Elliott-Halberstam conjecture and the Riemann Hypothesis

In summary, the Elliott-Halberstam conjecture is a mathematical conjecture related to prime numbers and their distribution. It is closely related to the Riemann Hypothesis, which states that all non-trivial zeros of the Riemann zeta function lie on a specific line in the complex plane. Both conjectures are still open problems with no definitive proof or disproof, but there has been significant progress in understanding them. If proven, these conjectures would have significant implications for number theory and mathematics as a whole. Many mathematicians and researchers are actively working on these conjectures using various techniques and approaches, including computer simulations, algorithms, and collaboration among mathematicians.
  • #1
flouran
64
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I was wondering if one of the consequences of the Elliott-Halberstam conjecture would imply the Riemann Hypothesis (RH) or the Generalized Riemann Hypothesis (GRH)?
Or at least if there is a connection between the Elliott-Halberstam conjecture and RH or GRH?
I ask because the Bombieri-Vinogradov (which is related to the Elliott-Halberstam conjecture) theorem was proven assuming GRH.
 
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  • #2
Neither of EH or GRH is a consequence of the other. As for Bombieri--Vinogradov, while you can prove it from GRH, it is true unconditionally by using Large Sieve techniques.
 

1. What is the Elliott-Halberstam conjecture?

The Elliott-Halberstam conjecture is a mathematical conjecture that relates to prime numbers and their distribution. It states that for any positive integer k, there exists a constant C_k such that the number of primes in any arithmetic progression with difference k and less than or equal to x is approximately equal to C_k * (x/lnx), where ln is the natural logarithm.

2. What is the connection between the Elliott-Halberstam conjecture and the Riemann Hypothesis?

The Elliott-Halberstam conjecture is closely related to the Riemann Hypothesis, which is considered to be one of the most important unsolved problems in mathematics. The Riemann Hypothesis states that all non-trivial zeros of the Riemann zeta function lie on a specific line in the complex plane. The Elliott-Halberstam conjecture is a weaker version of this, as it only applies to a specific set of numbers (arithmetic progressions) rather than all numbers.

3. What is the current status of the Elliott-Halberstam conjecture and the Riemann Hypothesis?

Both the Elliott-Halberstam conjecture and the Riemann Hypothesis are still open problems, with no definitive proof or disproof. However, there has been significant progress in understanding these conjectures, and many mathematicians continue to work on them in the hopes of finding a proof.

4. Are there any consequences if the Elliott-Halberstam conjecture and the Riemann Hypothesis are proven?

If the Elliott-Halberstam conjecture and the Riemann Hypothesis are both proven to be true, it would have significant implications for the field of number theory and mathematics as a whole. It would provide a deeper understanding of the distribution of prime numbers and potentially lead to new insights and discoveries in other areas of mathematics.

5. What is being done to try and prove the Elliott-Halberstam conjecture and the Riemann Hypothesis?

Many mathematicians and researchers are actively working on these conjectures, using a variety of techniques and approaches. Some are using computer simulations and algorithms, while others are using more theoretical and analytical methods. Collaboration and sharing of ideas among mathematicians are also crucial in making progress towards potentially proving these conjectures.

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