Connection of zeta to primes: less Euler than error or L-functions?

In summary, the Riemann Hypothesis (RH) is often associated with prime numbers due to the equivalence of the zeta function and Euler's product formula on Re(s)>1. However, the true significance of the RH to primes is either seen through the formulas for error terms in the prime number theorem, such as the von Koch/Schoenfeld's formula, or through the non-trivial zeros of the Riemann zeta function in explicit formulas for L-functions. These relationships ultimately stem from the Euler product and involve taking the logarithmic derivative and performing contour integration. The Riemann hypothesis plays a crucial role in these results, as it allows for a smaller error term in the prime number theorem.
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nomadreid
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The connection between the Riemann zeta function and primes is often stated as lying in Euler's product formula, but wouldn't it be more correct that it lies in the zeta function role in the error function for the prime number theorem, or perhaps the explicit formulas of L-functions?
Often I read that the Riemann Hypothesis (RH) is related to prime numbers because of the equivalence on Re(s)>1 of the zeta function and Eurler's product formula
Euler.PNG

, but is it more accurate that the relevance of the RH to primes (or vice-versa) is either that the RH implies formulas for the error terms of the prime number theorem, for example, the von Koch/Schoenfeld's

von Koch.PNG

(as given in https://en.wikipedia.org/wiki/Riemann_hypothesis#Distribution_of_prime_numbers)

or even more so that the ρ in some of the explicit formulas for L-functions ranges over the non-trivial zeros of the Riemann zeta function (as outlined in https://en.wikipedia.org/wiki/Explicit_formulae_for_L-functions#Riemann's_explicit_formula)?

Or do the latter two results simply hark back to the Euler result in some way?
 
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In general, all of relationships between the zeta function and distribution of primes come from the Euler product. Exactly how depends on which identity you want, but the general idea is that you take the logarithmic derivative of the Euler product and perform contour integration. The further you can push the region of integration to the left, the better estimates you get, which is why the Riemann hypothesis would imply a smaller error term in the prime number theorem (since if RH is true, then ##\zeta'/\zeta## doesn't have any poles until you reach the line of real part ##1/2##).
 
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Thank you, Infrared. Excellent explanation. :biggrin:
 

1. What is the connection between the zeta function and prime numbers?

The zeta function, denoted by ζ(s), is a mathematical function that is closely related to the distribution of prime numbers. In fact, the Riemann Hypothesis, one of the most famous unsolved problems in mathematics, states that all non-trivial zeros of the zeta function lie on the critical line Re(s) = 1/2. This means that the behavior of the zeta function is intimately connected to the distribution of prime numbers.

2. How is the zeta function used to study prime numbers?

The zeta function has several properties that make it a useful tool for studying prime numbers. One of the most important is the Euler product formula, which expresses the zeta function as an infinite product involving all prime numbers. This allows us to relate the behavior of the zeta function to the properties of prime numbers. Additionally, the zeta function is closely connected to other important mathematical functions, such as the Riemann xi function and the Dirichlet L-functions, which also have connections to prime numbers.

3. What is the role of Euler in the connection between zeta and primes?

Leonhard Euler was a Swiss mathematician who made significant contributions to the study of prime numbers and the zeta function. In particular, he discovered the Euler product formula for the zeta function, which is a key tool in understanding the connection between the two. Euler also made important contributions to the study of the Riemann Hypothesis, which is closely related to the behavior of the zeta function on the critical line.

4. How does the connection between zeta and primes relate to L-functions?

L-functions, named after the mathematician Peter Gustav Lejeune Dirichlet, are a class of mathematical functions that are closely related to the zeta function. In fact, the zeta function can be seen as a special case of an L-function. These functions are also closely connected to prime numbers, as they can be used to study the distribution of primes in arithmetic progressions. The Riemann Hypothesis also has implications for the behavior of L-functions.

5. What are some potential errors or limitations in using the zeta function to study primes?

While the zeta function has proven to be a powerful tool in studying prime numbers, there are still some limitations and potential errors that must be considered. For example, the Riemann Hypothesis, which is closely connected to the zeta function, has yet to be proven and remains one of the most famous unsolved problems in mathematics. Additionally, the zeta function only provides information about the distribution of primes on the critical line, and may not give a complete picture of the behavior of prime numbers in other areas.

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