---
abstract: |
    The function $S_n (t) =_{} \pi \left( \frac{3}{2} - {\ensuremath{\operatorname{frac}}} \left(
      \frac{\vartheta (t)}{\pi} \right) + \left( \lfloor \frac{t \ln \left(
      \frac{t}{2 \pi e} \right)}{2 \pi} + \frac{7}{8} \rfloor - n \right) \right)$ is conjectured to be equal to $S (t_n)_{} = \arg \zeta \left( \frac{1}{2} +
      i t_n \right)$ when $t = t_n$ is the imaginary part of the $n$-th zero of $\zeta$ on the critical line. If $S (t_n) = S_n (t_n)$ then the exact transcendental equation for the Riemann zeros has a solution for each positive integer $n$ which proves that Riemann’s hypothesis is true since the counting function for zeros on the critical line is equal to the counting function for zeros on the critical strip if the transcendental equation has a solution for each $n$.
author:
- 'Stephen Crowley &lt;stephencrowley214@gmail.com&gt;'
title: 'An Expression For The Argument of $\zeta$ at Zeros on the Critical Line'
---

Introduction
============

The Riemann-Siegel $\vartheta (t)$ Function
-------------------------------------------

The Riemann-Siegel $\vartheta$ function is defined by $$\begin{array}{cl}
    \vartheta (t) & = - \frac{i}{2} \left( \ln \Gamma \left( \frac{1}{4} +
    \frac{i t}{2} \right) - \ln \Gamma \left( \frac{1}{4} - \frac{i t}{2}
    \right) \right) - \frac{\ln (\pi) t}{2} \text{}\\
    & = \arg \left( \Gamma \left( \frac{1}{4} + \frac{i t}{2} \right) \right)
    - \frac{\ln (\pi) t}{2}
  \end{array}$$ Let $$\tilde{\vartheta} (t) = \frac{t \ln \left( \frac{t}{2 \pi e} \right)}{2} -
   \frac{\pi}{8}$$ be the approximate $\vartheta$ function where the $\Gamma$ function has been replaced with its Stirling approximation. $$\Gamma (s) \simeq \sqrt{2 \pi} s^{s - \frac{1}{2}} e^{- s}$$ The $\vartheta (t)$ function is not invertible but the inverse of its approximation $\tilde{\vartheta} (t)$ is defined by a linear function of the Lambert W function given by $$\tilde{\vartheta}^{- 1} (t) = \frac{\pi + 8 t}{4 W \left( \frac{\pi + 8 t}{8
  \pi e} \right)}$$ Let ${\ensuremath{\operatorname{frac}}} (x) = \left\{ \begin{array}{ll}
  x - \lfloor x \rfloor & x \geqslant 0\\
  x - \lceil x \rceil & x < 0
\end{array} \right. \forall x \in \mathbbm{R}$ be the function which gives the fractional part of a real number by subtracting either the floor $\lfloor x
\rfloor$ or the ceiling $\lceil x \rceil$ of $x$ from $x$, depending upon its sign. Furthermore, let $$S (t) = \arg \left( \zeta \left( \frac{1}{2} + {\ensuremath{\operatorname{it}}} \right) \right) =
  \lim_{\varepsilon \rightarrow 0}  \frac{1}{2} ((S {}(\rho + i
  \varepsilon) - S (\rho - i \varepsilon))$$ be the argument of $\zeta$ on the critical line.

\[ec\]The for the imaginary part of the $n$-th zero of $\zeta \left( \frac{1}{2} + i t \right)$[[@z0t Equation 20]]{} $$\vartheta (t_n) + S (t_n) = \left( n - \frac{3}{2} \right) \pi \label{ee}$$ has a solution for each integer $n \geqslant 1$ where $t_n$ enumerate the zeros of $Z$ on the real line and the zeros of $\zeta$ on the critical line $$\zeta \left( \frac{1}{2} + i t_n \right) = 0 \forall n \in \mathbbm{Z}^+$$ where $\mathbbm{Z}^+$ denotes the positive integers. [[@z0t Equation 14]]{}

