Enquiry on an equality involving the Riemann zeta function

Question

Let $\zeta$ denote the Riemann zeta function. Does there exist a $t\geq 0$ such that

$$\Re(1/4 + t^2)\zeta(1/2 + it)=2t\arg \zeta(1/2 + it) + 2(1/4 + t^2)\ ?$$

Answer 1

Your question is a little imprecise, because you use $\arg(\zeta(1/2+it))$ without explaining what value to take. I assume that you refer to $\pi S(t)$ as defined in the book of Titchmarsh (section 9.3).

Assuming the above value of the argument, there are many values of $t$ for which this equality is true. For example in the interval $0<t<100$ I find just 24 solutions. The first ones are situated at

\begin{align*} t&=0.16758273673627602\\ t&=1.1639847411644978\\ t&=17.121600816951701\\ t&=18.681304365748058\\ t&=26.871493076980211\\ t&=28.557756796709644 \end{align*}

I find them using mpmath in sage that has implementation for zeta, and S(t) as backlunds(). I plot two functions $\Re(\zeta(1/2+it))$ and $\frac{2\pi t}{1/4+t^2} S(t)+2$. Then use findroot() of mpmath to find the approximate values above, from approximate values taken from the graphs. Then I check some of these values with Mathematica.

When $t$ is large the second function I plotted tends to $2$. Therefore your equation is almost equivalent to $\Re\zeta(1/2+it)=2$, we know that this has an infinite number of solutions. This appear clearly in the plot of the functions in the interval $(10^6,10^6+5)$ which contains 8 solutions of your equation.