# On a Recursive Inductive Framework for the Riemann Hypothesis

Borros Arneth

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# On a Recursive Inductive Framework for the Riemann Hypothesis

Borros Arneth, Philipps University Marburg, Justus Liebig University Giessen, Germany
borros.arneth@staff.uni-marburg.de

## Abstract

We formulate a recursive induction scheme relating the phase-counting functions of the Riemann zeta function $\zeta(s)\zeta(s)\zeta(s)$ and Hardy's ZZZ-function. By showing that additional phase jumps in the discrepancy function between these two objects would necessarily correspond to non-critical zeros, we reduce the Riemann Hypothesis (RH) to the statement that no such discrepancies occur. We present the scheme formally, demonstrate its inductive consistency, and highlight the single analytic condition whose resolution would establish RH. The purpose of this note is to set the framework precisely and to invite the mathematical community to examine the feasibility of closing the remaining gap.

## 1. Introduction

The Riemann zeta function

$$ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}, \qquad \Re(s) > 1, $$

admits analytic continuation to the entire complex plane except at $s=1$. Its nontrivial zeros are conjectured by the **Riemann Hypothesis** (RH) to lie on the critical line $\Re(s) = 0.5$. Despite over a century of progress, RH remains unresolved.

Hardy (1914) introduced the real-valued function

$$ Z(t) = e^{i\theta(t)} \zeta(0.5 + it) $$

where $\theta(t)$ is the Riemann–Siegel theta function, to study the distribution of zeros on the critical line.

The real function $Z(t)$ vanishes precisely at those points where $\zeta(0.5 + it) = 0$. Thus Hardy's construction transforms the complex oscillation of $\zeta$ into a real oscillatory function with the same critical zeros.

The present paper formalizes a recursive induction principle: the counting of phase jumps of $\zeta$ and of $Z$ can be compared on successive intervals between zeros. Any discrepancy must correspond to a zero off the critical line. Therefore, proving the global absence of such discrepancies would establish RH.
## 2. Preliminaries

### 2.1. Zeta function and functional equation

The completed zeta function is

$$ \xi(s) = 0.5 s(s - 1)\pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s) $$

which satisfies $\xi(s) = \xi(1-s)$.

### 2.2. Hardy's Z-function

Let

$$ \theta(t) = \arg \Gamma \left( \frac{1}{4} + \frac{it}{2} \right) - t/2 \log \pi $$

defined continuously for $t \in \mathbb{R}$. Then

$$ Z(t) = e^{i\theta(t)} \zeta\left(\frac{1}{2} + it\right) \in \mathbb{R} $$

Zeros of $Z(t)$ are in bijection with zeros of $\zeta(s)$ on the critical line.

### 2.3. Counting functions

Define

$$ N(T) = \#\{\rho = \beta + i\gamma : \zeta(\rho) = 0, 0 < \gamma < T\} $$

the standard zero-counting function. Let

$$ N_0(T) = \#\{\rho = 1/2 + i\gamma : \zeta(\rho) = 0, 0 < \gamma < T\}, $$

the number of critical-line zeros. The Riemann–von Mangoldt formula states

$$ N(T) = \frac{T}{2\pi}\log\frac{T}{2\pi} - \frac{T}{2\pi} + O(\log T). $$

fine the discrepancy

$$ D(T) = N(T) - N_0(T) $$

The Riemann Hypothesis is equivalent to the statement $D(T) \equiv 0$.
### 3. Phase-Counting Functions

**Definition 3.1 (Phase function of $\zeta$).**

Let

$$ \phi_{\zeta}(t) = \arg \zeta \left( \frac{1}{2} + it \right) $$

defined by continuous variation along the critical line.

**Definition 3.2 (Phase function of Z).**

Since $Z(t)$ is real, its phase is restricted to $\{0, \pi\}$. Define the phase-counting function

$$ P_Z(T) = \#\{\text{sign changes of } Z(t) \text{ in } (0, T)\} $$

**Proposition 3.3.**

For each zero $\frac{1}{2} + iy$, both $\phi_{\zeta}(t)$ and $P_Z(t)$ undergo a phase jump.

*Sketch.* This follows from the factor $e^{i\theta(t)}$ in the definition of $Z(t)$, which exactly cancels the continuous phase of $\zeta(1/2 + it)$ leaving real sign oscillation.

### 4. Recursive Induction Scheme

**Theorem 4.1 (Conditional Induction).**

Assume that for some $T_0 > 0$, all zeros of $\zeta(s)$ with $0 < \Im(s) < T_0$ lie on the critical line. Suppose further that for each $n$, on the interval $(\gamma_n, \gamma_{n+1})$ between consecutive critical zeros, the discrepancy

$$ \Delta_n = (P_\zeta(\gamma_{n+1}) - P_\zeta(\gamma_n)) - (PP_Z(\gamma_{n+1}) - P_Z(\gamma_n)) $$

is zero. Then all zeros of $\zeta(s)$ up to height $\gamma_{n+1}$ lie on the critical line.
Proof.

Base step: For $0 < T < T_0$, the hypothesis guarantees coincidence.

Inductive step:

Assume all zeros up to $\gamma_n$ lie on the critical line and that $\Delta_n=0$. If a zero off the critical line existed with ordinate between $\gamma_n$ and $\gamma_{n+1}$, then $P_\zeta$ would record an additional phase jump not reflected in $P_\zeta$. This would force $\Delta_n \neq 0$. By assumption $\Delta_n=0$, hence no such zero exists. Thus, by induction, RH holds up to arbitrary height.

## 5. Discussion

The above theorem reduces RH to the claim:

**Key Condition.** For all $n$, $\Delta_n=0$.

Equivalently, the discrepancy function

$$S(T) = P_{\zeta}(T) - P_{\bar{\zeta}}(T)$$

remains constant (and equal to zero) for all T.

This condition is consistent with all numerical computations (Odlyzko's tables, Platt's verifications to heights beyond $10^{13}$), but remains unproven. Its resolution appears to hinge on deep control of the argument function of $\zeta$ in the critical strip, perhaps through automorphic or random-matrix-theoretic methods.

## 6. Conclusion

We have presented a recursive induction framework: if the phase-counting functions of $\zeta$ and Hardy's Z-function never diverge on intervals between critical zeros, then RH follows. This formulation isolates RH to a single analytic condition concerning phase discrepancy. The purpose of this note is not to claim a completed proof, but to formulate a rigorous reduction and to invite the mathematical community to test whether the key condition can be established.

## References

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2. H.M. Edwards, *Riemann's Zeta Function*, Academic Press (1974).
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4. Atle Selberg, *Contributions to the theory of the Riemann zeta-function*, Arch. Math. Naturvid. (1946).

5. Hugh Montgomery, *The pair correlation of zeros of the zeta function*, Analytic Number Theory, Proc. Symp. Pure Math. (1973).

6. A.M. Odlyzko, *The 102010^{20}1020-th zero of the Riemann zeta function and 175 million of its neighbors*, AT&T Report (1989).

7. T. Tao, *Structure and randomness in the zeta function*, blog notes (2008).

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10. D.A. Hejhal, *On the distribution of zeros of the Riemann zeta function*, C. R. Acad. Sci. Paris (1976).