu ^*$ -constant stratum if their projective tangent cones (defined by $f_0$ and $f_1$ , respectively) make a Zariski pair of curves in $\mathbb {P}^2$ , the singularities of which are Newton non-degenerate. In this case, we say that $V(g_0):=g_0^{-1}(0)$ and $V(g_1):=g_1^{-1}(0)$ make a $\mu ^*$ -Zariski pair of surface singularities. Being such a pair is a necessary condition for the germs $V(g_0)$ and $V(g_1)$ to have distinct embedded topologies. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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7. Relations of multiple t-values of general level.
- Author
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Li, Zhonghua and Wang, Zhenlu
- Subjects
- *
ZETA functions , *GENERATING functions , *HYPERGEOMETRIC functions - Abstract
We study the relations of multiple t -values of general level. The generating function of sums of multiple t -(star) values of level N with fixed weight, depth and height is represented by the generalized hypergeometric function 3 F 2 , which generalizes the results for multiple zeta(-star) values and multiple t -(star) values. As applications, we obtain formulas for the generating functions of sums of multiple t -(star) values of level N with height one and maximal height and a weighted sum formula for sums of multiple t -(star) values of level N with fixed weight and depth. Using the stuffle algebra, we also get the symmetric sum formulas and Hoffman's restricted sum formulas for multiple t -(star) values of level N. Some evaluations of multiple t -star values of level 2 with one–two–three indices are given. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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8. Functional equations and gamma factors of local zeta functions for the metaplectic cover of SL2.
- Author
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Oshita, Kazuki and Tsuzuki, Masao
- Subjects
- *
FUNCTIONAL equations , *ZETA functions , *VECTOR spaces , *SYMMETRIC spaces , *MELLIN transform , *BESSEL functions - Abstract
We introduce a local zeta-function for an irreducible admissible supercuspidal representation π of the metaplectic double cover of SL 2 over a non-archimedean local field of characteristic zero. We prove a functional equation of the local zeta-functions showing that the gamma factor is given by a Mellin type transform of the Bessel function of π. We obtain an expression of the gamma factor, which shows its entireness on C. Moreover, we show that, through the local theta-correspondence, the local zeta-function on the covering group is essentially identified with the local zeta-integral for spherical functions on PGL 2 ≅ SO 3 associated with the prehomogenous vector space of symmetric matrices of degree 2. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
9. Selberg Zeta function and hyperbolic Eisenstein series.
- Author
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Falliero, Thérèse
- Subjects
- *
EISENSTEIN series , *HYPERBOLIC functions , *ZETA functions , *GEODESICS , *RIEMANN surfaces , *HYPERBOLIC groups - Abstract
Let Γ be a Fuchsian group acting on the hyperbolic upper half-plane H , such that Γ \ H is a geometrically finite Riemann surface with respect to the natural hyperbolic metric induced from H. If γ is hyperbolic then following [J. Kudla and S. Millson, Harmonic differentials and closed geodesics on a Riemann surface, Invent. Math. 54 (1979) 193–211; J. D. Fay, Fourier coefficients of the resolvent for a Fuchsian group, J. Reine Angew. Math. 293 (1977) 143–203], there is a corresponding hyperbolic Eisenstein series. In this paper, we study the limiting behavior of hyperbolic Eisenstein series on a degenerating family of geometrically finite hyperbolic surfaces. In particular, we give a partial lightening to a question of Ji, concerning the approximation of Eisenstein series during degeneration (see Proposition 5.2 and Theorem 5.2). [ABSTRACT FROM AUTHOR]
- Published
- 2024
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10. AUTHOR INDEX FOR VOLUME 109.
- Subjects
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QUINTIC equations , *PLANE curves , *ZETA functions , *FINITE simple groups , *FINITE groups , *PARTIALLY ordered sets , *NONLINEAR Schrodinger equation - Abstract
This document is an author index for Volume 109 of the Bulletin of the Australian Mathematical Society. It lists the authors and titles of articles published in the journal. Some of the topics covered include birth-and-death processes, convex functions, plane curves, solvable groups, Diophantine equations, meromorphic functions, shape analysis, numerical semigroups, Brownian motion, and nonlinear Schrödinger equations. The index provides a comprehensive overview of the mathematical research published in the journal. [Extracted from the article]
- Published
- 2024
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11. Joint moments of derivatives of characteristic polynomials of random symplectic and orthogonal matrices.
- Author
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Andrade, Julio C and Best, Christopher G
- Subjects
- *
POLYNOMIALS , *SYMPLECTIC groups , *UNITARY groups , *HYPERGEOMETRIC functions , *ZETA functions , *RANDOM matrices - Abstract
We investigate the joint moments of derivatives of characteristic polynomials over the unitary symplectic group S p (2 N) and the orthogonal ensembles S O (2 N) and O − (2 N) . We prove asymptotic formulae for the joint moments of the n 1th and n 2th derivatives of the characteristic polynomials for all three matrix ensembles. Our results give two explicit formulae for each of the leading order coefficients, one in terms of determinants of hypergeometric functions and the other as combinatorial sums over partitions. We use our results to put forward conjectures on the joint moments of derivatives of L -functions with symplectic and orthogonal symmetry. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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12. On Some Multipliers Related to Discrete Fractional Integrals.
