اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn a conjecture of Kaneko and Ohno
91
0
0.0
(
0
)
تأليف
Zhong-hua Li
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Let $X_0^{star}(k,n,s)$ denote the sum of all multiple zeta-star values of weight $k$, depth $n$ and height $s$. Kaneko and Ohno conjecture that for any positive integers $m,n,s$ with $m,ngeqslant s$, the difference $(-1)^mX_0^{star}(m+n+1,n+1,s)-(-1)^nX_0^{star}(m+n+1,m+1,s)$ can be expressed as a polynomial of zeta values with rational coefficients. We give a proof of this conjecture in this paper.
قيم البحث
اقرأ أيضاً
By using various expansions of the parametric digamma function and the method of residue computations, we study three variants of the linear Euler sums, related Hoffmans double $t$-values and Kaneko-Tsumuras double $T$-values, and establish several s
ymmetric extensions of the Kaneko-Tsumura conjecture. Some special cases are discussed in detail to determine the coefficients of involved mathematical constants in the evaluations. In particular, it can be found that several general convolution identities on the classical Bernoulli numbers and Genocchi numbers are required in this study, and they are verified by the derivative polynomials of hyperbolic tangent.
It is well known that for any prime $pequiv 3$ (mod $4$), the class numbers of the quadratic fields $mathbb{Q}(sqrt{p})$ and $mathbb{Q}(sqrt{-p})$, $h(p)$ and $h(-p)$ respectively, are odd. It is natural to ask whether there is a formula for $h(p)/h(
-p)$ modulo powers of $2$. We show the formula $h(p) equiv h(-p) m(p)$ (mod $16$), where $m(p)$ is an integer defined using the negative continued fraction expansion of $sqrt{p}$. Our result solves a conjecture of Richard Guy.
In this short note we confirm the relation between the generalized $abc$-conjecture and the $p$-rationality of number fields. Namely, we prove that given K$/mathbb{Q}$ a real quadratic extension or an imaginary $S_3$-extension, if the generalized $ab
c$-conjecture holds in K, then there exist at least $c,log X$ prime numbers $p leq X$ for which K is $p$-rational, here $c$ is some nonzero constant depending on K. The real quadratic case was recently suggested by Bockle-Guiraud-Kalyanswamy-Khare.
We show that for a monic polynomial $f(x)$ over a number field $K$ containing a global permutation polynomial of degree $>1$ as its composition factor, the Newton Polygon of $fmodmathfrak p$ does not converge for $mathfrak p$ passing through all fini
te places of $K$. In the rational number field case, our result is the only if part of a conjecture of Wan about limiting Newton polygons.
We use logarithmic {ell}-class groups to take a new view on Greenbergs conjecture about Iwasawa {ell}-invariants of a totally real number field K. By the way we recall and complete some classical results. Under Leopoldts conjecture, we prove that Gre
enbergs conjecture holds if and only if the logarithmic classes of K principalize in the cyclotomic Z{ell}-extensions of K. As an illustration of our approach, in the special case where the prime {ell} splits completely in K, we prove that the sufficient condition introduced by Gras just asserts the triviality of the logarithmic class group of K.Last, in the abelian case, we provide an explicit description of the circular class groups in connexion with the so-called weak conjecture.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
