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Hey,

I remember that in an interview, Selberg said that what Conne did was essentially getting a "different access" to the explicit formula, but that his proof did not yield new "hard facts" about the riemann zeta function. However take me with a grain of salt: I am not an expert in algebraic fields.

At the same time analytic methods give us information that are of a statistical nature. So for example they are quite well suited to prove that there is a positive proportion of the zeros on the half-line (40%, which is impressive enough I think). They can also show that almost all zeroes are concentrated in a box of height T and width $10^{10}/\log T$ around the line $\sigma = 1/2$. So those results are very useful in their own rights: For example they allow us to prove that the prime number theorem holds in intervals of length $x^{1/2+1/10}$ something you would expect to know only assuming the RH or a quasi-RH. In fact for many arithmetic applications the resultresults we already know allow us (often/sometimes?) to by-pass the Riemann Hypothesis.

So I wouldn't say that all approaches failed hopelessly. We are armed to deal with the problem, and paraphrasing Montgomery: "Sometimes I have the impression that we are missing just a fundamental insight to prove the RH" .

Hey,

I remember that in an interview, Selberg said that what Conne did was essentially getting a "different access" to the explicit formula, but that his proof did not yield new "hard facts" about the riemann zeta function. However take me with a grain of salt: I am not an expert in algebraic fields.

At the same time analytic methods give us information that are of a statistical nature. So for example they are quite well suited to prove that there is a positive proportion of the zeros on the half-line (40%, which is impressive enough I think). They can also show that almost all zeroes are concentrated in a box of height T and width $10^{10}/\log T$ around the line $\sigma = 1/2$. So those results are very useful in their own rights: For example they allow us to prove that the prime number theorem holds in intervals of length $x^{1/2+1/10}$ something you would expect to know only assuming the RH or a quasi-RH. In fact for many arithmetic applications the result we already know allow us to by-pass the Riemann Hypothesis.

So I wouldn't say that all approaches failed hopelessly. We are armed to deal with the problem, and paraphrasing Montgomery: "Sometimes I have the impression that we are missing just a fundamental insight to prove the RH" .

Hey,

I remember that in an interview, Selberg said that what Conne did was essentially getting a "different access" to the explicit formula, but that his proof did not yield new "hard facts" about the riemann zeta function. However take me with a grain of salt: I am not an expert in algebraic fields.

At the same time analytic methods give us information that are of a statistical nature. So for example they are quite well suited to prove that there is a positive proportion of the zeros on the half-line (40%, which is impressive enough I think). They can also show that almost all zeroes are concentrated in a box of height T and width $10^{10}/\log T$ around the line $\sigma = 1/2$. So those results are very useful in their own rights: For example they allow us to prove that the prime number theorem holds in intervals of length $x^{1/2+1/10}$ something you would expect to know only assuming the RH or a quasi-RH. In fact for many arithmetic applications the results we already know allow us (often/sometimes?) to by-pass the Riemann Hypothesis.

So I wouldn't say that all approaches failed hopelessly. We are armed to deal with the problem, and paraphrasing Montgomery: "Sometimes I have the impression that we are missing just a fundamental insight to prove RH" .

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anon
anon
  • 143
  • 8

Hey,

I remember that in an interview, Selberg said that what Conne did was essentially getting a "different access" to the explicit formula, but that his proof did not yield new "hard facts" about the riemann zeta function. However take me with a grain of salt: I am not an expert in algebraic fields.

At the same time analytic methods give us information that are of a statistical nature. So for example they are quite well suited to prove that there is a positive proportion of the zeros on the half-line (40%, which is impressive enough I think). They can also show that almost all zeroes are concentrated in a box of height T and width $10^{10}/\log T$ around the line $\sigma = 1/2$. So those results are very useful in their own rights: For example they allow us to prove that the prime number theorem holds in intervals of length $x^{1/2+1/10}$ something you would expect to know only assuming the RH or a quasi-RH. In fact for many arithmetic applications the result we already know allow us to by-pass the Riemann Hypothesis.

So I wouldn't say that all approaches failed hopelessly. We are armed to deal with the problem, and paraphrasing Montgomery: "Sometimes I have the impression that we are missing just a fundamental insight to prove the RH" .