All Questions

Let: $\rho$ be a non-trivial zero of the Riemann zeta function, $\Lambda$ be the von-Mangoldt function and $\psi(x) =\sum_{n \leq x} \Lambda(n)$. What is the best known upper bound for $$f(x, T) := \...

When $t$ is an ordinate of a zero of Riemann zeta function, we define \begin{equation} f(t):=\frac{t}{2\pi}\log\left(\frac{t}{2\pi e}\right)+S(t)-\frac{1}{8}+\frac{1}{48 \pi t}+\frac{7}{5760 t^3}+...

Let $D$ denote a closed two dimensional figure as: $D=2+iT\to 2\to 2-\delta\to 2-\delta+i(T-\delta)\to \frac{1}{2}+\epsilon+i(T-\delta)\to\frac{1}{2}+\epsilon\to\frac{1}{2}-\epsilon\to \frac{1}{2}-\...

Let $N(T)$ be the number of zeros of Riemann zeta function upto height $T$ in the critical strip and $N_0(T)$ be the number of zeros on the critical line. What will be the significance of proving ...

The functions in question are $$L(s)=\sum_{k=1}^\infty \frac{\lambda(k)}{k^s}=\frac{\zeta(2s)}{\zeta(s)} \mbox{ and } L^*(s)=\frac{1}{2}\sum_{k=1}^\infty \frac{\lambda(k)+(-1)^{k+1}}{k^s}=\frac{L(s)+\...

It should be the case that, in some appropriate sense $$\pi (x)\sim \operatorname{Ri}(x)-\sum_{\rho}\operatorname{Ri}(x^{\rho}) \tag{4}\label{386213_4}$$ with $\operatorname{Ri}$ denoting the Riemann ...