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Look, I want to use the most standard approach: we have $$ X_{n+1}=\begin{pmatrix} a_{n+2} \\ a_{n+1} \end{pmatrix} = \begin{pmatrix} 14&-1 \\ 1&0 \end{pmatrix} \begin{pmatrix} a_{n+1} \\ a_{n} \end{ …

The Lindelöf hypothesis is: $$ \forall \epsilon >0,\exists C_\epsilon >0,\forall t\ge 1,\quad \vert\zeta(\frac12+it)\vert\le C_\epsilon t^\epsilon.\qquad \tag{LH}. $$ It is a weaker statement than the …

Let me reformulate your question. How can we control the $L^\infty$ norm of $u$ by some behavior of the Fourier transform? The most classical thing that could be said is $$ H^s(\mathbb R^n)\subset L^ …

The trivial zeroes of the Riemann Zeta function are located on $-2\mathbb N^*$ and they are simple. It is not difficult to see that, but the proof I have in mind is using the fact that $\xi(-2k)=\xi(1 …

The function $\text{Li}$ (logarithmic integral) is defined for $x>0$ by $$ \text{Li}(x)=\int_2^{x}\frac{dt}{\ln t}. $$ The prime number theorem, proven by Hadamard and de la Vallée-Poussin in 1896 ass …

Numerical calculations on the zeroes of the Riemann zeta function have reached a very high degree of refinement and sophistication and I think that the first $10^{20}$ (with positive imaginary part) o …

The function $\mathbb R\ni t\mapsto\zeta(\frac12+it)$ is analytic and smaller in absolute value than $C(1+\vert t\vert)^{1/6}$ (the $1/6$ may be replaced by $9/56$ and even by a slightly smaller numb …

Let me speak about the "Triumph of Fourier" according to the words of Laurent Schwartz in his autobiography. The Fourier transformation is a handy tool to characterize regularity of functions. Let $ …

Let me assume that $a=-\infty, b=+\infty, x_0=0$ and $f$ smooth and compactly supported near 0. Then after a suitable change of variable, you get that $ I(\lambda)=\int g(t) e^{i\lambda t^3/3} dt, $ …