Search Results

Any solution to $\zeta(s) = 1$ must have real part $\sigma \le \sigma(1)$ where $\sigma(1)$ is equal to the unique solution $\sigma>1$ of the equation $$\zeta(\sigma)=\frac{2^\sigma+1}{2^\sigma-1}$$ …

When I was writing my answer, Dan Romik answer appeared. Mine is the same but with more detail. We have $$\log\Gamma(1-x)=\gamma x+\sum_{n=2}^\infty\zeta(n)\frac{x^n}{n}$$ (this is known and is also a …

Equations of this type are known. You may see, for example, the classical book "Primzahlen" by Landau paragraph 67. "Continuation of the zeta function by partial integration" There it is proved the f …

This is not a completely satisfactory answer. I would like a simpler one. Nevertheless still probably a good exercise in Complex variables. I will only sketch it. What we want to show is equivalent t …

First observe that $t\coth(tz) e^{-t^2}$ is an even function of $t$. So the function can be written as $$f(t)={1\over 2}\int_{-\infty}^{+\infty} t\coth(tz) e^{-t^2} \,dt$$ The properties of the funct …

We prove that $$I=-\frac{\pi}{4}(\gamma+\log 4).$$ $$I=\int_0^\infty\frac{t\arg\Gamma(\frac14+\frac{it}{2})}{(\frac14+t^2)^2}\,dt.$$ $I$ is the imaginary part of the complex integral $$\int_0^\infty …

The function $$f(s):=\sin(\tfrac{\pi}{2}(s+1)) \zeta(s)+\sin(\tfrac{\pi s}{2}) L(\tfrac{s+1}{2})\tag{1}$$ where $L(s)$ is the second Dirichlet series $$L(s)=\sum\limits_{n=1}^{\infty } \frac{(-1)^{n-1 …

If $\varphi$ is in the class of Schwartz we have $$\int_{-\infty}^{+\infty}\varphi(t)\zeta(\frac12+it)\,dt= \sum_{n=0}^\infty\Bigl\{ \frac{1}{\sqrt{n+1}}\widehat{\varphi}\Bigl(\frac{1}{2\pi}\log (n+1) …

The function $\sigma\mapsto f(\sigma+it)$ is periodic with period $1$, for any value of $t$. Therefore $$f(\sigma+it)=\sum_{n\in Z} c_n(t)e^{2\pi i n\sigma}.$$ It follows that $$c_n(t)=\int_0^1 f(x+i …

In the paper: M. W. Coffey, "Asymptotic estimation of $\xi^{(2n)}(1/2)$: On a conjecture of Farmer and Rhoades", Mathematics of Computation, {\bf 78} (2009) 1147--1154 you may find the first terms o …

There is no simple equation that given $n$ gives us $\gamma_n$, with sufficient approximation. Given a constant $c>0$, the error in both equations are at times greater than $c/\log n$ for any given c …

The Riemann hypothesis implies that the function $\Xi(z)=\xi(1/2+iz)$ is in the Laguerre-Pólya class. Therefore it is a limit, uniformly on compact sets, of a sequence of polynomials with real roots. …