Dirichlet L-functions Research Articles

Riemann Zeta Function and Its Applications

Abstract. This paper focuses on the introduction and proof of the fundamental properties of (z), i.e. Riemann zeta function and explores its applications in algebra. We begin with a systematic derivation and proof of the basic characteristics of the zeta function. Following this, we examine its application in algebra, including the use of the Dirichlet L-function to prove Dirichlets theorem. Furthermore, we show the classical result for the subgroup growth rate of J-groups and the enumeration of n-dimensional irreducible representations of Heisenberg groups.

Bounds for moments of Dirichlet L-functions to a fixed modulus

Bounds for moments of Dirichlet L-functions to a fixed modulus

Characteristic polynomials of orthogonal and symplectic random matrices, Jacobi ensembles & L-functions

Starting from Montgomery’s conjecture, there has been a substantial interest on the connections of random matrix theory and the theory of L-functions. In particular, moments of characteristic polynomials of random matrices have been considered in various works to estimate the asymptotics of moments of L-function families. In this paper, we first consider joint moments of the characteristic polynomial of a symplectic random matrix and its second derivative. We obtain the asymptotics, along with a representation of the leading order coefficient in terms of the solution of a Painlevé equation. This gives us the conjectural asymptotics of the corresponding joint moments over families of Dirichlet L-functions. In doing so, we compute the asymptotics of a certain additive Jacobi statistic, which could be of independent interest in random matrix theory. Finally, we consider a slightly different type of joint moment that is the analogue of an average considered over [Formula: see text] in various works before. We obtain the asymptotics and the leading order coefficient explicitly.

On the Universal Presence of Mathematics for Incompletely Predictable Problems in Rigorous Proofs for Riemann Hypothesis, Modified Polignac’s and Twin Prime Conjectures

We validly ignore even prime number 2. Based on all arbitrarily large number of even Prime gaps 2, 4, 6, 8, 10...; the complete set and its derived subsets of Odd Primes fully comply with the Prime number theorem for Arithmetic Progressions, and our derived Generic Squeeze theorem and Theorem of Divergent-to-Convergent series conversion for Prime numbers. With these conditions being satisfied by all Odd Primes, we argue Polignac's and Twin prime conjectures are proven to be true when they are usefully treated as Incompletely Predictable Problems. In so doing [and with Riemann hypothesis being a special case], this action also support the generalized Riemann hypothesis formulated for Dirichlet L-function. By broadly applying Hodge conjecture, Grothendieck period conjecture and Pi-Circle conjecture to Dirichlet eta function (proxy function for Riemann zeta function), Riemann hypothesis is separately proven to be true when it is usefully treated as Incompletely Predictable Problem. Crucial connections exist between these Incompletely Predictable Problems and Quantum field theory whereby eligible (sub-)functions and (sub-)algorithms are treated as infinite series.

The resolvent kernel on the discrete circle and twisted cosecant sums

In this paper we present a general unifying principle for computing finite trigonometric sums of types that arise in physics and number theory. We obtain formulas that are more general than previous expressions and deduce linear recursions, which are computationally more efficient than the degree two recursions proved by Zagier. As an application, we provide an answer to a question recently posed by Xie, Zhao and Zhao concerning special values of Dirichlet L-functions. The proofs use the combinatorial Laplacian on cyclic graphs and their twisted coverings. The techniques therefore connect the trigonometric sums to spectral invariants of graphs and open up for future investigations.

Discrete mean values of the product of derivatives of shifted Dirichlet L-functions

Assuming the Generalized Riemann Hypothesis, we calculate ∑0<γχ≤TL(μ)(ρχ+iδ,χ)L(ν)(1−ρχ−iδ,χ‾), which extends the result proposed by Gonek in 1984.

