Can Riemann's Function Prove an Equation for Prime Numbers?

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In summary, the conversation discusses the function \psi(x) defined as the sum of exponential terms and its relation to an equation involving x and its reciprocal. The speaker mentions the possibility of proving this equation and suggests reading a book by Harold Edwards for more information.
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jostpuur
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Assuming that I understood correctly one claim from the Riemann's On the Number of Prime Numbers less than a Given Quantity, then if we define a function

[tex]
\psi:]0,\infty[\to\mathbb{R},\quad \psi(x) = \sum_{n=1}^{\infty} e^{-n^2\pi x},
[/tex]

it satisfies an equation

[tex]
2\psi(x) + 1 = x^{-\frac{1}{2}}\Big(2\psi\big(\frac{1}{x}\big) + 1\Big).
[/tex]

Anyone knowing how to prove that?
 
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well just glancing at it, doubling that function and adding one would appear to result from just summing over all integers n. see if that helps.
doesn't seem to.

or read harold edwards book studying this paper in detail.
 

What is Riemann's paper?

Riemann's paper, also known as "On the Number of Prime Numbers Less Than a Given Quantity", is a mathematical paper written by German mathematician Bernhard Riemann in 1859. It is considered one of the most influential papers in the field of number theory.

What is the main idea behind Riemann's paper?

The main idea behind Riemann's paper is the connection between the distribution of prime numbers and the behavior of the Riemann zeta function. Riemann proposed a hypothesis that has become one of the most important unsolved problems in mathematics, known as the Riemann Hypothesis.

What is the significance of Riemann's paper?

Riemann's paper has had a profound impact on many areas of mathematics, including number theory, analysis, and geometry. It has also influenced the development of other branches of science, such as physics and computer science. The Riemann Hypothesis, in particular, has been a major focus of research for over 150 years.

What are some key concepts discussed in Riemann's paper?

Riemann's paper covers a wide range of topics, including the prime number theorem, the Riemann zeta function, and the Riemann Hypothesis. It also introduces the concept of the Riemann surface, a geometric object that provides a way to visualize the behavior of the Riemann zeta function.

What is the current status of the Riemann Hypothesis?

The Riemann Hypothesis remains one of the most famous unsolved problems in mathematics. Many mathematicians have attempted to prove or disprove it, but so far, no one has been successful. It is still an active area of research, and many mathematicians believe that solving the Riemann Hypothesis could have significant implications for other areas of mathematics.

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