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This is my third question on this site regarding Montgomery's conjecture -- and I apologize if this is too much -- but I am still not understanding well why this conjecture is believed to be true.

The conjecture I am talking about is as follows (I am giving the slightly corrected version of Freidlander-Granville). et $q>1$ be an integer, $a$ an integer coprime to $q$, $\psi(q,a,x) = \sum_{p^\alpha < x, p^\alpha \equiv a \pmod{q}} \Lambda(n)$. Then:

Conjecture for $x>q$, one has $\psi(x,q) = \frac{x}{\phi(q)} + O(x^{1/2+\epsilon} q^{-1/2})$, with an implied constant depending only on $\epsilon$.

In his answer to my preceding question, Matt Young gives the following heuristic: For $\chi$ a non-principal Dirichlet character of $(\mathbb Z/q\mathbb Z)^\ast$, one has under GRH $\psi(\chi,x) = O(x^{1/2+\epsilon})$. Now $\psi(q,a,x)$ is the arithmetic average of the (approximately $q$) terms $\chi^{-1}(a) \psi(\chi,x)$, and if those (weighted) terms are in random position one should expect their sum to have a norm roughly $q^{1/2}$ the norm of the individual term (by Einstein's Brownian motion theorem if you like), giving the conjecture.

This heuristic helped me a lot then but now I'd like to go further.

Is there a good reason to believe that the $\chi^{-1}(a) \psi(\chi,x)$ are kind of randomly distributed, for every $a$ relatively prime to $q$? Could this expectation have some link, through the explicit formulas, with the conjectures on the position of zeros on the critical line of Dirichlet L-functions (assuming GRH)?

This is my third question on this site regarding Montgomery's conjecture -- and I apologize if this is too much -- but I am still not understanding well why this conjecture is believed to be true.

The conjecture I am talking about is as follows (I am giving the slightly corrected version of Freidlander-Granville). et $q>1$ be an integer, $a$ an integer coprime to $q$, $\psi(q,a,x) = \sum_{p^\alpha < x, p^\alpha \equiv a \pmod{q}} \Lambda(n)$. Then:

Conjecture for $x>q$, one has $\psi(x,q) = \frac{x}{\phi(q)} + O(x^{1/2+\epsilon} q^{-1/2})$, with an implied constant depending only on $\epsilon$.

In his answer to my preceding question, Matt Young gives the following heuristic: For $\chi$ a non-principal Dirichlet character of $(\mathbb Z/q\mathbb Z)^\ast$, one has under GRH $\psi(\chi,x) = O(x^{1/2+\epsilon})$. Now $\psi(q,a,x)$ is the arithmetic average of the (approximately $q$) terms $\chi^{-1}(a) \psi(\chi,x)$, and if those (weighted) terms are in random position one should expect their sum to have a norm roughly $q^{1/2}$ the norm of the individual term (by Einstein's Brownian motion theorem if you like), giving the conjecture.

This heuristic helped me a lot then but now I'd like to go further.

Is there a good reason to believe that the $\chi^{-1}(a) \psi(\chi,x)$ are kind of randomly distributed, for every $a$ relatively prime to $q$? Could this expectation have some link, through the explicit formulas, with the conjectures on the position of zeros on the critical line of Dirichlet L-functions (assuming GRH)?

This is my third question on this site regarding Montgomery's conjecture -- and I apologize if this is too much -- but I am still not understanding well why this conjecture is believed to be true.

The conjecture I am talking about is as follows (I am giving the slightly corrected version of Freidlander-Granville). et $q>1$ be an integer, $a$ an integer coprime to $q$, $\psi(q,a,x) = \sum_{p^\alpha < x, p^\alpha \equiv a \pmod{q}} \Lambda(n)$. Then:

Conjecture for $x>q$, one has $\psi(x,q) = \frac{x}{\phi(q)} + O(x^{1/2+\epsilon} q^{-1/2})$, with an implied constant depending only on $\epsilon$.

In his answer to my preceding question, Matt Young gives the following heuristic: For $\chi$ a non-principal Dirichlet character of $(\mathbb Z/q\mathbb Z)^\ast$, one has under GRH $$\psi(\chi,x) = O(x^{1/2+\epsilon})$. Now $\psi(q,a,x)$ is the arithmetic average of the (approximately $q$) terms $\chi^{-1}(a) \psi(\chi,x)$, and if those (weighted) terms are in random position one should expect their sum to have a norm roughly $q^{1/2}$ the norm of the individual term (by Einstein's Brownian motion theorem if you like), giving the conjecture.

This heuristic helped me a lot then but now I'd like to go further.

Is there a good reason to believe that the $\chi^{-1}(a) \psi(\chi,x)$ are kind of randomly distributed, for every $a$ relatively prime to $q$? Could this expectation have some link, through the explicit formulas, with the conjectures on the position of zeros on the critical line of Dirichlet L-functions (assuming GRH)?

