arXiv subjects

Roland Quême

Publications and source records attributed to Roland Quême.

5 recordsLinked to original sources

arXiv2020

An attempt of proof of Riemann Hypothesis

This paper deals with an attempt of proof of the Riemann Hypothesis (RH). Let $T>10^{10}$ arbitrarily large. Let the region $Ω_T=\Big\{z=x+i y\ \Big|\ \frac{1}{2} 0$. There exists at least one root $ρ=\frac{1}{2}+{\bf u}+iγ$ whose real part is greater or equal to the real part of all the other roots in $Ω_T$. Let $v\geq \frac{3}{2}$. Let $\varepsilon>0$ arbitrarily small. We prove that $f(z)=\frac{ζ'(z)}{ζ(z)}$ is analytic in the open disk $Ω_\varepsilon=\Big\{ \Big|z-\Big(ρ+\frac{\varepsilon}{2}+v\Big)\Big|\Big\}< v.$ Let $s=ρ+\varepsilon$. We prove, from the Taylor series of $ζ(s)$, that $f(s)\sim \frac{1}{\varepsilon}\rightarrow \infty$ when $\varepsilon\rightarrow 0$, and that, through the representation of $f(s)$ as a Taylor series, $f(s)=f(c_0)-(v-\frac{\varepsilon}{2})f'(c_0) +\frac{(v-\frac{\varepsilon}{2})^2}{2!}f''(c_0)-\frac{(v-\frac{\varepsilon}{2})^3}{3!}f^{(3)}(c_0)+\dots\mbox{\ for\ }c_0=ρ+\frac{\varepsilon}{2}+v,$ in $Ω_\varepsilon$, that $f(s)\not\rightarrow \infty$ when $\varepsilon\rightarrow 0$, a contradiction which allows us to prove RH.

math.GM↗

arXiv2013

On second case of Strong Fermat's Last Theorem conjecture

This article deals with a conjecture, introduced in [GQ] (hereinafter $SFLT2$), which generalizes the second case of Fermat's Last Theorem: {\it Let $p>3$ be a prime. The diophantine equation $\frac{u^p+v^p}{u+v}=w_1^p$ with $u,v,u+v, w_1\in\Z\backslash\{0\}$, $u,v$ coprime and $v\equiv 0 \mod p$ has no solution.} Let $ζ$ be a $p$th primitive root of unity and $K:=\Q(ζ)$. A prime $q$ is said {\it $p$-principal} if the class of any prime ideal $\mathfrak q_K$ of $K$ over $q$ is a $p$-power of a class. Assume that $SFLT2$ fails for $(p,u,v)$. Let $q$ be any odd prime coprime with $puv$, $f$ the order of $q\mod p$, $n$ the order of $\frac{v}{u}\mod q$, $ξ$ a primitive $n$th root of unity, $\mathfrak q$ the prime ideal $(q,uξ-v)$ of $\Q(ξ)$. In this complement of the article [GQ] revisiting some works of Vandiver, we prove that, if $q$ is {\it $p$-principal} and $n\not=2p$ then $$\Big(\frac{1+ξζ^k}{1+ξζ}\Big)^{(q^f-1)/p}\equiv 1\mod \mathfrak q for k=1,\dots,p-1.$$ We shall derive, by example, of this congruence that, for $p$ sufficiently large, a very large number of primes should divide $v$. In an other hand we shall show that if $q$ is any prime of order $f\mod p$ dividing $(u^p+v^p)$ then $$(1-ζ)^{(q^f-1)/p}\equiv p^{-(q^f-1)/p}\mod q, $$ and a result of same nature if $q$ divides $u^p-v^p$, which reinforces strongly the first and second theorem of Furtwängler. The principle of proof relies on the $p$-Hilbert class field theory. Keywords: Fermat's Last Theorem; cyclotomic fields; cyclotomic units; class field theory; Vandiver's and Furtwängler's theorems

