Riemann Zeta Zeros: Sum Formula & Proof Request

Hey guys! I've been diving deep into the fascinating world of the Riemann zeta function, and I've stumbled upon a formula that I'm trying to nail down. Specifically, I'm looking for a solid reference or a detailed proof for the following asymptotic behavior:

Assuming the Riemann Hypothesis (RH) and the simple zeros conjecture, we want to show that

$$\sum_{0<\Im(\rho)\le T}\frac{1}{\zeta'(\rho)}\sim\frac{T}{2\pi}.$$

Where the sum is over the non-trivial zeros $\rho$ of the Riemann zeta function, and $T$ goes to infinity.

The Riemann Hypothesis and Simple Zeros Conjecture

Before we dive into the sum, let's quickly recap what the Riemann Hypothesis and the simple zeros conjecture are all about, just to make sure we're all on the same page. These are crucial for understanding the context of the formula we're trying to prove.

Riemann Hypothesis (RH)

The Riemann Hypothesis is arguably one of the most famous unsolved problems in mathematics. It states that all non-trivial zeros of the Riemann zeta function have a real part equal to 1/2. In other words, if $\rho$ is a non-trivial zero, then $\Re(\rho) = \frac{1}{2}$.

The implications of the Riemann Hypothesis are vast, touching upon the distribution of prime numbers and many other areas of number theory. Proving (or disproving) it would have profound consequences for our understanding of the mathematical universe. It is one of the million-dollar millennium problems.

Simple Zeros Conjecture

The simple zeros conjecture posits that all non-trivial zeros of the Riemann zeta function are simple, meaning that the zeta function has a non-zero derivative at these points. Mathematically, this means that if $\zeta(\rho) = 0$, then $\zeta'(\rho) \neq 0$. This conjecture is closely related to the Riemann Hypothesis and is often assumed to hold when studying the properties of the zeta function's zeros.

Importance of These Assumptions

Both the Riemann Hypothesis and the simple zeros conjecture are critical for the formula we're examining. The Riemann Hypothesis provides a precise location for the zeros, which simplifies many calculations. The simple zeros conjecture ensures that the derivative of the zeta function at these zeros is well-behaved, allowing us to work with $\frac{1}{\zeta'(\rho)}$ without running into singularities or other issues. These assumptions allow us to make significant progress in analyzing the sum over the zeros.

Understanding these foundational concepts is essential before delving into the intricacies of the sum formula. These assumptions provide the necessary framework for exploring the deeper properties of the Riemann zeta function and its zeros. With these concepts in mind, let's proceed to explore how they influence the behavior of the sum over zeros.

Understanding the Asymptotic Formula

Now, let's break down the asymptotic formula itself. The formula essentially tells us something about the average behavior of the reciprocals of the derivative of the Riemann zeta function at its zeros. The left-hand side of the formula sums $\frac{1}{\zeta'(\rho)}$ over all zeros $\rho$ such that their imaginary part lies between 0 and $T$. As $T$ becomes very large, this sum approaches $\frac{T}{2\pi}$.

Asymptotic Behavior

The term "asymptotic" means that the ratio of the two sides of the equation approaches 1 as $T$ goes to infinity. In other words:

$$\lim_{T \to \infty} \frac{\sum_{0<\Im(\rho)\le T}\frac{1}{\zeta'(\rho)}}{\frac{T}{2\pi}} = 1$$

This indicates that for large $T$, the sum behaves approximately like $\frac{T}{2\pi}$. This kind of result is incredibly useful because it gives us a way to estimate the behavior of a complex sum by a simpler function.

Implications and Significance

This formula has significant implications for our understanding of the distribution of zeros of the Riemann zeta function. It suggests that the average value of $\frac{1}{\zeta'(\rho)}$ is closely tied to the density of zeros along the critical line. The factor of $\frac{1}{2\pi}$ is particularly interesting because it is related to the average spacing between the zeros.

Specifically, the number of zeros with imaginary part between 0 and $T$ is approximately $\frac{T}{2\pi} \log(\frac{T}{2\pi}) - \frac{T}{2\pi}$. Therefore, the formula implies that the sum of the reciprocals of the derivatives at these zeros is, on average, proportional to the number of zeros, but without the logarithmic factor. This gives us a more refined understanding of how the values of $\zeta'(\rho)$ are distributed.

Furthermore, understanding the asymptotic behavior of this sum can help in various calculations and estimations in analytic number theory. It provides a valuable tool for studying the fine-grained properties of the Riemann zeta function and its zeros, contributing to our broader understanding of prime number distribution and related concepts.

Why is This Formula Important?

The importance of this formula lies in its ability to connect the microscopic behavior of the zeta function (at its individual zeros) to its macroscopic behavior (as described by the asymptotic formula). It provides a bridge between the discrete and continuous aspects of the zeta function, allowing mathematicians to make informed predictions and estimations.

Seeking a Reference or Proof

Alright, so here's where I'm at. I've been digging through the literature, but I haven't been able to find a clear, self-contained proof or a definitive reference for this specific asymptotic formula. I'm hoping someone here might be able to point me in the right direction.

What I'm Looking For

1. A Reference: If you know of a paper, book, or set of lecture notes where this result is explicitly stated and proven, that would be amazing!
2. A Proof Sketch: If a full proof is too lengthy, even a sketch of the main ideas and techniques involved would be incredibly helpful. I'm particularly interested in understanding what tools from analytic number theory are typically used to tackle this kind of problem.
3. Related Results: If you're not aware of this exact formula, but you know of similar results or techniques that might be relevant, please share! Even partial information can help me piece together the puzzle.

Possible Approaches and Techniques

From what I've gathered, some potential approaches might involve:

• The Argument Principle: This is a powerful tool from complex analysis that relates the number of zeros and poles of a function inside a contour to the integral of its logarithmic derivative around the contour.
• Explicit Formulas: These are formulas that relate sums over the zeros of the zeta function to sums over prime numbers. They often involve intricate analysis and careful handling of error terms.
• Techniques from Random Matrix Theory: There are deep connections between the distribution of zeros of the zeta function and the eigenvalues of random matrices. These connections might provide insights into the behavior of the sum in question.

Why This is a Challenge

The reason this problem is challenging is that it requires a delicate balance between complex analysis, number theory, and potentially probabilistic methods. The Riemann zeta function is notoriously difficult to work with, and even small changes in the assumptions or the form of the sum can lead to significant complications.

Community Input Needed!

So, guys, if you have any insights, references, or ideas related to this problem, please share them! I'm really eager to understand this result better, and I believe that the collective knowledge of this community can help me make significant progress. Any help would be greatly appreciated!

Thanks in advance for your time and expertise. Let's crack this zeta function nut together!