Articles of riemann zeta

Do partial sums of the Riemann zeta function lie on a curve?

This answer shows a diagram representing a series of partial sums of the zeta function in the complex plane, and says that it shows figures looking like Euler spirals. I have plotted a similar diagram: an enlargement at the origin shows the partial sums (black) superimposed on a Euler spiral (green) plotted between appropriate centres. […]

How does $\zeta(1 – s)$ become $(-1/s + \cdots)$?

Why is $$\zeta(1 – s) = -\frac{1}{s} + \cdots$$ for small negative values of $s$? A detailed explanation would be appreciated.

Asymptotics for zeta zeros?

What are the best known asymptotics for the nth zeta zero (imaginary part)? Is there anything similar to $p_n\sim n\log n$, ie where $\rho$ is in form $\sigma+it$, $t_n\sim\dots?$

Why $\zeta (1/2)=-1.4603545088…$?

I saw $\zeta (1/2)=-1.4603545088…$ in this link. But how can that be? Isn’t $\zeta (1/2)$ divergent since $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+..>\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+..$ ?

Alternative form to express the second derivative of $\zeta (2) $

It is known that the first derivative of the Riemann zeta function $\zeta’ (x)$ can be espressed, for $x=2$, as $$\zeta'(2) = -\frac {\pi^2}{6} [ 12 \log(A)- \gamma -\log(2 \pi)]$$ where $A $ is the Glaisher-Kinkelin constant and $\gamma $ is the Euler-Mascheroni constant. I would be interested to know whether it is possible to […]

counterexample to RH; how big would it have to be?

If the Riemann hypothesis is false, then there has to be a first counterexample for $\zeta(z)=0$ in the critical strip with $\Re(z) \ne \frac{1}{2}$. For such a counterexample, how large would $T=|\Im(z)|$ have to be? On the one hand, we’ve only computed the first 10 trillion or so zeros, so it could be the very […]

Are there iterative formulas to find zeta zeros?

I am wondering whether one could find Riemann zeta zeros iteratively by using relationships such as this one: $$\rho _1=\lim_{s\to 1} \, \frac{\zeta (s) \zeta \left(s \cdot \rho _1\right)}{\zeta ‘(\rho _1)}$$ where $\rho _1 \approx 0.5 + 14.1347i$ is the first zeta zero. Are there better relationships like Newton iteration or something like that? The […]

Explicit example of a non-trivial zero of Riemann zeta function

Assume we are given the Riemann zeta function on $\mathrm{Re}(s) > 0$ by: $$\zeta(s) = \dfrac{s}{s-1} – s\int_1^{\infty} \dfrac{\{u\}}{u^{s+1}}du$$ My question is: can you give me explicitely a real number $t>0$ such that $$\zeta(1/2 + it) = 0$$ (and providing a proof that this is exactly a zero of $\zeta$). I saw questions like Show […]

Does the series $\sum_{n=1}^\infty (-1)^n \frac{\cos(\ln(n))}{n^{\epsilon}},\,\epsilon>0$ converge?

Numerical results showed that the series $$Q=\sum_{n=1}^{\infty} (-1)^n \frac{\cos(\ln(n))}{n^{\epsilon}},\epsilon>0\tag{1}$$ with $\epsilon=10^{-6}$ converged to $-0.53259554096828…$ We can rewrite this series as: $$Q=\sum_{k=0}^{\infty} a_k\tag{2}$$ $$a_k=\frac{\cos(\ln(2k+1))}{(2k+1)^{\epsilon}}-\frac{\cos(\ln(2k+2))}{(2k+2)^{\epsilon}}\tag{3}$$ Does this series converge? Using a method in a related question, we can show that (with $m=2k+1$) $$-a_k=m^{-1-\epsilon}\left(\epsilon\cos(\ln m)+\sin(\ln m)\right)+O(m^{-2-\epsilon}).\qquad m\to\infty \tag{4}$$ Thus $$|a_k|\le m^{-1-\epsilon}\left(\epsilon|\cos(\ln m)|+|\sin(\ln m)|\right)+O(m^{-2-\epsilon})=O(m^{-1-\epsilon})=O(k^{-1-\epsilon})\tag{5}$$ Therefore the series in (1) […]

regularization of sum $n \ln(n)$

I was testing out a few summation using my previous descriped methodes when i found an error in my reasoning. I’m really hoping someone could help me out. The function which i was evaluating was $\sum_{n=1}^{\infty} n\ln(n)$ which turns out to be $-\zeta'(-1)$. This made me hope i could confirm my previous summation methode for […]