math.NT↗

arXiv2013

On Furtwängler's theorems and second case of Fermat's Last Theorem

This article, complement to the article [Que], deals with some generalizations of Futwängler's theorems for the second case of Fermat's Last Theorem (FLT2). Let $p$ be an odd prime, $ζ$ a $p$th primitive root of unity, $K:=\Q(ζ)$ and $C\ell_K$ the class group of $K$. A prime $q$ is said $p$-principal if the class $c\ell_K (\mk q_K)\in C\ell_K$ of any prime ideal $\mk q_K$ of $\Z_K$ over $q$ is the $p$th power of a class. Assume that FLT2 fails for $(p,x,y,z)$ where $x, y, z$ are mutually coprime integers, $p$ divides $y$ and $x^p+y^p+z^p=0$. Let $q$ be a prime dividing $\frac{(x^p+y^p)(y^p+z^p)(z^p+x^p)}{(x+y)(y+z)(z+x)}$ and $\mk q_K$ be any prime ideal of $K$ over $q$. We obtain the $p$-power residue symbols relations: $$(\frac{p}{\mk q_K})_K=(\frac{1-ζ^j}{\mk q_K})_K for j=1,\dots,p-1.$$ As an application, we prove that: if Vandiver's conjecture holds for $p$ then $q$ is a $p$-principal prime. Similarly, let $q$ be a prime dividing $\frac{(x^p-y^p)(y^p-z^p)(z^p-x^p)}{(x-y)(y-z)(z-x)}$ and $\mk q_K$ be the prime ideal of $K$ over $q$ dividing $(xζ-y)(zζ-y)(xζ-z)$. We give an explicit formula for the $p$-power residue symbols $(\frac{ε_{k}}{\mk q_K})_K$ for all $k$ with $1<k\leq\frac{p-1}{2},$ where $ε_k$ is the cyclotomic unit given by $ε_k=:ζ^{(1-k)/2}\cdot\frac{1+ζ^k}{1+ζ}.$ The principle of proofs rely on the $p$-Hilbert class field theory.

math.NT↗

arXiv2011

Some works of Furtwängler and Vandiver revisited and Fermat's last theorem

From some works of P. Furtwängler and H.S. Vandiver, we put the basis of a new cyclotomic approach to Fermat's last theorem for p>3 and to a stronger version called SFLT, by introducing governing fields of the form Q(exp(2 i pi/q-1)) for prime numbers q. We prove for instance that if there exist infinitely many primes q, q not congruent to 1 mod p, q^(p-1) not congruent to 1 mod p^2, such that for Q dividing q in Q(exp(2 i pi /q-1)), we have Q^(1-c) = A^p . (alpha), with alpha congruent to 1 mod p^2 (where c is the complex conjugation), then Fermat's last theorem holds for p. More generally, the main purpose of the paper is to show that the existence of nontrivial solutions for SFLT implies some strong constraints on the arithmetic of the fields Q(exp(2 i pi /q-1)). From there, we give sufficient conditions of nonexistence that would require further investigations to lead to a proof of SFLT, and we formulate various conjectures. This text must be considered as a basic tool for future researchs (probably of analytic or geometric nature) - This second version includes some corrections in the English language, an in depth study of the case p=3 (especially Theorem 8), further details on some conjectures, and some minor mathematical improvements.

math.NT↗

arXiv2011

The Hilbert's class field and the p-class group of the cyclotomic fields

Let $p$ be an irregular prime and $K=\Q(ζ)$ the $p$-cyclotomic field. Let $σ$ be a $\Q$-isomorphism of $K$ generating $Gal(K/\Q)$. Let $S/K$ be a cyclic unramified extension of degree $p$, defined by $S= K(A^{1/p})$ where $A\in K\backslash K^p$, $A\Z_K=\mk a^p$ with $\mk a$ non-principal ideal of $\Z_K$, $A^{σ-μ}\in K^p$ and $μ\in{\bf F}_p$. We compute explicitly the decomposition of the prime $p$ in the subfields $M$ of $S$ of degree $[M:\Q]=p$.

math.NT↗