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edit approved Apr 29, 2014 at 0:07
Alex Becker
edit approved Apr 29, 2014 at 0:07
Alex Becker
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I have seen justifications for various conjectures in analytic number theory, e.g. the (generalized) Riemann hypothesis, the Chowla conjectures, etc. justified by a heuristics in which the relevant arithmetic functions are modeled by random variables.

In the setting of primes in arithmetic progression, these heuristics say that the error term $E(x;q,a)$ for primes in particular primitive residue class should be approximately on the order of magnitude of $\frac{\sqrt{x}}{\varphi(q)}$, which is approximately GRH. If one sums this for $q$ up to $\sqrt{x}$, then the accumulated error is approximately on the order of magnitude of $x$, and the Bombieri-Vinogradov theorem actually establishes a boundedbound of this nature.

The Elliott-Halberstam conjecture asserts that we actually get an error bound of $\frac{x}{(\log x)^A}$ for any fixed $A$ if we sum over moduli all the way up to $x^{1-\epsilon}$. My question is ifwhether there is a convincing heuristic model that this should be the case, since the standard one that suggests GRH doesn't seem to accommodate such large moduli.

I have seen justifications for various conjectures in analytic number theory, e.g. the (generalized) Riemann hypothesis, the Chowla conjectures, etc. justified by a heuristics in which the relevant arithmetic functions are modeled by random variables.

In the setting of primes in arithmetic progression, these heuristics say that the error term $E(x;q,a)$ for primes in particular primitive residue class should be approximately on the order of magnitude of $\frac{\sqrt{x}}{\varphi(q)}$, which is approximately GRH. If one sums this for $q$ up to $\sqrt{x}$, then the accumulated error is approximately on the order of magnitude of $x$, and the Bombieri-Vinogradov theorem actually establishes a bounded of this nature.

The Elliott-Halberstam conjecture asserts that we actually get an error bound of $\frac{x}{(\log x)^A}$ for any fixed $A$ if we sum over moduli all the way up to $x^{1-\epsilon}$. My question is if there is a convincing heuristic model that this should be the case, since the standard one that suggests GRH doesn't seem to accommodate such large moduli.

I have seen justifications for various conjectures in analytic number theory, e.g. the (generalized) Riemann hypothesis, the Chowla conjectures, etc. justified by a heuristics in which the relevant arithmetic functions are modeled by random variables.

In the setting of primes in arithmetic progression, these heuristics say that the error term $E(x;q,a)$ for primes in particular primitive residue class should be approximately on the order of magnitude of $\frac{\sqrt{x}}{\varphi(q)}$, which is approximately GRH. If one sums this for $q$ up to $\sqrt{x}$, then the accumulated error is approximately on the order of magnitude of $x$, and the Bombieri-Vinogradov theorem actually establishes a bound of this nature.

The Elliott-Halberstam conjecture asserts that we actually get an error bound of $\frac{x}{(\log x)^A}$ for any fixed $A$ if we sum over moduli all the way up to $x^{1-\epsilon}$. My question is whether there is a convincing heuristic model that this should be the case, since the standard one that suggests GRH doesn't seem to accommodate such large moduli.

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Why believe the Elliott-Halberstam conjecture?

I have seen justifications for various conjectures in analytic number theory, e.g. the (generalized) Riemann hypothesis, the Chowla conjectures, etc. justified by a heuristics in which the relevant arithmetic functions are modeled by random variables.

In the setting of primes in arithmetic progression, these heuristics say that the error term $E(x;q,a)$ for primes in particular primitive residue class should be approximately on the order of magnitude of $\frac{\sqrt{x}}{\varphi(q)}$, which is approximately GRH. If one sums this for $q$ up to $\sqrt{x}$, then the accumulated error is approximately on the order of magnitude of $x$, and the Bombieri-Vinogradov theorem actually establishes a bounded of this nature.

The Elliott-Halberstam conjecture asserts that we actually get an error bound of $\frac{x}{(\log x)^A}$ for any fixed $A$ if we sum over moduli all the way up to $x^{1-\epsilon}$. My question is if there is a convincing heuristic model that this should be the case, since the standard one that suggests GRH doesn't seem to accommodate such large moduli.