The Gram Points
---------------

The $n$-th Gram point is defined to be the solution of the equation $$\vartheta (t) = (n - 1) \pi$$ A very accurate approximation $\tilde{g} (n)$ to the Gram points is $g (n)$ is found by inverting $\tilde{\vartheta} (t)$ to get the exact solution $$\begin{array}{cl}
    \tilde{g} (n) & = \{ t : \tilde{\vartheta} (t) - (n - 1) \pi = 0 \}\\
    & = \left\{ t : \left( \frac{t \ln \left( \frac{t}{2 \pi e} \right)}{2} -
    \frac{\pi}{8} \right) - (n - 1) \pi = 0 \right\}\\
    & = \frac{(8 n - 7) \pi}{4 W \left( \frac{8 n - 7}{8 e} \right)}\\
    & = g (n) + O (\delta_n)
  \end{array}$$ where $W$ is the Lambert W function, and the approximation bounds $\delta_n$ when $n = 1$ is $\delta_1 = 0.00223698 \ldots$, followed by $\delta_2 =
0.00137812 \ldots$ and decreases monotincally with increasing $n$, that is, $\delta_{n + 1} < \delta_n$.  The inverse of $\tilde{g} (n)$ is given by $$\begin{array}{ll}
    \tilde{g}^{- 1} (n) & = \{ t : \tilde{g} (n) = 0 \}\\
    & = \frac{t \ln \left( \frac{t}{2 \pi e} \right)}{2 \pi} + \frac{7}{8}
  \end{array}$$ Now define the infinite sequence of functions indexed by $n \in \mathbbm{Z}^+$ $$\begin{array}{cl}
    T_n (t) & = 1 + \lfloor \tilde{g}^{- 1} (n) \rfloor - n\\
    & = 1 + \lfloor \frac{t \ln \left( \frac{t}{2 \pi e} \right)}{2 \pi} +
    \frac{7}{8} \rfloor - n
  \end{array}$$ Near each “bad” Gram point there will be a corresponding zero on the critical line which has an argument not on the principal branch. The function $T_n (t)$ determines how many multiples of $\pi$ to add or subtract to $-
\frac{1}{2} - \lfloor \frac{\vartheta (s)}{\pi} \rfloor$ so that it agrees with the argument of $\zeta$ at a zero on the critical line where it is discontinuous, having the value $\lim_{\varepsilon \rightarrow 0}  \frac{1}{2}
((S {}(\rho + i \varepsilon) - S (\rho - i \varepsilon))$ when $\zeta
(\rho) = 0$.

Let $$\begin{array}{ll}
      S_n (t_{}) & = \pi \left( \frac{1}{2} - {\ensuremath{\operatorname{frac}}} \left(
      \frac{\vartheta (t)}{\pi} \right) - T_n (t) \right)\\
      & = \pi \left( \frac{1}{2} - {\ensuremath{\operatorname{frac}}} \left( \frac{\vartheta
      (t)}{\pi} \right) - (\lfloor \tilde{g}^{- 1} (n) \rfloor - n + 1)
      \right)\\
      & =_{} \pi \left( \frac{3}{2} - {\ensuremath{\operatorname{frac}}} \left( \frac{\vartheta
      (t)}{\pi} \right) - (\lfloor \tilde{g}^{- 1} (n) \rfloor - n) \right)\\
      & =_{} \pi \left( \frac{3}{2} - {\ensuremath{\operatorname{frac}}} \left( \frac{\vartheta
      (t)}{\pi} \right) + \left( \lfloor \frac{t \ln \left( \frac{t}{2 \pi e}
      \right)}{2 \pi} + \frac{7}{8} \rfloor - n \right) \right)
    \end{array}$$

Let $s_{\vartheta} (t) = \frac{\vartheta (t)}{| \vartheta (t) |}$ be the sign of $\vartheta (t)$.

\[c\]The argument $S (t)$ of $\zeta \left( \frac{1}{2} + i t \right)$ at the n-th non-trivial zero  $\zeta \left( \frac{1}{2} + i t_n \right) = 0
  \forall n \geqslant 1$ on the critical strip is equal to $s_{\vartheta} (t)
  S_n (t)$, that is $$\begin{array}{cl}
      S (t_n) & = s_{\vartheta} (t) S_n (t_n) = \frac{1}{2} (\lim_{t
      \rightarrow t^-_n} S (t_n) + \lim_{t \rightarrow t^+_n} S (t_n))\\
      & = \frac{\vartheta (t_n)}{| \vartheta (t_n) |}_{} \pi \left(
      \frac{3}{2} - {\ensuremath{\operatorname{frac}}} \left( \frac{\vartheta (t_n)}{\pi} \right) +
      \left( \lfloor \frac{t \ln \left( \frac{t_n}{2 \pi e} \right)}{2 \pi} +
      \frac{7}{8} \rfloor - n \right) \right)
    \end{array} \label{een}$$

If Conjecture \[c\] is true then Conjecture \[ec\] is true and, due to ${\ensuremath{\operatorname{Lemma}}}$ \[fl\], so is Conjecture \[RH\], the Riemann hypothesis.

