اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدIrrationality measure and lower bounds for pi(x)
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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
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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.
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 generalize the classical theorem by Jarnik and Besicovitch on the irrationality exponents of real numbers and Hausdorff dimension. Let a be any real number greater than or equal to 2 and let b be any non-negative real less than or equal to 2/a. We
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We prove that the Hausdorff dimension of the set $mathbf{x}in [0,1)^d$, such that $$ left|sum_{n=1}^N expleft(2 pi ileft(x_1n+ldots+x_d n^dright)right) right|ge c N^{1/2} $$ holds for infinitely many natural numbers $N$, is at least $d-1/2d$ for $d g
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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 $k geq 0$.
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