- Author
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Cheng, Jinhua
- Subjects
- *
FRACTIONAL integrals , *HARDY-Littlewood method , *NUMBER theory , *QUADRATIC fields , *FOURIER transforms , *HARMONIC analysis (Mathematics) , *ZETA functions - Abstract
This paper explores the properties of multipliers associated with discrete analogues of fractional integrals, revealing intriguing connections with Dirichlet characters, Euler's identity, and Dedekind zeta functions of quadratic imaginary fields. Employing Fourier transform techniques, the Hardy–Littlewood circle method, and a discrete analogue of the Stein–Weiss inequality on product space through implication methods, we establish ℓ p → ℓ q bounds for these operators. Our results contribute to a deeper understanding of the intricate relationship between number theory and harmonic analysis in discrete domains, offering insights into the convergence behavior of these operators. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
13. A NEW PROGRAM FOR THE ENTIRE FUNCTIONS IN NUMBER THEORY.
- Author
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YANG, XIAO-JUN
- Subjects
- *
NUMBER theory , *ZETA functions , *AUTOMORPHIC functions , *HEAT equation , *FUNCTIONAL equations , *INTEGRAL functions , *L-functions - Abstract
In this paper, we propose a new program for introducing the sign of the functional equation to present the entire functions of order one in number theory. We suggest some open problems for the zeros of these entire functions related to the completed Dedekind zeta function, completed quadratic Dirichlet L-functions, completed Ramanujan zeta function and completed automorphic L-function. These lead to the contribution to giving the deep understanding for diffusion equations associated with the entire functions of order one in number theory. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
14. Two types of Witten zeta functions.
- Author
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Levin, A. and Olshanetsky, M.
- Subjects
- *
ZETA functions , *SYMMETRIC spaces , *LIE groups , *MAXIMAL subgroups , *PARTITION functions , *SYMMETRY breaking - Abstract
We define two types of Witten's zeta functions according to Cartan's classification of compact symmetric spaces. The type II is the original Witten zeta function constructed by means of irreducible representations of the simple compact Lie group U. The type I Witten zeta functions, we introduce here, are related to the irreducible spherical representations of U. They arise in the harmonic analysis on compact symmetric spaces of the form U/K, where K is the maximal subgroup of U. To construct the type I zeta function we calculate the partition functions of 2d YM theory with broken gauge symmetry using the Migdal–Witten approach. We prove that for the rank one symmetric spaces the generating series for the values of the type I functions with integer arguments can be defined in terms of the generating series of the Riemann zeta-function. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
15. The uniform distribution modulo one of certain subsequences of ordinates of zeros of the zeta function.
- Author
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ÇİÇEK, FATMA and GONEK, STEVEN M.
- Subjects
- *
RIEMANN hypothesis , *ZETA functions , *REAL numbers , *MULTIPLICITY (Mathematics) - Abstract
On the assumption of the Riemann hypothesis and a spacing hypothesis for the nontrivial zeros $1/2+i\gamma$ of the Riemann zeta function, we show that the sequence \begin{equation*}\Gamma_{[a, b]} =\Bigg\{ \gamma : \gamma>0 \quad \mbox{and} \quad \frac{ \log\big(| \zeta^{(m_{\gamma })} (\frac12+ i{\gamma }) | / (\!\log{{\gamma }})^{m_{\gamma }}\big)}{\sqrt{\frac12\log\log {\gamma }}} \in [a, b] \Bigg\},\end{equation*} where the ${\gamma }$ are arranged in increasing order, is uniformly distributed modulo one. Here a and b are real numbers with $a , and $m_\gamma$ denotes the multiplicity of the zero $1/2+i{\gamma }$. The same result holds when the ${\gamma }$ 's are restricted to be the ordinates of simple zeros. With an extra hypothesis, we are also able to show an equidistribution result for the scaled numbers $\gamma (\!\log T)/2\pi$ with ${\gamma }\in \Gamma_{[a, b]}$ and $0. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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16. The Barnes–Hurwitz zeta cocycle at s=0 and Ehrhart quasi-polynomials of triangles.
- Author
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Espinoza, Milton
- Subjects
- *
TRIANGLES , *ZETA functions , *GENERALIZATION - Abstract
Following a theorem of Hayes, we give a geometric interpretation of the special value at s = 0 of certain 1 -cocycle on PGL 2 (ℚ) previously introduced by the author. This work yields three main results: an explicit formula for our cocycle at s = 0 , a generalization and a new proof of Hayes' theorem, and an elegant summation formula for the zeroth coefficient of the Ehrhart quasi-polynomial of certain triangles in ℝ 2 . [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
17. Dense clusters of zeros near the zero-free region of ζ(s).
- Author
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Banks, William D.
