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I am studying Sieve theory from Iwaniec's notes. I have come across a theorem which estimates $\varphi(x,N)=\#\{1\leq n \leq x:(n,N)=1\}$, where $N$ is product of distinct primes.

Let's define $R(x,N)=\sum_{d|N}\mu(d)\{\frac{x}{d}\}=x\sum_{d|N}\frac{\mu(d)}{d}-\sum_{d|N}\mu(d)[\frac{x}{d}]=x\frac{\varphi(N)}{N}-\varphi(x,N)$.

Also say, $K(N)=\displaystyle\sum_{n=1}^{\infty}(\frac{R(n.N)}{n})^2$.

Now let's introduce a weighted $l^2$ space $\mathcal{H}$ with norm defined as $||x||^2:=\sum_{n=1}^{\infty}\frac{x(n)^2}{n^2}$ (provided convergent). Consider a Hilbert space $\mathcal{M}$ generated by $\left\{\gamma_n|\gamma_n(k)=\left\{\frac{k}{n}\right\},k=1,2,...;n>1\right\}$ inside $\mathcal{H}$.

It is known from Bagchi's result that the RH is true if and only if $\gamma=(1,1,...)\in\mathcal{M}$. It can be deduced that, above is also equivalent to the statement that if $x_m=\sum_{n=2}^{m}\mu(n)\gamma_n$ then $x_m\to_{strongly} \gamma$ as $m\to\infty$.

If we define, $\bar x_m=\sum_{n|m}\mu(n)\gamma_n$ we see $K(N)=||x_N||^2$.

I think that it will be easy to show, $\bar x_m\to_{strongly} \gamma$ if and only if $x_m\to_{strongly} \gamma$.

My questions are:

1. Can this methodology create some other reformulation of RH? Like, "RH is true if and only if $K(N)\to1$"?

2. Does there exist any sieve method which gives a good bound of $K(N)$? Honestly, I know a very little of sieve methods. Whatever sieve bounds I have seen those force upper bounds of $K(N)$ to go to infinity.

3. Has any work been done in this way? If not, will it be a good reformulation? If yes, how one can approch further? As, $K(N)=\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^2}[\frac{\varphi(n,N)}{n}-\frac{\varphi(N)}{N}]^2$ looks like a 'good' arithmetical as well as a probabilistic function.

I am studying Sieve theory from Iwaniec's notes. I have come across a theorem which estimates $\varphi(x,N)=\#\{1\leq n \leq x:(n,N)=1\}$, where $N$ is product of distinct primes.

Let's define $R(x,N)=\sum_{d|N}\mu(d)\{\frac{x}{d}\}=x\sum_{d|N}\frac{\mu(d)}{d}-\sum_{d|N}\mu(d)[\frac{x}{d}]=x\frac{\varphi(N)}{N}-\varphi(x,N)$.

Also say, $K(N)=\displaystyle\sum_{n=1}^{\infty}(\frac{R(n.N)}{n})^2$.

Now let's introduce a weighted $l^2$ space $\mathcal{H}$ with norm defined as $||x||^2:=\sum_{n=1}^{\infty}\frac{x(n)^2}{n^2}$ (provided convergent). Consider a Hilbert space $\mathcal{M}$ generated by $\left\{\gamma_n|\gamma_n(k)=\left\{\frac{k}{n}\right\},k=1,2,...;n>1\right\}$ inside $\mathcal{H}$.

It is known from Bagchi's result that the RH is true if and only if $\gamma=(1,1,...)\in\mathcal{M}$. It can be deduced that, above is also equivalent to the statement that if $x_m=\sum_{n=2}^{m}\mu(n)\gamma_n$ then $x_m\to_{strongly} \gamma$ as $m\to\infty$.

If we define, $\bar x_m=\sum_{n|m}\mu(n)\gamma_n$ we see $K(N)=||x_N||^2$.

I think that it will be easy to show, $\bar x_m\to_{strongly} \gamma$ if and only if $x_m\to_{strongly} \gamma$.

My questions are:

1. Can this methodology create some other reformulation of RH? Like, "RH is true if and only if $K(N)\to1$"?

2. Does there exist any sieve method which gives a good bound of $K(N)$? Honestly, I know a very little of sieve methods. Whatever sieve bounds I have seen those force upper bounds of $K(N)$ to go to infinity.

3. Has any work been done in this way? If not, will it be a good reformulation? If yes, how one can approch further? As, $K(N)=\displaystyle\sum_{n=1}^{\infty}[\frac{\varphi(n,N)}{n}-\frac{\varphi(N)}{N}]^2$ looks like a 'good' arithmetical as well as a probabilistic function.