This is my third question on this site regarding Montgomery's conjecture -- and I apologize if this is too much -- but I am still not understanding well why this conjecture is believed to be true.

The conjecture I am talking about is as follows (I am giving the slightly corrected version of Freidlander-Granville). et $q>1$ be an integer, $a$ an integer coprime to $q$, $\psi(q,a,x) = \sum_{p^\alpha < x, p^\alpha \equiv a \pmod{q}} \Lambda(n)$. Then:

Conjecture for $x>q$, one has $\psi(x,q) = \frac{x}{\phi(q)} + O(x^{1/2+\epsilon} q^{-1/2})$, with an implied constant depending only on $\epsilon$.

In his answer to my preceding question, Matt Young gives the following heuristic: For $\chi$ a non-principal Dirichlet character of $(\mathbb Z/q\mathbb Z)^\ast$, one has under GRH $\psi(\chi,x) = O(x^{1/2+\epsilon})$. Now $\psi(q,a,x)$ is the arithmetic average of the (approximately $q$) terms $\chi^{-1}(a) \psi(\chi,x)$, and if those (weighted) terms are in random position one should expect their sum to have a norm roughly $q^{1/2}$ the norm of the individual term (by Einstein's Brownian motion theorem if you like), giving the conjecture.

This heuristic helped me a lot then but now I'd like to go further.

Is there a good reason to believe that the $\chi^{-1}(a) \psi(\chi,x)$ are kind of randomly distributed, for every $a$ relatively prime to $q$? Could this expectation have some link, through the explicit formulas, with the conjectures on the position of zeros on the critical line of Dirichlet L-functions (assuming GRH)?

This is my third question on this site regarding Montgomery's conjecture -- and I apologize if this is too much -- but I am still not understanding well why this conjecture is believed to be true.

The conjecture I am talking about is as follows (I am giving the slightly corrected version of Freidlander-Granville). et $q>1$ be an integer, $a$ an integer coprime to $q$, $\psi(q,a,x) = \sum_{p^\alpha < x, p^\alpha \equiv a \pmod{q}} \Lambda(n)$. Then:

Conjecture for $x>q$, one has $\psi(x,q) = \frac{x}{\phi(q)} + O(x^{1/2+\epsilon} q^{-1/2})$, with an implied constant depending only on $\epsilon$.

In his answer to my preceding [question][1], Matt Young gives the following heuristic: For $\chi$ a non-principal Dirichlet character of $(\mathbb Z/q\mathbb Z)^\ast$, one has under GRH $$\psi(\chi,x) = O(x^{1/2+\epsilon})$. Now $\psi(q,a,x)$ is the arithmetic average of the (approximately $q$) terms $\chi^{-1}(a) \psi(\chi,x)$, and if those (weighted) terms are in random position one should expect their sum to have a norm roughly $q^{1/2}$ the norm of the individual term (by Einstein's Brownian motion theorem if you like), giving the conjecture.

This heuristic helped me a lot then but now I'd like to go further.

Is there a good reason to believe that the $\chi^{-1}(a) \psi(\chi,x)$ are kind of randomly distributed, for every $a$ relatively prime to $q$? Could this expectation have some link, through the explicit formulas, with the conjectures on the position of zeros on the critical line of Dirichlet L-functions (assuming GRH)?

This is my third question on this site regarding Montgomery's conjecture -- and I apologize if this is too much -- but I am still not understanding well why this conjecture is believed to be true.

The conjecture I am talking about is as follows (I am giving the slightly corrected version of Freidlander-Granville). et $q>1$ be an integer, $a$ an integer coprime to $q$, $\psi(q,a,x) = \sum_{p^\alpha < x, p^\alpha \equiv a \pmod{q}} \Lambda(n)$. Then:

Conjecture for $x>q$, one has $\psi(x,q) = \frac{x}{\phi(q)} + O(x^{1/2+\epsilon} q^{-1/2})$, with an implied constant depending only on $\epsilon$.

In his answer to my preceding question, Matt Young gives the following heuristic: For $\chi$ a non-principal Dirichlet character of $(\mathbb Z/q\mathbb Z)^\ast$, one has under GRH $$\psi(\chi,x) = O(x^{1/2+\epsilon})$. Now $\psi(q,a,x)$ is the arithmetic average of the (approximately $q$) terms $\chi^{-1}(a) \psi(\chi,x)$, and if those (weighted) terms are in random position one should expect their sum to have a norm roughly $q^{1/2}$ the norm of the individual term (by Einstein's Brownian motion theorem if you like), giving the conjecture.

This heuristic helped me a lot then but now I'd like to go further.

Is there a good reason to believe that the $\chi^{-1}(a) \psi(\chi,x)$ are kind of randomly distributed, for every $a$ relatively prime to $q$? Could this expectation have some link, through the explicit formulas, with the conjectures on the position of zeros on the critical line of Dirichlet L-functions (assuming GRH)?