- Subjects
- *
ZETA functions , *SAWLOGS - Abstract
The methods of Korobov and Vinogradov produce a zero-free region for the Riemann zeta function ζ (s) of the form σ > 1 − c (log τ) 2 / 3 (log log τ) 1 / 3 (τ : = | t | + 1 0 0). For many decades, the general shape of the zero-free region has not changed (although explicit known values for c have improved over the years). In this paper, we show that if the zero-free region cannot be widened substantially, then there exist infinitely many distinct dense clusters of zeros of ζ (s) lying close to the edge of the zero-free region. Our proof provides specific information about the location of these clusters and the number of zeros contained in them. To prove the result, we introduce and apply a variant of the original method of de la Vallée Poussin combined with ideas of Turán to control the real parts of power sums. We also prove similar results for L -functions associated to nonquadratic Dirichlet characters χ modulo q ≥ 2. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
18. On real zeros of the Hurwitz zeta function.
- Author
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Ikeda, Karin
- Subjects
- *
ZETA functions , *BERNOULLI polynomials , *POLYNOMIALS - Abstract
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 the intervals (0 , 1) and (− N , − N + 1) for N ≥ 5 by Endo-Suzuki and Matsusaka, respectively. We prove the uniqueness of the real zeros in the remaining intervals by examining the behavior of certain associated polynomials. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
19. Mean of the product of derivatives of Hardy's Z-function with Dirichlet polynomial.
- Author
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Das, Mithun Kumar and Pujahari, Sudhir
- Subjects
- *
POLYNOMIALS , *ZETA functions , *POLYNOMIAL time algorithms - Abstract
Inspired by the work of Balasubramanian, Conrey and Heath-Brown [1] , we obtain an asymptotic expression for the mean of the product of any two finite order derivatives of Hardy's Z -function times Dirichlet polynomials in short intervals. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
20. The Generalized Eta Transformation Formulas as the Hecke Modular Relation.
- Author
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Wang, Nianliang, Kuzumaki, Takako, and Kanemitsu, Shigeru
- Subjects
- *
FUNCTIONAL equations , *MODULAR construction , *MODULAR forms , *LARGE space structures (Astronautics) , *ZETA functions , *VECTOR spaces - Abstract
The transformation formula under the action of a general linear fractional transformation for a generalized Dedekind eta function has been the subject of intensive study since the works of Rademacher, Dieter, Meyer, and Schoenberg et al. However, the (Hecke) modular relation structure was not recognized until the work of Goldstein-de la Torre, where the modular relations mean equivalent assertions to the functional equation for the relevant zeta functions. The Hecke modular relation is a special case of this, with a single gamma factor and the corresponding modular form (or in the form of Lambert series). This has been the strongest motivation for research in the theory of modular forms since Hecke's work in the 1930s. Our main aim is to restore the fundamental work of Rademacher (1932) by locating the functional equation hidden in the argument and to reveal the Hecke correspondence in all subsequent works (which depend on the method of Rademacher) as well as in the work of Rademacher. By our elucidation many of the subsequent works will be made clear and put in their proper positions. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
21. On the extension of the Voronin universality theorem for the Riemann zeta-function.
- Author
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Laurinčikas, Antanas
- Subjects
- *
ZETA functions , *ANALYTIC functions , *ANALYTIC spaces , *FUNCTION spaces , *PROBABILITY measures - Abstract
In the paper, we prove a general universality theorem for the Riemann zeta-function ζ (s) on approximation of a class of analytic functions by generalized shifts ζ (s + ia(τ)). Here a(τ) is a real-valued continuous increasing to +∞ function, uniformly distributed modulo 1 and such that |ζ(σ + ia(τ) + it)|2, for σ > 1/2, has a traditional mean estimate for every t ∈ ℝ. For example, the function tv(τ), v > 0, where t(τ) is the Gram function, satisfies the hypotheses of the proved theorem. For the proof, the method of weak convergence of probabilistic measures in the space of analytic functions is developed. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
22. FAMILIES OF LOG LEGENDRE CHI FUNCTION INTEGRALS.
- Author
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Sofo, Anthony
- Subjects
- *
LEGENDRE'S functions , *INTEGRAL functions , *SPECIAL functions , *INTEGRAL representations , *ZETA functions - Abstract
In this paper we investigate the representation of integrals involving the product of the Legendre Chi function, polylogarithm function and log function. We will show that in many cases these integrals take an explicit form involving the Riemann zeta function, the Dirichlet Eta function, Dirichlet lambda function and many other special functions. Some examples illustrating the theorems will be detailed. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
23. The Ihara expression of a generalization of the weighted zeta function on a finite digraph.
- Author
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Ishikawa, Ayaka
- Subjects
- *
ZETA functions , *WEIGHTED graphs , *GENERALIZATION - Abstract
We define a new weighted zeta function for a finite digraph and obtain its determinant expression called the Ihara expression. The graph zeta function is a generalization of the weighted graph zeta function introduced in previous research. That is, our result makes it possible to derive the Ihara expressions of the previous graph zeta functions for any finite digraphs. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
24. A Further Study of the Analytical Properties of the Generalized Hurwitz-Lerch Zeta Function and Pathway Fractional Integral Operator.
