Odd Characters Research Articles

Odd degree rational characters and the order of rational elements in finite groups

We prove that, in a finite group, if every rational irreducible character has odd degree, then all rational elements are 2-elements, as it was originally conjectured by P. H. Tiep and H. P. Tong-Viet.

The strong Stark conjecture for totally odd characters

We prove the p p -part of the strong Stark conjecture for every totally odd character and every odd prime p p . Let L / K L/K be a finite Galois CM-extension with Galois group G G , which has an abelian Sylow p p -subgroup for an odd prime p p . We give an unconditional proof of the minus p p -part of the equivariant Tamagawa number conjecture for the pair ( h 0 ( Spec ⁡ ( L ) ) , Z [ G ] ) (h^0(\operatorname {Spec}(L)), \mathbb {Z}[G]) under certain restrictions on the ramification behavior in L / K L/K .

Mean square value of $L$-functions at $s=1$ for non-primitive characters, Dedekind sumsand bounds on relative class numbers

Let $d_0$ be a given square-free integer. We give an explicit formula $M_{d_0}(p) =A(d_0)(1+B_{d_0}(p)/p)$ for the quadratic mean value at $s=1$ of the Dirichlet $L$-functions associated with the non-primitive odd Dirichlet characters modulo $d_0p$ induced by the odd Dirichlet characters modulo an odd prime $p$. Here $d_0\mapsto A(d_0)$ is an explicit multiplicative arithmetic function and $B_{d_0}(f)$ is a twisted sum over the divisors $d$ of $d_0$ of Dedekind sums $s(h,df)$. To prove that $f\mapsto B_{d_0}(f)$ is $d_0$-periodic, we find a new and closed formula for the Dedekind sums $f\mapsto s(a+bf,c+df)$, for fixed integers $a$, $b$, $c$ and $d$. We deduce explicit upper bounds for relative class numbers of cyclotomic number fields.

Mean square values of L-functions over subgroups for nonprimitive characters, Dedekind sums and bounds on relative class numbers

AbstractAn explicit formula for the mean value of $\vert L(1,\chi )\vert ^2$ is known, where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors p. Bounds on the relative class number of the cyclotomic field ${\mathbb Q}(\zeta _p)$ follow. Lately, the authors obtained that the mean value of $\vert L(1,\chi )\vert ^2$ is asymptotic to $\pi ^2/6$ , where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors $p\equiv 1\ \ \pmod {2d}$ which are trivial on a subgroup H of odd order d of the multiplicative group $({\mathbb Z}/p{\mathbb Z})^*$ , provided that $d\ll \frac {\log p}{\log \log p}$ . Bounds on the relative class number of the subfield of degree $\frac {p-1}{2d}$ of the cyclotomic field ${\mathbb Q}(\zeta _p)$ follow. Here, for a given integer $d_0>1$ , we consider the same questions for the nonprimitive odd Dirichlet characters $\chi '$ modulo $d_0p$ induced by the odd primitive characters $\chi $ modulo p. We obtain new estimates for Dedekind sums and deduce that the mean value of $\vert L(1,\chi ')\vert ^2$ is asymptotic to $\frac {\pi ^2}{6}\prod _{q\mid d_0}\left (1-\frac {1}{q^2}\right )$ , where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors p which are trivial on a subgroup H of odd order $d\ll \frac {\log p}{\log \log p}$ . As a consequence, we improve the previous bounds on the relative class number of the subfield of degree $\frac {p-1}{2d}$ of the cyclotomic field ${\mathbb Q}(\zeta _p)$ . Moreover, we give a method to obtain explicit formulas and use Mersenne primes to show that our restriction on d is essentially sharp.

THEMATIC AMPLIFICATION: A STUDY OF PETER OMOKO’S CHARACTERISATION CRAFT IN CRUDE NIGHTMEN AND MAJESTIC REVOLT

A literary writer communicates a wide range of ideas and concerns about his society, whether past or present. This is done with the ultimate aim of making his readers benefit from his ideological standpoint. But the manner of expression is crucial to the reception accorded, and the successful interpretation of the literary text by the intended reader. A writer considers a number of literary tools to enable him to express his ideas effectively. The consideration for the deployment of the different literary tools in communicating these ideas makes a writer the successful artist he desires to be and equally acknowledged by the reader. This we can say of Peter Omoko who has written a number of plays. His works are loaded with different shades of literary themes suitable for critical scrutiny. There are few critical works done on his plays but mostly focus on the surface thematic analysis. It suffices to say that no work has been dedicated to investigating how the playwright uses the tool of characterization to reinforce his thematic concerns. This paper examines how the writer creates characters for the purpose of furthering the themes in the chosen play texts. The analytical procedure is anchored on new criticism which supports analyzing the intrinsic elements of the texts. The Paper finds out that the playwright slightly deploys odd characters to explain abnormal situations created by regular characters. It concludes that this particular method is the hallmark of the playwright’s characterization craft that makes him stand unique as a writer among his contemporaries.

