Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userLower bounds for the modified Szpiro ratio
86
0
0.0
(
0
)
Authors
Alexander J. Barrios
Ask ChatGPT about the research
No Arabic abstract
Let $E/mathbb{Q}$ be an elliptic curve. The modified Szpiro ratio of $E$ is the quantity $sigma_{m}( E) =logmaxleft{ leftvert c_{4}^{3}rightvert ,c_{6}^{2}right} /log N_{E}$ where $c_{4}$ and $c_{6}$ are the invariants associated to a global minimal model of $E$, and $N_{E}$ denotes the conductor of $E$. In this article, we show that for each of the fifteen torsion subgroups $T$ allowed by Mazurs Torsion Theorem, there is a rational number $l_{T}$ such that if $Thookrightarrow E(mathbb{Q})_{text{tors}}$, then $sigma_{m}(E) >l_{T}$. We also show that this bound is sharp if the $ABC$ Conjecture holds.
rate research
Read More
In this note we show how the irrationality measure of $zeta(s) = pi^2/6$ can be used to obtain explicit lower bounds for $pi(x)$. We analyze the key ingredients of the proof of the finiteness of the irrationality measure, and show how to obtain good lower bounds for $pi(x)$ from these arguments as well. Whi
We study the $2k$-th discrete moment of the derivative of the Riemann zeta-function at nontrivial zeros to establish sharp lower bounds for all real $k geq 0$ under the Riemann hypothesis (RH).
We establish in this paper sharp lower bounds for the $2k$-th moment of the derivative of the Riemann zeta function on the critical line for all real $k geq 0$.
For relatively prime positive integers $u_0$ and $r$, we consider the least common multiple $L_n:=mathrm{lcm}(u_0,u_1,ldots, u_n)$ of the finite arithmetic progression ${u_k:=u_0+kr}_{k=0}^n$. We derive new lower bounds on $L_n$ which improve upon those obtained previously when either $u_0$ or $n$ is large. When $r$ is prime, our best bound is sharp up to a factor of $n+1$ for $u_0$ properly chosen, and is also nearly sharp as $ntoinfty$.
For relatively prime positive integers u_0 and r, we consider the arithmetic progression {u_k := u_0+k*r} (0 <= k <= n). Define L_n := lcm{u_0,u_1,...,u_n} and let a >= 2 be any integer. In this paper, we show that, for integers alpha,r >= a and n >= 2*alpha*r, we have L_n >= u_0*r^{alpha+a-2}*(r+1)^n. In particular, letting a = 2 yields an improvement to the best previous lower bound on L_n (obtained by Hong and Yang) for all but three choices of alpha,r >= 2.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