- Author
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Panwar, Savita, Pandey, Rupakshi Mishra, Rai, Prakriti, and Mathur, Pankaj
- Subjects
- *
ZETA functions , *INTEGRAL operators , *FRACTIONAL integrals , *BETA functions , *GENERATING functions , *INTEGRAL representations - Abstract
This article introduces a novel generalization of the Hurwitz-Lerch Zeta function, which is precisely reducible to several remarkable extensions of the Hurwitz-Lerch Zeta function. Our new version of the Hurwitz-Lerch Zeta function is defined with the help of the generalized Beta function. Analytical characteristics such as differential formulas, generating functions, and multiple integral representations have been studied in further detail. We study the pathway fractional integral formulas for our newly generalized formation of the extended Hurwitz-Lerch Zeta functions. We attain several particular and limiting cases of our main results. We additionally look at some statistical uses of our defined Hurwitz-Lerch Zeta function in probability distribution theory. [ABSTRACT FROM AUTHOR]
- Published
- 2024
25. Generalized Krätzel functions: an analytic study.
- Author
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Kabeer, Ashik A. and Kumar, Dilip
- Subjects
- *
ANALYTIC functions , *DIFFERENTIAL operators , *LIPSCHITZ continuity , *OPERATOR functions , *INTEGRAL operators , *ZETA functions - Abstract
The paper is devoted to the study of generalized Krätzel functions, which are the kernel functions of type-1 and type-2 pathway transforms. Various analytical properties such as Lipschitz continuity, fixed point property and integrability of these functions are investigated. Furthermore, the paper introduces two new inequalities associated with generalized Krätzel functions. The composition formulae for various fractional operators, such as Grünwald-Letnikov fractional differential operator, Riemann-Liouville fractional integral and differential operators with these functions are also obtained. Additionally, it has been demonstrated that these kernel functions are related to the Heaviside step function, the Dirac delta function, and the Riemann zeta function. A computable series representation of the kernel function is also obtained. Further scope of the work is pointed out by modifying the kernel function with the Mittag-Leffler function. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
26. A Dirichlet series related to the error term in the Prime Number Theorem.
- Author
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Elma, Ertan
- Subjects
- *
PRIME number theorem , *DIRICHLET series , *ZETA functions , *NATURAL numbers , *RIEMANN hypothesis , *MEROMORPHIC functions - Abstract
For a natural number n , let Z 1 (n) : = ∑ ρ n ρ ρ where the sum runs over the nontrivial zeros of the Riemann zeta function. For a primitive Dirichlet character χ modulo q ≥ 3 , we define Z 1 (s , χ) : = ∑ n = 1 ∞ χ (n) Z 1 (n) n s for ℜ (s) > 2 and obtain the meromorphic continuation of the function Z 1 (s , χ) to the region ℜ (s) > 1 2 . Our main result indicates that the poles of Z 1 (s , χ) in the region 1 2 < ℜ (s) < 1 , if they exist, are related to the zeros of many Dirichlet L -functions in the same region. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
27. A bound for Stieltjes constants.
- Author
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Pauli, S. and Saidak, F.
- Subjects
- *
ZETA functions - Abstract
The main goal of this note is to improve the best known bounds for the Stieltjes constants, using the method of steepest descent that was applied in 2011 by Coffey and Knessl in order to approximate these constants. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
28. A Discrete Version of the Mishou Theorem Related to Periodic Zeta-Functions.
- Author
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Balčiūnas, Aidas, Jasas, Mindaugas, and Rimkevičienė, Audronė
- Subjects
- *
ZETA functions , *DIRICHLET series , *ANALYTIC functions - Abstract
In the paper, we consider simultaneous approximation of a pair of analytic functions by discrete shifts ζuN (s + ikh1; a) and ζuN (s + ikh2, α; b) of the absolutely convergent Dirichlet series connected to the periodic zeta-function with multiplicative sequence a, and the periodic Hurwitz zeta-function, respectively. We suppose that uN → ∞ and uN ≪ N2 as N → ∞, and the set {(h1 log p: p ∈ P), (h2 log(m + α): m ∈ N0), 2π} is linearly independent over Q. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
29. Generalized Limit Theorem for Mellin Transform of the Riemann Zeta-Function.
- Author
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Laurinčikas, Antanas and Šiaučiūnas, Darius
- Subjects
- *
MELLIN transform , *ZETA functions , *PROBABILITY measures , *DIFFERENTIABLE functions , *LIMIT theorems - Abstract
In the paper, we prove a limit theorem in the sense of the weak convergence of probability measures for the modified Mellin transform Z (s) , s = σ + i t , with fixed 1 / 2 < σ < 1 , of the square | ζ (1 / 2 + i t) | 2 of the Riemann zeta-function. We consider probability measures defined by means of Z (σ + i φ (t)) , where φ (t) , t ⩾ t 0 > 0 , is an increasing to + ∞ differentiable function with monotonically decreasing derivative φ ′ (t) satisfying a certain normalizing estimate related to the mean square of the function Z (σ + i φ (t)) . This allows us to extend the distribution laws for Z (s) . [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