The mean value of the class numbers of cubic function fields

We compute the mean value of |L(s,χ)|2 evaluated at s=1 when χ goes through the primitive cubic Dirichlet odd characters of A:=Fq[T], where Fq is a finite field with q elements and q≡1(mod3). Furthermore, we find the mean value of the class numbers for the cubic function fields Km=k(m3), where k:=Fq(T) is the rational function field, m∈A is a cube-free polynomial, and deg⁡(m)≡1(mod3).

The Powers of Certain Number-Theoretic Matrices

For each integer we define a specific symmetric integer matrix of size . We then explicitly describe all positive and negative integer powers, as well as fractional powers, of each of these matrices; their eigenvalues also appear and play an important role. All this is based on an orthogonality relation for odd Dirichlet characters which, in turn, leads to a diagonalization. In the course of this article, various other number-theoretic objects make their appearance, including the Euler and Jordan totient functions, Möbius inversion, and the divisor function.

Quasi-solvable lattice models for and Demazure atoms and characters

Abstract We construct coloured lattice models whose partition functions represent symplectic and odd orthogonal Demazure characters and atoms. We show that our lattice models are not solvable, but we are able to show the existence of sufficiently many solutions of the Yang–Baxter equation that allow us to compute functional equations for the corresponding partition functions. From these functional equations, we determine that the partition function of our models are the Demazure atoms and characters for the symplectic and odd orthogonal Lie groups. We coin our lattice models as quasi-solvable. We use the natural bijection of admissible states in our models with Proctor patterns to give a right key algorithm for reverse King tableaux and Sundaram tableaux.

Partial Gaussian sums and the Pólya–Vinogradov inequality for primitive characters

In this paper we obtain a new fully explicit constant for the Pólya–Vinogradov inequality for primitive characters. Given a primitive character \chi modulo q , we prove the following upper bound: \Big| \sum_{1 \le n\le N} \chi(n) \Big|\le c \sqrt{q} \log q, where c=3/(4\pi^2)+o_q(1) for even characters and c=3/(8\pi)+o_q(1) for odd characters, with explicit o_q(1) terms. This improves a result of Frolenkov and Soundararajan for large q . We proceed, following Hildebrand, to obtain the explicit version of a result by Montgomery–Vaughan on partial Gaussian sums and an explicit Burgess-like result on convoluted Dirichlet characters.

Short Character Sums and the Pólya–Vinogradov Inequality

Abstract We show in a quantitative way that any odd primitive character χ modulo q of fixed order g ≥ 2 satisfies the property that if the Pólya–Vinogradov inequality for χ can be improved to $$\begin{equation*} \max_{1 \leq t \leq q} \left|\sum_{n \leq t} \chi(n)\right| = o_{q \rightarrow \infty}(\sqrt{q}\log q) \end{equation*}$$ then for any ɛ > 0 one may exhibit cancellation in partial sums of χ on the interval [1, t] whenever $t \gt q^{\varepsilon}$, i.e., $$\begin{equation*} \sum_{n \leq t} \chi(n) = o_{q \rightarrow \infty}(t)\ \text{for all } t \gt q^{\varepsilon}. \end{equation*}$$ We also prove a converse implication, to the effect that if all odd primitive characters of fixed order dividing g exhibit cancellation in short sums then the Pólya–Vinogradov inequality can be improved for all odd primitive characters of order g. Some applications are also discussed.

The integrated fourth moment of Dirichlet L-functions over rational function fields

We consider the hybrid fourth shifted moment of Dirichlet L-functions over rational function fields, where the moment average is taken over all odd primitive characters of modulus Q∈Fq[t] and over the critical circle, which is the symmetry line of the corresponding functional equation. We obtain an asymptotic formula for this moment with the full main term for arbitrary modulus Q, as deg⁡Q→∞ and q is fixed. Moreover, in case of an irreducible modulus Q we get an exact formula for this hybrid moment.

Harmonic Maass forms associated to real quadratic fields

In this paper, we explicitly construct harmonic Maass forms that map to the holomorphic weight 1 theta series associated by Hecke to odd ray class group characters of real quadratic fields. From this construction, we give precise arithmetic information contained in the Fourier coefficients of the holomorphic part of the harmonic Maass form, establishing the main part of a conjecture of the second author.