If $s_{\vartheta} (t) S_n (t_n)$=$S (t_n)$ then $S (t_n)$ is well-defined $\forall n \geqslant 1$ since $s_{\vartheta} (t) S_n (t_n)$ is well-defined $\forall n \geqslant 1$.

$\arg \left( \zeta \left( \frac{1}{2} + i g (n) \right) \right) = 0 \forall
  n \in \mathbbm{Z}^+$

The argument of any positive number $x$ with ${\ensuremath{\operatorname{Im}}} (x) = 0$ is equal to $0$ and ${\ensuremath{\operatorname{Im}}} \left( \zeta \left( \frac{1}{2} + i g (n) \right) \right)
  = 0$

$S (t) - f_0 (t) \in \{ - 1, 0, 1 \} \forall t \in \mathbbm{R}$. That is $${\ensuremath{\operatorname{frac}}} \left( \frac{\vartheta (t)}{\pi} \right) + \frac{1}{\pi} \arg
    \left( \zeta \left( \frac{1}{2} + {\ensuremath{\operatorname{it}}} \right) \right) \in \{ - 1, 0,
    1 \} \forall 0 < t \in \mathbbm{R}$$

Appendix
========

Transcendental Equations Satisifed By The Nontrivial Riemann Zeros
------------------------------------------------------------------

The is the line in the complex plane defined by ${\ensuremath{\operatorname{Re}}} (t) = \frac{1}{2}$.

The is the strip in the complex plane defined by $0 < {\ensuremath{\operatorname{Re}}} (t) < 1$.

The for the $n$-th zero of the Hardy $Z$ function $$\frac{t_n}{2 \pi} \ln \left( \frac{t_n}{2 \pi t} \right) + S (t_n) = n -
    \frac{11}{8} \label{ae}$$ [[@z0t Equation 20]]{}

\[le\]If the limit $$\lim_{\delta \rightarrow 0^+} \arg \left( \zeta \left( \frac{1}{2} +
    \delta + i t \right) \right)$$ is exists and is well-defined $\forall t$ then the left-hand side of Equation (\[ae\]) is well-defined $\forall t$, and due to monotonicity, there must be a unique solution for every $n \in \mathbbm{Z}^+$. [[@z0t II.A]]{}

The number of solutions of Equation (\[ae\]) over the interval $[0, t]$ is given by $$N_0 (t) = \frac{t}{2 \pi} \ln \left( \frac{t}{2 \pi e} \right) +
    \frac{7}{8} + S (t) + O (t^{- 1}) \label{N0}$$

which counts the number of zeros .

\[RH\](The Riemann hypothesis) All solutions $t$ of the equation $$\zeta (t) = 0$$ besides the trivial solutions $t = - 2 n$ with $n \in \mathbbm{Z}^+$ have real-part $\frac{1}{2}$, that is, ${\ensuremath{\operatorname{Re}}} (t) = \frac{1}{2}$ when $\zeta
  (t) = 0$ and $t \neq - 2 n$.

The Riemann-von-Mangoldt formula makes use of Cauchy’s argument principle to count the number of zeros $0 <
  {\ensuremath{\operatorname{Im}}} (\rho_n) < t$ where $\zeta (\sigma + i \rho_n)$ with $0 < \sigma <
  1$ $$N (t) = \frac{t}{2 \pi} \ln \left( \frac{t}{2 \pi e} \right) + \frac{7}{8}
    + S (t) + O (t^{- 1})$$ and this definition does not depend on the Riemann hypothesis(Conjecture \[RH\]). This equation has exactly the same form as the asymptotic Equation \[ae\]. [[@z0t Equation 15]]{}

\[fl\]If the exact Equation (\[ee\]) has a unique solution for each $n
  \in \mathbbm{Z}^+$ then Conjecture \[RH\], the Riemann hypothesis, follows.

If the exact equation has a unique solution for each $n$, then the zeros obtained from its solutions on the can be counted since they are enumerated by the integer $n$, leading to the counting function $N_0 (t)$ in Equation (\[N0\]). The number of solutions obtained on the would saturate counting function of the number of solutions on the so that $N (t) = N_0
  (t)$ and thus all of the non-trivial zeros of $\zeta$ would be enumerated in this manner. If there are zeros off of the critical line, or zeros with multiplicity $m \geqslant 2$, then the exact Equation (\[ee\]) would fail to capture all the zeros on the critical strip which would mean $N_0 (t) < N
  (t)$.  [[@z0t IX]]{}

[1]{} Guilherme Fran[ç]{}a and Andr[é]{} LeClair. Transcendental equations satisfied by the individual zeros of riemann zeta, dirichlet and modular l-functions. [[*[Communications in Number Theory and Physics]{}*]{}]{}, 2015.