30. Exploring Explicit Definite Integral Formulae with Trigonometric and Hyperbolic Functions.
- Author
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Chen, Yulei and Guo, Dongwei
- Subjects
- *
HYPERBOLIC functions , *DEFINITE integrals , *BETA functions , *TRIGONOMETRIC functions , *ZETA functions - Abstract
Making use of integration by parts and variable replacement methods, we derive some interesting explicit definite integral formulae involving trigonometric or hyperbolic functions, whose results are expressed in terms of Catalan's constant, Dirichlet's beta function, and Riemann's zeta function, as well as π in the denominator. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
31. SUM OF VALUES OF THE IDEAL CLASS ZETA-FUNCTION OVER NONTRIVIAL ZEROS OF THE RIEMANN ZETA-FUNCTION.
- Author
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WANANIYAKUL, SAEREE, STEUDING, JÖRN, and RUNGTANAPIROM, NITHI
- Subjects
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ZETA functions , *LOGICAL prediction - Abstract
We prove an upper bound for the sum of values of the ideal class zeta-function over nontrivial zeros of the Riemann zeta-function. The same result for the Dedekind zeta-function is also obtained. This may shed light on some unproved cases of the general Dedekind conjecture. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
32. NEW EFFECTIVE RESULTS IN THE THEORY OF THE RIEMANN ZETA-FUNCTION.
- Author
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SIMONIČ, ALEKSANDER
- Subjects
- *
PRIME number theorem , *ANALYTIC number theory , *RIEMANN hypothesis , *ZETA functions , *ARITHMETIC series , *NUMBER theory - Abstract
The article informs about new effective results in the theory of the Riemann zeta-function, focusing on four groups providing estimates for the zeta-function and associated functions under the assumption of the Riemann hypothesis. Topic include explicit and RH estimates for various functions related to the zeta-function, including their applications to the distribution of prime numbers and other arithmetic properties, emphasizing the importance of these findings in mathematical research.
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- 2024
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33. INFINITE SERIES CONCERNING HARMONIC NUMBERS AND QUINTIC CENTRAL BINOMIAL COEFFICIENTS.
- Author
-
LI, CHUNLI and CHU, WENCHANG
- Subjects
- *
BINOMIAL coefficients , *INFINITE series (Mathematics) , *HYPERGEOMETRIC series , *QUINTIC equations , *ZETA functions - Abstract
By examining two hypergeometric series transformations, we establish several remarkable infinite series identities involving harmonic numbers and quintic central binomial coefficients, including five conjectured recently by Z.-W. Sun ['Series with summands involving harmonic numbers', Preprint, 2023, arXiv:2210.07238v7]. This is realised by 'the coefficient extraction method' implemented by Mathematica commands. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
34. A generalized Selberg zeta function for flat space cosmologies.
- Author
-
Bagchi, Arjun, Keeler, Cynthia, Martin, Victoria, and Poddar, Rahul
- Subjects
- *
ZETA functions , *FUNCTION spaces , *COSMOLOGICAL constant , *PHYSICAL cosmology , *ORBIFOLDS , *SPACE-time symmetries , *RIEMANN hypothesis - Abstract
Flat space cosmologies (FSCs) are time dependent solutions of three-dimensional (3D) gravity with a vanishing cosmological constant. They can be constructed from a discrete quotient of empty 3D flat spacetime and are also called shifted-boost orbifolds. Using this quotient structure, we build a new and generalized Selberg zeta function for FSCs, and show that it is directly related to the scalar 1-loop partition function. We then propose an extension of this formalism applicable to more general quotient manifolds M /ℤ, based on representation theory of fields propagating on this background. Our prescription constitutes a novel and expedient method for calculating regularized 1-loop determinants, without resorting to the heat kernel. We compute quasinormal modes in the FSC using the zeroes of a Selberg zeta function, and match them to known results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
35. Pro-isomorphic zeta functions of some D^\ast Lie lattices of even rank.
- Author
-
Moadim-Lesimcha, Yifat and Schein, Michael M.
- Subjects
- *
ZETA functions , *POLYNOMIALS - Abstract
We compute the local pro-isomorphic zeta functions at all but finitely many primes for a family of class-two-nilpotent Lie lattices of even rank, parametrized by irreducible monic non-linear polynomials f(x) \in \mathbb {Z}[x]. These Lie lattices correspond to a family of groups introduced by Grunewald and Segal. The result is expressed in terms of a combinatorially defined family of rational functions. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
36. On arithmetic nature of a q-Euler-double zeta values.
- Author
-
Chatterjee, Tapas and Garg, Sonam
- Subjects
- *
ZETA functions , *LAURENT series , *ARITHMETIC - Abstract
Chatterjee and Garg [Proc. Amer. Math. Soc. 151 (2023), pp. 2011-2022] established closed form for a q-analogue of the Euler-Stieltjes constants. In this article, we aim to build upon their work by extending it to a q-analogue of the double zeta function. Specifically, we derive a closed form expression for \gamma _{0,0}(q) which is a q-analogue of Euler's constant of height 2 and appear as the constant term in the Laurent series expansion of a q-analogue of the double zeta function around s_1 = 1 and s_2=1. Moreover, we examine the linear independence of a set of numbers involving the constant \gamma _0^{\prime *}(q^i), where 1 \leq i \leq r for any integer r \geq 1, that appears in the Laurent series expansion of a q-double zeta function. Finally, we discuss the irrationality of certain numbers involving a 2-double Euler-Stieltjes constant (\gamma _{0,0}(2)). [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