A mishap of “odd characters” disposed as tormenting obsessional ruminations in the psyche of an adolescent boy: The etiological design and empirical consideration as per Ayurveda principles Vis-a-Vis the biopsychosocial model

A mishap of “odd characters” disposed as tormenting obsessional ruminations in the psyche of an adolescent boy: The etiological design and empirical consideration as per Ayurveda principles Vis-a-Vis the biopsychosocial model

Cotangent sums, quantum modular forms, and the generalized Riemann hypothesis

We show that an asymptotic property of the determinants of certain matrices whose entries are finite sums of cotangents with rational arguments is equivalent to the GRH for odd Dirichlet characters. This is then connected to the existence of certain quantum modular forms related to Maass Eisenstein series.

Sam's Love for Sam

A re-examination of Beckett's notes for Human Wishes reveals that his original intention of writing about Dr. Johnson's love for Mrs. Thrale was abandoned at an earlier stage than hitherto believed. Using a later edition of Boswell, he began a fresh set of notes about a new idea for a play about the odd characters living in Johnson's house. The influence of Dr. Johnson and of Jeremy Taylor's Rules and Exercises of Holy Living and Dying on Beckett is shown to be so deep that it pervades Beckett's work.

Infinite products involving Dirichlet characters and cyclotomic polynomials

Using some basic properties of the gamma function, we evaluate a simple class of infinite products involving Dirichlet characters as a finite product of gamma functions and, in the case of odd characters, as a finite product of sines. As a consequence we obtain evaluations of certain multiple L-series. In the final part of this paper we derive expressions for infinite products of cyclotomic polynomials, again as finite products of gamma or of sine functions.

The frequency and the structure of large character sums

Let M(\chi) denote the maximum of |\sum_{n\le N}\chi(n)| for a given non-principal Dirichlet character \chi modulo q , and let N_\chi denote a point at which the maximum is attained. In this article we study the distribution of M(\chi)/\sqrt{q} as one varies over characters modulo q , where q is prime, and investigate the location of N_\chi . We show that the distribution of M(\chi)/\sqrt{q} converges weakly to a universal distribution \Phi , uniformly throughout most of the possible range, and get (doubly exponential decay) estimates for \Phi 's tail. Almost all \chi for which M(\chi) is large are odd characters that are 1-pretentious. Now, M(\chi)\ge |\sum_{n\le q/2}\chi(n)| = \frac{|2-\chi(2)|}\pi \sqrt{q} |L(1,\chi)| , and one knows how often the latter expression is large, which has been how earlier lower bounds on \Phi were mostly proved. We show, though, that for most \chi with M(\chi) large, N_\chi is bounded away from q/2 , and the value of M(\chi) is little bit larger than \frac{\sqrt{q}}{\pi} |L(1,\chi)| .

“Depraved subjects and the maliciousness of objects,” i.e. quirky objects

AbstractThe authors deal with a range of phenomena characterized by anomalies with respect to the mismatch between the surface realization of nominal phrases on the morphosyntactic level, i.e. the selection of cases, and their grammatical function. In connection with the subject function, traditional scholars generally regarded this kind of non-canonical relation between sentence structure and semantic interpretation as a categorial deviation and treated it under the cover terms grammatical versus logical subject. The eye-catching title of this paper makes allusions to rather dangerous, odd and sinister characters and topics in real life. On the one hand, therefore it should express the “irrational” use of well-established terms and categories. On the other hand, it may point at our attempt to group together and “telescope” different and mostly separate constructions such as

Restriction of Odd Degree Characters of S<sub>n</sub>

Let $n$ and $k$ be natural numbers such that $2^k < n$. We study the restriction to $\mathfrak{S}_{n-2^k}$ of odd-degree irreducible characters of the symmetric group $\mathfrak{S}_n$. This analysis completes the study begun in [Ayyer A., Prasad A., Spallone S., Sem. Lothar. Combin. 75 (2015), Art. B75g, 13 pages] and recently developed in [Isaacs I.M., Navarro G., Olsson J.B., Tiep P.H., J. Algebra 478 (2017), 271-282].

On the non-vanishing of certain Dirichlet series

Given k∈N, we study the vanishing of the Dirichlet seriesDk(s,f):=∑n≥1dk(n)f(n)n−s at the point s=1, where f is a periodic function modulo a prime p. We show that if (k,p−1)=1 or if (k,p−1)=2 and p≡3 (mod 4), then there are no odd rational-valued functions f≢0 such that Dk(1,f)=0, whereas in all other cases there are examples of odd functions f such that Dk(1,f)=0.As a consequence, we obtain, for example, that the set of values L(1,χ)2, where χ ranges over odd characters mod p, are linearly independent over Q.