I am studying Sieve theory from Iwaniec's notes. I have come across a theorem which estimates $\varphi(x,N)=\#\{1\leq n \leq x:(n,N)=1\}$, where $N$ is product of distinct primes.

Let's define $R(x,N)=\sum_{d|N}\mu(d)\{\frac{x}{d}\}=x\sum_{d|N}\frac{\mu(d)}{d}-\sum_{d|N}\mu(d)[\frac{x}{d}]=x\frac{\varphi(N)}{N}-\varphi(x,N)$.

Also say, $K(N)=\displaystyle\sum_{n=1}^{\infty}(\frac{R(n.N)}{n})^2$.

Now let's introduce a weighted $l^2$ space $\mathcal{H}$ with norm defined as $||x||^2:=\sum_{n=1}^{\infty}\frac{x(n)^2}{n^2}$ (provided convergent). Consider a Hilbert space $\mathcal{M}$ generated by $\left\{\gamma_n|\gamma_n(k)=\left\{\frac{k}{n}\right\},k=1,2,...;n>1\right\}$ inside $\mathcal{H}$.

It is known from Bagchi's result that the RH is true if and only if $\gamma=(1,1,...)\in\mathcal{M}$. It can be deduced that, above is also equivalent to the statement that if $x_m=\sum_{n=2}^{m}\mu(n)\gamma_n$ then $x_m\to_{strongly} \gamma$ as $m\to\infty$.

If we define, $\bar x_m=\sum_{n|m}\mu(n)\gamma_n$ we see $K(N)=||x_N||^2$.

I think that it will be easy to show, $\bar x_m\to_{strongly} \gamma$ if and only if $x_m\to_{strongly} \gamma$.

My questions are:

1. Can this methodology create some other reformulation of RH? Like, "RH is true if and only if $K(N)\to1$"?

2. Does there exist any sieve method which gives a good bound of $K(N)$? Honestly, I know a very little of sieve methods. Whatever sieve bounds I have seen those force $K(N)$ to go to infinity.

3. Has any work been done in this way? If not, will it be a good reformulation? If yes, how one can approch further? As, $K(N)=\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^2}[\frac{\varphi(n,N)}{n}-\frac{\varphi(N)}{N}]^2$ looks like a 'good' arithmetical as well as a probabilistic function.

I am studying Sieve theory from Iwaniec's notes. I have come across a theorem which estimates $\varphi(x,N)=\#\{1\leq n \leq x:(n,N)=1\}$, where $N$ is product of distinct primes.

Let's define $R(x,N)=\sum_{d|N}\mu(d)\{\frac{x}{d}\}=x\sum_{d|N}\frac{\mu(d)}{d}-\sum_{d|N}\mu(d)[\frac{x}{d}]=x\frac{\varphi(N)}{N}-\varphi(x,N)$.

Also say, $K(N)=\displaystyle\sum_{n=1}^{\infty}(\frac{R(n.N)}{n})^2$.

Now let's introduce a weighted $l^2$ space $\mathcal{H}$ with norm defined as $||x||^2:=\sum_{n=1}^{\infty}\frac{x(n)^2}{n^2}$ (provided convergent). Consider a Hilbert space $\mathcal{M}$ generated by $\left\{\gamma_n|\gamma_n(k)=\left\{\frac{k}{n}\right\},k=1,2,...;n>1\right\}$ inside $\mathcal{H}$.

It is known from Bagchi's result that the RH is true if and only if $\gamma=(1,1,...)\in\mathcal{M}$. It can be deduced that, above is also equivalent to the statement that if $x_m=\sum_{n=2}^{m}\mu(n)\gamma_n$ then $x_m\to_{strongly} \gamma$ as $m\to\infty$.

If we define, $\bar x_m=\sum_{n|m}\mu(n)\gamma_n$ we see $K(N)=||x_N||^2$.

I think that it will be easy to show, $\bar x_m\to_{strongly} \gamma$ if and only if $x_m\to_{strongly} \gamma$.

My questions are:

1. Can this methodology create some other reformulation of RH? Like, "RH is true if and only if $K(N)\to1$"?

2. Does there exist any sieve method which gives a good bound of $K(N)$? Honestly, I know a very little of sieve methods. Whatever sieve bounds I have seen those force upper bounds of $K(N)$ to go to infinity.

3. Has any work been done in this way? If not, will it be a good reformulation? If yes, how one can approch further? As, $K(N)=\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^2}[\frac{\varphi(n,N)}{n}-\frac{\varphi(N)}{N}]^2$ looks like a 'good' arithmetical as well as a probabilistic function.