37. On an exponential sum related to the M\"{o}bius function.
- Author
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Zhang, Wei
- Subjects
- *
EXPONENTIAL sums , *MOBIUS function , *ZETA functions , *MATHEMATICS - Abstract
Let \mu (n) be the Möbius function and e(\alpha)=e^{2\pi i\alpha }. In this paper, we study upper bounds of the classical sum \[ S(x,\alpha)≔\sum _{1\leq n\leq x}\mu (n)e(\alpha n). \] We can improve some classical results of Baker and Harman [J. London Math. Soc. (2) 43 (1991), pp. 193–198]. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
38. An explicit sub-Weyl bound for ζ(1/2 + it).
- Author
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Patel, Dhir and Yang, Andrew
- Subjects
- *
ZETA functions , *EXPONENTIAL sums - Abstract
In this article we prove an explicit sub-Weyl bound for the Riemann zeta function ζ (s) on the critical line s = 1 / 2 + i t. In particular, we show that | ζ (1 / 2 + i t) | ≤ 66.7 t 27 / 164 for t ≥ 3. Combined, our results form the sharpest known bounds on ζ (1 / 2 + i t) for t ≥ exp (61). [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
39. A note on the two variable Artin's conjecture.
- Author
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Hazra, S.G., Ram Murty, M., and Sivaraman, J.
- Subjects
- *
RIEMANN hypothesis , *LOGICAL prediction , *ZETA functions , *ARTIN algebras , *RATIONAL numbers , *DIOPHANTINE approximation , *INTEGERS - Abstract
In 1927, Artin conjectured that any integer a which is not −1 or a perfect square is a primitive root for a positive density of primes p. While this conjecture still remains open, there has been a lot of progress in last six decades. In 2000, Moree and Stevenhagen proposed what is known as the two variable Artin's conjecture and proved that for any multiplicatively independent rational numbers a and b , the set { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } has positive density under the Generalised Riemann Hypothesis for certain Dedekind zeta functions. While the infinitude of such primes is known, the only unconditional lower bound for the size of the above set is due to Ram Murty, Séguin and Stewart who in 2019 showed that for infinitely many pairs (a , b) # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log 2 x. In this paper we improve this lower bound. In particular we show that given any three multiplicatively independent integers S = { m 1 , m 2 , m 3 } such that m 1 , m 2 , m 3 , − 3 m 1 m 2 , − 3 m 2 m 3 , − 3 m 1 m 3 , m 1 m 2 m 3 are not squares, there exists a pair of elements a , b ∈ S such that # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log log x log 2 x. Further, under the assumption of a level of distribution greater than x 2 3 in a theorem of Bombieri, Friedlander and Iwaniec (as modified by Heath-Brown), we prove the following conditional result. Given any two multiplicatively independent integers S = { m 1 , m 2 } such that m 1 , m 2 , − 3 m 1 m 2 are not squares, there exists a pair of elements a , b ∈ { m 1 , m 2 , − 3 m 1 m 2 } such that # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log log x log 2 x. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
40. Arithmetic of Hecke L-functions of quadratic extensions of totally real fields.
- Author
-
Tomé, Marie-Hélène
- Subjects
- *
L-functions , *ARITHMETIC , *ZETA functions , *CLASS groups (Mathematics) , *QUADRATIC fields - Abstract
Deep work by Shintani in the 1970's describes Hecke L -functions associated to narrow ray class group characters of totally real fields F in terms of what are now known as Shintani zeta functions. However, for [ F : Q ] = n ≥ 3 , Shintani's method was ineffective due to its crucial dependence on abstract fundamental domains for the action of totally positive units of F on R + n , so-called Shintani sets. These difficulties were recently resolved in independent work of Charollois, Dasgupta, and Greenberg and Diaz y Diaz and Friedman. For those narrow ray class group characters whose conductor is an inert rational prime in a totally real field F with narrow class number 1, we obtain a natural combinatorial description of these sets, allowing us to obtain a simple description of the associated Hecke L -functions. As a consequence, we generalize earlier work of Girstmair, Hirzebruch, and Zagier, that offer combinatorial class number formulas for imaginary quadratic fields, to real and imaginary quadratic extensions of totally real number fields F with narrow class number 1. For CM quadratic extensions of F , our work may be viewed as an effective affirmative answer to Hecke's Conjecture that the relative class number has an elementary arithmetic expression in terms of the relative discriminant. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
