---
# System prepended metadata

title: Ramanujan's Hidden Clue to the Riemann Hypothesis
tags: [millennium problem, Euler's totient, Gronwall function, sigma function, highly composite numbers, Robin's criterion, Guy Robin, primorial numbers, Ramanujan, Riemann Hypothesis, unsolved problems, mathematical history, arithmetic functions, Jean-Louis Nicolas, number theory, divisor function]

---

# Ramanujan's Hidden Clue to the Riemann Hypothesis

**Author:** Frank Vega
**Affiliation:** Information Physics Institute, 840 W 67th St, Hialeah, FL 33012, USA
**Email:** vega.frank@gmail.com
**ORCID:** 0000-0001-8210-4126

---

## The Century-Old Mathematical Mystery

What do a self-taught genius from India, an unsolved million-dollar problem, and a special kind of number have in common? More than you might think! Today, we're exploring a story that spans a century, linking the work of Srinivasa Ramanujan directly to one of mathematics' greatest challenges: **The Riemann Hypothesis**.

## The Cast of Characters: σ(n) and φ(n)

Our mathematical journey begins with two fundamental arithmetic functions:

- **The divisor sum function, σ(n)**: The sum of all divisors of a number *n*
  - Example: σ(6) = 1 + 2 + 3 + 6 = 12

- **Euler's totient function, φ(n)**: Counts numbers less than *n* that are relatively prime to it
  - Example: φ(6) = 2 (only 1 and 5 are coprime with 6)

These seemingly simple functions hold profound secrets that would eventually connect to one of mathematics' most famous unsolved problems.

## The Gronwall Ratio & Ramanujan's Brilliant Insight

Enter the **Gronwall function**:
```
G(n) = σ(n) / (n log log n)
```

This ratio measures how "abundant" a number is relative to its size. In his lost notebooks, Ramanujan made a stunning discovery:

> **If the Riemann Hypothesis is true, then for all sufficiently large n, G(n) is always less than e^γ**

where *e* is Euler's number and *γ* is the Euler-Mascheroni constant. This was the **first direct link** between elementary number theory and the profound Riemann Hypothesis.

## The Lost Letters and Hardy's Famous Underestimation

Ramanujan was fascinated by **highly composite numbers** - numbers with more divisors than any smaller number. Years later, Paul Erdos independently rediscovered them, calling them "superabundant" and "colossally abundant."

Much of this work was actually buried in Ramanujan's lost letters, only recovered decades later by mathematician Bruce Berndt. Ironically, G.H. Hardy once remarked:

> **"Even Ramanujan could not make highly composite numbers interesting."**

History would prove this assessment spectacularly wrong.

## The Modern Criteria: Nicolas and Robin's Breakthroughs

In the 1980s, the plot thickened dramatically:

- **Jean-Louis Nicolas** used **primorial numbers** (products of the first k primes: 2, 2×3=6, 2×3×5=30, etc.) to prove the Riemann Hypothesis is true **if and only if** a certain inequality involving φ(n) holds for all primorials

- **Guy Robin** (Nicolas' student) refined this into the famous **Robin's Criterion**:
  
> **The Riemann Hypothesis is true if and only if G(n) < e^γ for every integer n > 5040**

## The Modern Hunt Continues

The mathematical community has been actively working to prove Robin's inequality:

- **2022**: My research, entitled "[Robin’s criterion on divisibility](https://doi.org/10.1007/s11139-022-00574-4)", showed the inequality holds for all numbers not divisible by a huge set of primes
- Mathematicians like Caveney, Nicolas, Sondow, and Axler have pushed boundaries further and get published in the same journal
- **November 2025**: A new paper by Fan, Kobayashi, and Molnar on arXiv presents a new analogue of Robin's criterion

Despite this progress, Nicolas himself has expressed doubt that this path alone will lead to a final proof. The question remains **wide open**.

## The Story Continues...

From Ramanujan's lost notebooks to modern-day preprints, the quest to understand primes through simple functions continues. It's a beautiful demonstration of how elementary ideas can connect to mathematics' deepest mysteries.

The last word on the Riemann Hypothesis has not been written. Will the final proof use Robin's criterion, or come from an entirely different direction? **Only the future will tell.**

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*What are your thoughts on the Riemann Hypothesis? Share your insights in the comments below!*