OT Conjecture about harmonic numbers and Riemann hypothesis

[Georgi Guninski <gguninski@gmail.com>]

Apologies for offtopic.

With latex formatting:  https://mathoverflow.net/q/499603/12481

We found conjecture about harmonic numbers related to RH.

Slightly rephrasing [wikipedia on
RH](https://en.wikipedia.org/wiki/Riemann_hypothesis#Growth_of_arithmetic_functions):

Let $\sigma(n)$ denote the sum of divisors on $n$ and $H_n$ denote the
harmonic numbers.

Define $\Delta(n)= H_n+\log{(H_n)} e^{H_n}-\sigma(n)$.

$\lfloor \Delta(n) \rfloor$[is OEIS A057641](https://oeis.org/A057641)


RH is equivalent to $\Delta(n)>0$ for all integers $n>1$ proved by  by
Jeffrey Lagarias in 2002.

**joro's conjecture about harmonic numbers** Let $a_1,a_2$ be integers
satsifying $a_1>1,a_2>1,(a_1,a_2) \ne (3,4), (a_1,a_2) \ne (4,3)$.
We have $\Delta(a_1 a_2) > \Delta(a_1)$ or $\Delta(a_1 a_2)>\Delta(a_2)$

>Q1 Is the conjecture true?

>Q2 In case Q1 is hopeless, can we get lower bound for the smallest counterexamples by fixing $a_1$ and let $a_2$ vary?

It holds for $ 1 < a_1,a_2 < 10^4$, but computation is slow.

We compute $H_n$ as `mpmath.polygamma(0,n+1)+mpmath.euler`


```
import mpmath
def delta2(n):
    h=mpmath.polygamma(0,n+1)+mpmath.euler #harmonic number H_n
    T=h+mpmath.log(h)*mpmath.exp(h)-sigma(n)
    return T


def delta1main(L=10**2,pre=20):
    """
    Author: Georgi Guninski Mon Aug 25 02:38:30 PM UTC 2025
    L=10^3 Wall time: 1min 32s
    L=10^4 Wall time: 2h 30min
    """
    import mpmath
    mpmath.mp.dps=pre
    mpmath.mp.pretty=True
    for a1 in range(2,L):
        for a2 in range(2,L):
            a3=a1*a2
            d1,d2,d3=[delta2(i) for i in (a1,a2,a3)]
            if not (d3>d1 or d3>d2):  print(a1,a2)

```