41. Riemann zeta functions for Krull monoids.
- Author
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Gotti, Felix and Krause, Ulrich
- Subjects
- *
MONOIDS , *ZETA functions , *ALGEBRAIC numbers , *ALGEBRAIC fields , *ARITHMETIC , *SET functions - Abstract
The primary purpose of this paper is to generalize the classical Riemann zeta function to the setting of Krull monoids with torsion class groups. We provide a first study of the same generalization by extending Euler's classical product formula to the more general scenario of Krull monoids with torsion class groups. In doing so, the Decay Theorem is fundamental as it allows us to use strong atoms instead of primes to obtain a weaker version of the Fundamental Theorem of Arithmetic in the more general setting of Krull monoids with torsion class groups. Several related examples are exhibited throughout the paper, in particular, algebraic number fields for which the generalized Riemann zeta function specializes to the Dedekind zeta function. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
42. FRACTIONAL CALCULUS OPERATORS OF THE GENERALIZED HURWITZ-LERCH ZETA FUNCTION.
- Author
-
KUMAWAT, SHILPA and SAXENA, HEMLATA
- Subjects
- *
ZETA functions , *FRACTIONAL calculus , *INTEGRAL transforms , *ANALYTIC functions - Abstract
In this paper, our aim is to establish certain generalized Marichev-Saigo-Maeda fractional integral and derivative formulas involving generalized p–extended Hurwitz-Lerch zeta function by using the Hadamard product (or the convolution) of two analytic functions. We then obtain their composition formulas by using fractional integral and derivative formulas and certain Integral transforms associated with Beta, Laplace and Whittaker transforms involving generalized p–extended Hurwitz-Lerch Zeta function. [ABSTRACT FROM AUTHOR]
- Published
- 2024
43. Reciprocal eigenvalue properties using the zeta and Möbius functions.
- Author
-
Kadu, Ganesh S., Sonawane, Gahininath, and Borse, Y.M.
- Subjects
- *
MOBIUS function , *FUNCTION algebras , *EIGENVALUES , *ZETA functions , *LINEAR operators , *VECTOR spaces , *BOOLEAN algebra - Abstract
In this paper, we develop a new approach to study the spectral properties of Boolean graphs using the zeta and Möbius functions on the Boolean algebra B n of order 2 n. This approach yields new proofs of the previously known results about the reciprocal eigenvalue property of Boolean graphs. Further, this approach allows us to extend the results to a more general setting of the zero-divisor graphs Γ (P) of complement-closed and convex subposets P of B n. To do this, we consider the left linear representation of the incidence algebra of a poset P on the vector space of all real-valued functions V (P) on P. We then write down the adjacency operator A of the graph Γ (P) as the composition of two linear operators on V (P) , namely, the operator that multiplies elements of V (P) on the left by the zeta function ζ of P and the complementation operator. This allows us to obtain the determinant of A and the inverse of A in terms of the Möbius function μ of the complement-closed posets P. Additionally, if we impose convexity on the poset P , then we obtain the strong reciprocal or strong anti-reciprocal eigenvalue property of Γ (P) and also obtain the absolute palindromicity of the characteristic polynomial of A. This produces a large family of examples of graphs having the strong reciprocal or strong anti-reciprocal eigenvalue property. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
44. Arithmetic equivalence under isoclinism.
- Author
-
Kida, Masanari
- Subjects
- *
ARITHMETIC , *ALGEBRAIC numbers , *ALGEBRAIC fields , *ZETA functions - Abstract
Two algebraic number fields are called arithmetically equivalent if the Dedekind zeta functions of the fields coincide. We show that if a G-extension contains non-conjugate arithmetically equivalent fields and there is an injection from G to another group H inducing an isoclinism between G and H, then there are non-conjugate arithmetically equivalent fields inside an H-extension. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
45. Shifted convolution sums motivated by string theory.
- Author
-
Fedosova, Ksenia and Klinger-Logan, Kim
- Subjects
- *
STRING theory , *SCATTERING amplitude (Physics) , *DIVISOR theory , *MOTIVATION (Psychology) , *LOGICAL prediction , *ZETA functions - Abstract
In [2] , it was conjectured that a particular shifted sum of even divisor sums vanishes, and in [11] , a formal argument was given for this vanishing. Shifted convolution sums of this form appear when computing the Fourier expansion of coefficients for the low energy scattering amplitudes in type IIB string theory [7] and have applications to subconvexity bounds of L -functions. In this article, we generalize the argument from [11] and rigorously evaluate shifted convolution of the divisor functions of the form ∑ n 1 , n 2 ∈ Z ∖ { 0 } n 1 + n 2 = n σ r 1 (n 1) σ r 2 (n 2) | n 1 | P and ∑ n 1 , n 2 ∈ Z ∖ { 0 } n 1 + n 2 = n σ r 1 (n 1) σ r 2 (n 2) | n 1 | Q log | n 1 | where σ ν (n) = ∑ d | n d ν. In doing so, we derive exact identities for these sums and conjecture that particular sums similar to but different from the one found in [2] will also vanish. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