Riemann zeros as quantized energies of scattering with impurities

We construct an integrable physical model of a single particle scattering with impurities spread on a circle. The S-matrices of the scattering with the impurities are such that the quantized energies of this system, coming from the Bethe Ansatz equations, correspond to the imaginary parts of the non-trivial zeros of the the Riemann ζ(s) function along the axis \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak{R}\\left(s\\right)=\\frac{1}{2}$$\\end{document} of the complex s-plane. A simple and natural generalization of the original scattering problem leads instead to Bethe Ansatz equations whose solutions are the non-trivial zeros of the Dirichlet L-functions again along the axis \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak{R}\\left(s\\right)=\\frac{1}{2}$$\\end{document}. The conjecture that all the non-trivial zeros of these functions are aligned along this axis of the complex s-plane is known as the Generalised Riemann Hypothesis (GRH). In the language of the scattering problem analysed in this paper the validity of the GRH is equivalent to the completeness of the Bethe Ansatz equations. Moreover the idea that the validity of the GRH requires both the duality equation (i.e. the mapping s → 1 – s) and the Euler product representation of the Dirichlet L-functions finds additional and novel support from the physical scattering model analysed in this paper. This is further illustrated by an explicit counterexample provided by the solutions of the Bethe Ansatz equations which employ the Davenport-Heilbronn function \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{D}\\left(s\\right)$$\\end{document}, i.e. a function whose completion satisfies the duality equation χ(s) = χ(1 – s) but that does not have an Euler product representation. In this case, even though there are infinitely many solutions of the Bethe Ansatz equations along the axis \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak{R}\\left(s\\right)=\\frac{1}{2}$$\\end{document}, there are also infinitely many pairs of solutions away from this axis and symmetrically placed with respect to it.

First moment of central values of some primitive Dirichlet L-functions with fixed order characters

We evaluate asymptotically the smoothed first moment of central values of families of primitive cubic, quartic and sextic Dirichlet L-functions, using the method of double Dirichlet series. Quantitative non-vanishing results for these L-values are also proved.

An Explicit Vinogradov–Korobov Zero-Free Region for Dirichlet L-Functions

Abstract We establish the first explicit form of the Vinogradov–Korobov zero-free region for Dirichlet L-functions.

An explicit version of Bombieri’s log-free density estimate and Sárközy’s theorem for shifted primes

Abstract We make explicit Bombieri’s refinement of Gallagher’s log-free “large sieve density estimate near σ = 1 {\sigma=1} ” for Dirichlet L-functions. We use this estimate and recent work of Green to prove that if N ≥ 2 {N\geq 2} is an integer, A ⊆ { 1 , … , N } {A\subseteq\{1,\ldots,N\}} , and for all primes p no two elements in A differ by p - 1 {p-1} , then | A | ≪ N 1 - 10 - 18 {|A|\ll N^{1-10^{-18}}} . This strengthens a theorem of Sárközy.

One-level density of zeros of Dirichlet L-functions over function fields

TextWe compute the one-level density of zeros of order ℓ Dirichlet L-functions over function fields Fq[t] for ℓ=3,4 in the Kummer setting (q≡1(modℓ)) and for ℓ=3,4,6 in the non-Kummer setting (q≢1(modℓ)). In each case, we obtain a main term predicted by Random Matrix Theory (RMT) and lower order terms not predicted by RMT. We also confirm the symmetry type of the families is unitary, supporting Katz and Sarnak's philosophy. VideoFor a video summary of this paper, please visit https://youtu.be/0tGmXHvVKF0.

The inverse of tails of Riemann zeta function, Hurwitz zeta function and Dirichlet L-function

&lt;abstract&gt;&lt;p&gt;In this paper, we derive the asymptotic formulas $ B^*_{r, s, t}(n) $ such that&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \mathop{\lim} \limits_{n \rightarrow \infty} \left\{ \left( \sum\limits^{\infty}_{k = n} \frac{1}{k^r(k+t)^s} \right)^{-1} - B^*_{r,s,t}(n) \right\} = 0, $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;where $ Re(r+s) &amp;gt; 1 $ and $ t \in \mathbb{C} $. It is evident that the asymptotic formulas for the inverses of the tails of both the Riemann zeta function and the Hurwitz zeta function on the half-plane $ Re(s) &amp;gt; 1 $ are its corollaries. Subsequently we provide the asymptotic formulas for the Riemann zeta function and the Hurwitz zeta function on the half-plane $ Re(s) &amp;lt; 0 $. Finally, we study the asymptotic formulas of the inverse of the tails of the Dirichlet L-function for $ Re(s) &amp;gt; 1 $ and $ Re(s) &amp;lt; 0 $.&lt;/p&gt;&lt;/abstract&gt;

A NEW PROGRAM FOR THE ENTIRE FUNCTIONS IN NUMBER THEORY

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.

