We study how often exceptional configurations of irreducible polynomials over finite fields occur in the context of prime number races and Chebyshev's bias. In particular, we show that three types of ...biases, which we call "complete bias", "lower order bias" and "reversed bias", occur with probability going to zero among the family of all squarefree monic polynomials of a given degree in $\mathbb{F}_qx$ as $q$, a power of a fixed prime, goes to infinity. The bounds given improve on a previous result of Kowalski, who studied a similar question along particular $1$-parameter families of reducible polynomials. The tools used are the large sieve for Frobenius developed by Kowalski, an improvement of it due to Perret-Gentil and considerations from the theory of linear recurrence sequences and arithmetic geometry.
We recall the construction by B. Enriquez of the elliptic associator A τ , a power series in two noncommutative variables a, b defined as an iterated integral of the Kronecker function, and turn our ...attention to a family of Fay relations satisfied by A τ , derived from the original well-known Fay relation satisfied by the Kronecker function. The Fay relations of A τ were studied by Broedel, Matthes and Schlotterer, and determined up to non-explicit correction terms that arise from the necessity of regularizing the nonconvergent integral. In this article, we study a reduced version Āτ of the elliptic associator mod 2πi. We recall a different construction of Āτ in three steps, due to Matthes, Lochak and the author: first one defines the reduced elliptic generating series Ēτ which comes from the reduced Drinfel'd associator Φ KZ and whose coefficients generate the same ring R as those of Āτ ; then one defines Ψ to be the automorphism of the free associative ring R a, b defined by Ψ(a) = Ēτ and Ψ(a, b) = a, b; finally one shows that the reduced elliptic associator Āτ is equal to Ψ ad(b) e ad(b) −1 (a). Using this construction and mould theory and working with Lie-like versions of the elliptic generating series and associator, we prove the following results: first, a mould satisfies the Fay relations if and only if a closely related mould satisfies the well-known "swap circneutrality" relations defining the elliptic Kashiwara-Vergne Lie algebra krv ell , second, the reduced elliptic generating series satisfies a family of Fay relations with extremely simple correction terms coming directly from those of the Drinfel'd associator, and third, the correction terms for the Fay relations satisfied by the reduced elliptic associator can be deduced explicitly from these.
The journal, of which Mochizuki is chief editor, is published by Japan's Research Institute for Mathematical Sciences (RIMS) at Kyoto University, where he works. The conjecture roughly states that if ...a lot of small primes divide two numbers, a and b, then only a few, large ones divide their sum, c. A confirmed proof could change number theory by, for example, providing an innovative approach to proving Fermat's last theorem, the legendary problem formulated by Pierre de Fermat in 1637 and solved only in 1994. In the world of mathematics, a journal's seal of approval is often not the end of the peer-review process.
We study the arithmetic of degree $N-1$ Eisenstein cohomology classes for
locally symmetric spaces associated to $\mathrm{GL}_N$ over an imaginary
quadratic field $k$. Under natural conditions we ...evaluate these classes on
$(N-1)$-cycles associated to degree $N$ extensions $F/k$ as linear combinations
of generalised Dedekind sums. As a consequence we prove a remarkable conjecture
of Sczech and Colmez expressing critical values of $L$-functions attached to
Hecke characters of $F$ as polynomials in Kronecker--Eisenstein series
evaluated at torsion points on elliptic curves with multiplication by $k$. We
recover in particular the algebraicity of these critical values.
Smarandache proposed the approach of neutrosophic sets. Neutrosophic sets deals with uncertain data. This paper defines the notion of neutrosophic b-open sets and neutrosophic b-closed sets and their ...properties are investigated. Further neutrosphic b-interior and neutrosphic b-closure operators are studied and their relationship with other operators are also discussed.
We investigate the large values of the derivatives of the Riemann zeta function
$\zeta (s)$
on the 1-line. We give a larger lower bound for
$\max _{t\in T,2T}|\zeta ^{(\ell )}(1+{i} t)|$
, which ...improves the previous result established by Yang ‘Extreme values of derivatives of the Riemann zeta function’, Mathematika 68 (2022), 486–510.
Let X be a smooth compactification of a homogeneous space of a linear algebraic group G over a number field k. We establish the conjecture of Colliot-Thélène, Sansuc, Kato and Saito on the image of ...the Chow group of zero-cycles of X in the product of the same groups over all the completions of k. When G is semisimple and simply connected and the geometric stabiliser is finite and supersolvable, we show that rational points of X are dense in the Brauer-Manin set. For finite supersolvable groups, in particular for finite nilpotent groups, this yields a new proof of Shafarevich's theorem on the inverse Galois problem, and solves, at the same time, Grunwald's problem, for these groups.
Soit X une compactification lisse d'un espace homogène d'un groupe algébrique linéaire G sur un corps de nombres k. Nous établissons la conjecture de Colliot-Thélène, Sansuc, Kato et Saito sur l'image du groupe de Chow des zéro-cycles de X dans le produit des mêmes groupes sur tous les complétés de k. Lorsque G est semi-simple et simplement connexe et que le stabilisateur géométrique est fini et hyper-résoluble, nous montrons que les points rationnels de X sont denses dans l'ensemble de Brauer-Manin. Pour les groupes finis hyper-résolubles, en particulier pour les groupes finis nilpotents, cela donne une nouvelle preuve du théorème de Shafarevich sur le problème de Galois inverse et résout en même temps, pour ces groupes, le problème de Grunwald.
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