46. Weil zeta functions of group representations over finite fields.
- Author
-
Corob Cook, Ged, Kionke, Steffen, and Vannacci, Matteo
- Subjects
- *
ZETA functions , *PROFINITE groups , *FREE groups , *REAL numbers , *FINITE groups , *REPRESENTATIONS of groups (Algebra) , *GROUP rings , *FINITE fields - Abstract
In this article we define and study a zeta function ζ G —similar to the Hasse-Weil zeta function—which enumerates absolutely irreducible representations over finite fields of a (profinite) group G. This Weil representation zeta function converges on a complex half-plane for all UBERG groups and admits an Euler product decomposition. Our motivation for this investigation is the observation that the reciprocal value ζ G (k) - 1 at a sufficiently large integer k coincides with the probability that k random elements generate the completed group ring of G. The explicit formulas obtained so far suggest that ζ G is rather well-behaved. A central object of this article is the Weil abscissa, i.e., the abscissa of convergence a(G) of ζ G . We calculate the Weil abscissae for free abelian, free abelian pro-p, free pro-p, free pronilpotent and free prosoluble groups. More generally, we obtain bounds (and sometimes explicit values) for the Weil abscissae of free pro- C groups, where C is a class of finite groups with prescribed composition factors. We prove that every real number a ≥ 1 is the Weil abscissa a(G) of some profinite group G. In addition, we show that the Euler factors of ζ G are rational functions in p - s if G is virtually abelian. For finite groups G we calculate ζ G using the rational representation theory of G. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
47. Fractional calculus of the Lerch zeta function – part II.
- Author
-
Guariglia, Emanuel
- Subjects
- *
ZETA functions , *FRACTIONAL calculus , *FUNCTIONAL equations - Abstract
This paper concerns the fractional derivative of the Lerch zeta function. The author already dealt with its functional equation. He reduced its computational cost and proved an approximate functional equation for this fractional derivative. Here, we study the mean square of this fractional derivative. Moreover, we deal with the distribution of zeros, showing some zero‐free regions. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
48. The modularity of Siegel's zeta functions.
- Author
-
Sugiyama, Kazunari
- Subjects
- *
HOLOMORPHIC functions , *MODULAR forms , *ZETA functions , *QUADRATIC forms , *FUNCTIONAL equations , *VECTOR spaces - Abstract
Siegel defined zeta functions associated with indefinite quadratic forms, and proved their analytic properties such as analytic continuations and functional equations. Coefficients of these zeta functions are called measures of representations, and play an important role in the arithmetic theory of quadratic forms. In a 1938 paper, Siegel made a comment to the effect that the modularity of his zeta functions would be proved with the help of a suitable converse theorem. In the present paper, we accomplish Siegel's original plan by using a Weil-type converse theorem for Maass forms, which has appeared recently. It is also shown that "half" of Siegel's zeta functions correspond to holomorphic modular forms. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
49. Spirals of Riemann's Zeta-Function — Curvature, Denseness and Universality.
- Author
-
SOURMELIDIS, ATHANASIOS and STEUDING, JÖRN
- Subjects
- *
PLANE curves , *CURVATURE , *ZETA functions - Abstract
This paper deals with applications of Voronin's universality theorem for the Riemann zeta-function $\zeta$. Among other results we prove that every plane smooth curve appears up to a small error in the curve generated by the values $\zeta(\sigma+it)$ for real t where $\sigma\in(1/2,1)$ is fixed. In this sense, the values of the zeta-function on any such vertical line provides an atlas for plane curves. In the same framework, we study the curvature of curves generated from $\zeta(\sigma+it)$ when $\sigma>1/2$ and we show that there is a connection with the zeros of $\zeta'(\sigma+it)$. Moreover, we clarify under which conditions the real and the imaginary part of the zeta-function are jointly universal. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
50. L-functions of elliptic curves modulo integers.
- Author
-
Baril Boudreau, Félix
- Subjects
- *
L-functions , *ELLIPTIC functions , *INTEGERS , *ELLIPTIC curves , *FINITE fields , *ZETA functions , *POLYNOMIALS - Abstract
In 1985, Schoof devised an algorithm to compute zeta functions of elliptic curves over finite fields by directly computing the numerators of these rational functions modulo sufficiently many primes (see [9]). If E / K is an elliptic curve with nonconstant j -invariant defined over a function field K of characteristic p ≥ 5 , we know that its L -function L (T , E / K) is a polynomial in Z [ T ] (see [5, p. 11]). Inspired by Schoof, we study the reduction of L (T , E / K) modulo integers. We obtain three main results. Firstly, if E / K has non-trivial K -rational N -torsion for some positive integer N coprime with p , we extend a formula for L (T , E / K) mod N due to Hall (see [4, p. 133, Theorem 4]) to all quadratic twists E f / K with f ∈ K × ∖ K × 2. Secondly, without any condition on the 2-torsion subgroup of E (K) , we give a formula for the quotient modulo 2 of L -functions of any two quadratic twists of E / K. Thirdly, we use these results to compute the global root numbers of an infinite family of quadratic twists of an elliptic curve and, under some assumption, find in most cases the exact analytic rank of these twists. We also illustrate that in favourable situations, our second main result allows one to compute much more efficiently L (T , E f / K) mod 2 than an algorithm of Baig and Hall (see [1]). Finally, we use our formulas to compute some degree 2 L -functions directly. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
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