Conjectures involving $\Lambda(n)$

Question

As the title suggests, I am looking for conjectures involving the Von Mangoldt function, $\Lambda(n)$. I understand this is not a rigorous mathematical question, however if reference requests for conjectures and theorems are permitted, then I humbly ask this question to be considered and approved here. I will make sure to include appropriate tags. I am looking specifically for conjectures that can be written in terms of $\Lambda(n)$, like for example: Goldbach's conjecture rewritten, or conjectures about the nature of $\Lambda(n)$. The reason is to attempt solving something in number theory, and find $\Lambda(n)$ to have some nice analytic properties/connections to the broader number theory. Thank you

Answer 1

The following has been proven by joriki and GH from MO: $$n>1:$$ $$\Lambda(n)=\lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n} \frac{\mu(d)}{d^{(s-1)}}$$

$$a(n) = -\lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n} \mu(d)(e^{d})^{(s-1)}$$

$$T(n,k) = a(GCD(n,k))$$

$$\displaystyle T = \begin{bmatrix} +1&+1&+1&+1&+1&+1&+1&\cdots \\ +1&-1&+1&-1&+1&-1&+1 \\ +1&+1&-2&+1&+1&-2&+1 \\ +1&-1&+1&-1&+1&-1&+1 \\ +1&+1&+1&+1&-4&+1&+1 \\ +1&-1&-2&-1&+1&+2&+1 \\ +1&+1&+1&+1&+1&+1&-6 \\ \vdots&&&&&&&\ddots \end{bmatrix}$$

which gives a symmetric expansion of the von Mangoldt function:

$$\displaystyle \begin{bmatrix} \frac{T(1,1)}{1 \cdot 1}&+\frac{T(1,2)}{1 \cdot 2}&+\frac{T(1,3)}{1 \cdot 3}+&\cdots&+\frac{T(1,k)}{1 \cdot k} \\ \frac{T(2,1)}{2 \cdot 1}&+\frac{T(2,2)}{2 \cdot 2}&+\frac{T(2,3)}{2 \cdot 3}+&\cdots&+\frac{T(2,k)}{2 \cdot k} \\ \frac{T(3,1)}{3 \cdot 1}&+\frac{T(3,2)}{3 \cdot 2}&+\frac{T(3,3)}{3 \cdot 3}+&\cdots&+\frac{T(3,k)}{3 \cdot k} \\ \vdots&\vdots&\vdots&\ddots&\vdots \\ \frac{T(n,1)}{n \cdot 1}&+\frac{T(n,2)}{n \cdot 2}&+\frac{T(n,3)}{n \cdot 3}+&\cdots&+\frac{T(n,k)}{n \cdot k} \end{bmatrix} = \begin{bmatrix} \frac{\infty}{1} \\ +\frac{\Lambda(2)}{2} \\ +\frac{\Lambda(3)}{3} \\ \vdots \\ +\frac{\Lambda(n)}{n} \end{bmatrix}$$

$$=\;\;\;\;\;\;\;\;\;\;\;$$

$$\displaystyle \begin{bmatrix} \frac{\infty}{1}&+\frac{\Lambda(2)}{2}&+\frac{\Lambda(3)}{3}+&\cdots&+\frac{\Lambda(k)}{k} \end{bmatrix}\;\;\;\;\;\;\;\;\;\;\;$$

The generating function for the matrix $T$ is:

$$\sum\limits_{k=1}^{\infty}\sum\limits_{n=1}^{\infty} \frac{T(n,k)}{n^c \cdot k^s} = \sum\limits_{n=1}^{\infty} \frac{\lim\limits_{z \rightarrow s} \zeta(z)\sum\limits_{d|n} \frac{\mu(d)}{d^{(z-1)}}}{n^c} = \frac{\zeta(s) \zeta(c)}{\zeta(s + c - 1)}$$

The generating function for the main diagonal of the matrix $T$ is: $$\sum_{n=1}^{\infty} \frac{T(n,n)}{n^s}=\frac{\zeta (s)}{\zeta (s-1)}$$

Set $c=s$ and add the generating functions together and take the limit:

$$\underset{s\to 1}{\text{lim}}\left(\frac{\zeta (s) \zeta (s)}{\zeta (s+s-1)}+\frac{\zeta (s)}{\zeta (s-1)}\right)=-2 \gamma+2 \log (2 \pi )$$

But that is not the same as the reciprocal sum over the zeta zeros:

RH Equivalence 5.3, page 6. The Riemann Hypothesis is equivalent to the equality:

$$\sum_{\rho} \frac{1}{|\rho|^2} =\sum_{\rho} \frac{1}{\rho (1{-}\rho)}= 2 + \gamma - \log (4\pi)$$

where the sum is over all complex zeros $$ρ = β + iγ$$ of $$ζ(s)$$ in the critical strip $$0 < β < 1$$.

Edit 21.8.2024:

The Riemann hypothesis is equivalent to:

$$\underset{\text{factor}}{-2}\left(\underset{\text{non-trivial zeros}}{\sum_{\rho} \frac{1}{\rho (1{-}\rho)}} + \underset{\text{trivial zeros}}{\sum_{k \geq 1} \frac{1}{-2k(1-(-2k))}} - \underset{\text{the pole}}{1}\right)= \underset{s\to 1}{\text{lim}}\left(\underset{\text{the whole matrix}}{\frac{\zeta (s) \zeta (s)}{\zeta (s+s-1)}}+\underset{\text{main diagonal}}{\frac{\zeta (s)}{\zeta (s-1)}}\right)$$

Where: $$\underset{\text{the whole matrix}}{\frac{\zeta (s) \zeta (s)}{\zeta (s+s-1)}}$$ is a Dirichlet generating function which when evaluated at $s=1$ as in the limit (above) is the symmetric expansion of the von Mangoldt function above.

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Series expansions in Mathematica 14: