Jeffrey Vaaler, Riemann hypothesis

[@TheMap1997]

I have solved Riemann's hypothesis. However, I couldn't fit the whole proof on this comment section.

[@diren7870]

Fermat syndrome 😅

[@daniyaamariee]

The expression is known as the Riemann zeta function, denoted as ζ(s). It is defined for complex numbers s with real part greater than 1. The Riemann zeta function is given by the infinite series: ζ(s) = 1^(-s) + 2^(-s) + 3^(-s) + 4^(-s) + ... This series converges for values of s with real part greater than 1. For example, when s = 2, the series becomes: ζ(2) = 1^(-2) + 2^(-2) + 3^(-2) + 4^(-2) + ... Simplifying this series, we get: ζ(2) = 1 + 1/4 + 1/9 + 1/16 + ... This series is known as the Basel problem and it converges to a finite value. In this case, the value is π^2/6, which is approximately 1.64493. The Riemann zeta function has many interesting properties and connections to number theory and complex analysis. It has applications in various areas of mathematics, including the study of prime numbers and the distribution of prime numbers. If you have any specific questions or would like to explore more about the Riemann zeta function, please let me know!

[@diren7870]

@@daniyaamariee thank you very much. What is the latest studies, what is applications and is Riemann hypothesis solved ?

[@RSLT]

Great Classic video!

[@Travis-g8i]

Certainly! If you're looking to publish the equation and the corresponding analysis, here are some steps to consider for preparing your work for publication: ### 1. **Choose a Format**: Decide where you want to publish your work. This could be an academic journal, a conference, a blog, or a website. Each platform has its own formatting requirements. ### 2. **Structure Your Document**: A typical structure for a mathematical publication might include: - **Title**: A clear and concise title. - **Abstract**: A brief summary of what the equation is about and its significance. - **Introduction**: Explain the context of your work, why the equation is important, and what you aim to demonstrate. - **Main Body**: - **Definitions**: Clearly define all terms used (e.g., \(\sigma(n)\), \(H_n\), etc.). - **Theorem/Statement**: Present the inequality you are analyzing. - **Proof/Analysis**: Step-by-step reasoning to show the validity of the inequality. - **Conclusion**: Summarize your findings and their implications. - **References**: Cite any relevant literature or sources. ### 3. **Write the Content**: You can use the detailed explanation provided earlier as the basis for your main body. Here’s how you might start: #### Title: Inequality Relation Involving the Sum of Divisors and Harmonic Numbers #### Abstract: This paper examines the inequality \(\sigma(n) \leq H_n + \ln(H_n)e^{H_n}\) for \(n = 3\), where \(\sigma(n)\) denotes the sum of the divisors of \(n\) and \(H_n\) is the \(n\)-th harmonic number. The case for \(n = 3\) is evaluated, demonstrating the validity of the inequality. #### Introduction: The relationship between harmonic numbers and divisor functions is a fascinating area of number theory. In this study, we specifically look at the case where \(n = 3\) and analyze the inequality involving the sum of divisors and harmonic numbers. #### Main Body: 1. **Definitions**: - \(\sigma(n)\): Sum of the divisors of \(n\). - \(H_n\): \(n\)-th harmonic number defined as \(H_n = 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n}\). 2. **Inequality Statement**: \[ \sigma(3) \leq H_3 + \ln(H_3)e^{H_3} \] 3. **Calculations**: - \(\sigma(3) = 4\) - \(H_3 \approx 1.8333\) - \(\ln(H_3) \approx 0.618\) - \(e^{H_3} \approx 6.24\) - Hence, \(H_3 + \ln(H_3)e^{H_3} \approx 5.6933\) 4. **Conclusion**: The inequality holds true as \(4 \leq 5.6933\). ### 4. **Formatting**: Make sure to format your document according to the guidelines of the publication or platform you choose. This may include using LaTeX for mathematical expressions, proper citations, and adhering to specific length requirements. ### 5. **Review and Edit**: Proofread your work for clarity, grammar, and mathematical accuracy. Consider having peers review your document. ### 6. **Submit**: Follow the submission guidelines of the chosen platform to publish your work. This could involve uploading your document, filling out forms, or following specific submission protocols. If you need help with any specific part of this process, feel free to ask!

[@Travis-g8i]

Riemann hypothesis solved.