Mollified moments of cubic Dirichlet L-functions over the Eisenstein field

We prove, assuming the generalized Riemann Hypothesis (GRH) that there is a positive density of L-functions associated with primitive cubic Dirichlet characters over the Eisenstein field that do not vanish at the central point s=1/2. This is achieved by computing the first mollified moment, which is obtained unconditionally, and finding a sharp upper bound for the higher mollified moments for these L-functions, under GRH. The proportion of non-vanishing is explicit, but extremely small.

Remark on a General Zero Density Theorem

Recently [3] we proved a general zero density theorem for a large class of functions which included among others the Riemann zeta function, Dedekind zeta functions, Dirichlet 𝐿-functions. The goal of the present work is a (slight) improvement of this general theorem which might lead to stronger results in some applications.

Special values of spectral zeta functions of graphs and Dirichlet L-functions

In this paper, we establish relations between special values of Dirichlet L-functions and those of spectral zeta functions or L-functions of cycle graphs. In fact, they determine each other in a natural way. These two kinds of special values are bridged together by a combinatorial derivative formula obtained from studying spectral zeta functions of the first order self-adjoint differential operators on the unit circle.

Mean square values of Dirichlet L-functions associated to fixed order characters

The main purpose of this paper is to establish bounds on the second moment of L(12+it,χ), averaged over families of fixed order characters. A discrete version of the main result is also stated, from which zero-density estimates pertaining to fixed order characters are derived.

An explicit upper bound for L(1,chi ) when chi is quadratic

We consider Dirichlet L-functions L(s, chi ) where chi is a non-principal quadratic character to the modulus q. We make explicit a result due to Pintz and Stephens by showing that |L(1, chi )|leqslant frac{1}{2}log q for all qgeqslant 2cdot 10^{23} and |L(1, chi )|leqslant frac{9}{20}log q for all qgeqslant 5cdot 10^{50}.

Dirichlet Series with Periodic Coefficients, Riemann’s Functional Equation, and Real Zeros of Dirichlet L-Functions

ABSTRACT In this paper, we provide Dirichlet series with periodic coefficients that have Riemann’s functional equation and real zeros of Dirichlet L-functions. The details are as follows. Let L(s, χ) be the Dirichlet L-function and G(χ) be the Gauss sum associated with a primitive Dirichlet character χ (mod q). We define f ( s , χ ) : = q s L ( s , χ ) + i − κ ( χ ) G ( χ ) L ( s , χ ¯ ) , where χ ¯ is the complex conjugate of χ and κ(χ) := (1 – χ(−1))/2. Then, we prove that f (s, χ) satisfies Riemann’s functional equation in Hamburger’s theorem if χ is even. In addition, we show that f (σ, χ) ≠ 0 for all σ ≥ 1. Moreover, we prove that f(σ, χ) ≠ 0 for all 1/2 ≤ σ &lt; 1 if and only if L(σ, χ) ≠ 0 for all 1/2 ≤ σ &lt; 1. When χ is real, all zeros of f (s, χ) with ℜ ( s ) &gt; 0 are on the line σ = 1/2 if and only if the generalized Riemann hypothesis for L(s, χ) is true. However, f (s, χ) has infinitely many zeros off the critical line σ = 1/2 if χ is non-real.

Extreme values of derivatives of zeta and L-functions.

It is proved that as , uniformly for all positive integers , we have where. Here, is the Dickman function. We have and when , which significantly improves previous results in [17, 40]. Similar results are established for Dirichlet L-functions. On the other hand, when assuming the Riemann hypothesis and the generalized Riemann hypothesis, we establish upper bounds for and . Furthermore, when assuming the Granville-Soundararajan conjecture is true, we establish the following asymptotic formulas: where is prime and isgiven.