Exact Projections of the Real Part of ζ(s) Arithmetic Grids, Identities, Bounds, and a Mean-Square Formula in ℜ(s) > 1

1. Introduction

For $s=\sigma +it$ with $\sigma >1$, the Dirichlet expansion

$$\zeta \left(s\right)=\sum _{n\ge 1}{n}^{-s}=\sum _{n\ge 1}{\displaystyle \frac{1}{n^{\sigma }}}{e}^{-itlogn}$$

(1.1)

converges absolutely. Taking the real part,

$$Re\zeta (\sigma +it)=\sum _{n\ge 1}{\displaystyle \frac{cos(tlogn)}{n^{\sigma }}}.$$

(1.2)

This observation enables an exact decomposition of $Re\zeta$ by regrouping terms according to disjoint subsets of $\mathbb{N}$ (“arithmetic grids”). In this note, we formalize this decomposition and provide useful identities and bounds for classical grids (prime powers, multiples of primes) and for their double grid (products $p^{m}n$).

2. Notation

• $\mathbb{P}$ denotes the set of prime numbers; $\mathbb{N}=\{1,2,3,\dots \}$.
• $\omega \left(n\right)$: number of distinct prime factors of n; $\Lambda \left(n\right)$: von Mangoldt function.
• $P\left(s\right)={\sum }_{p\in \mathbb{P}}{p}^{-s}$; $\zeta \left(s\right)={\sum }_{n\ge 1}{n}^{-s}$ for $\Re \left(s\right)>1$.
• $1_S$ denotes the indicator function of a subset $S\subset \mathbb{N}$.
• For a prime P, $M_P\left(s\right)={\prod }_{p\le P}(1-p^{-s})$ (primorial mollifier).

3. Basic Results and Decomposition

Lemma 3.1 

(Decomposition by partitions). Let $\sigma >1$ and a disjoint partition $\mathbb{N}={⨆}_{j=1}^k{R}_j$. Then

$$Re\zeta (\sigma +it)=\sum _{j=1}^k\sum _{n\in R_j}{\displaystyle \frac{cos(tlogn)}{n^{\sigma }}}.$$

(3.1)

Proof. 

The series (1.1) converges absolutely for $\sigma >1$, allowing for rearrangement by blocks $R_j$; taking the real part yields the identity. □

Lemma 3.2 

(Additive layer as $\omega \left(n\right)$). For $\Re \left(s\right)>1$,

$$\sum _{n\ge 1}{\displaystyle \frac{\omega \left(n\right)}{n^s}}=\zeta \left(s\right)P\left(s\right),P\left(s\right)=\sum _{p\in \mathbb{P}}{\displaystyle \frac{1}{p^s}}.$$

(3.2)

Proof. 

Each integer n has $\omega \left(n\right)$ decompositions of the form $n=pm$ with p a prime and $m\in \mathbb{N}$. By direct counting, ${\sum }_n\omega \left(n\right)n^{-s}={\sum }_p{\sum }_m{\left(pm\right)}^{-s}=P\left(s\right)\zeta \left(s\right)$. □

Lemma 3.3 

(Multiplicative spine). For $\Re \left(s\right)>1$,

$$\sum _{n\ge 1}{\displaystyle \frac{\Lambda \left(n\right)}{n^s}}=-{\displaystyle \frac{{\zeta }^{\prime }\left(s\right)}{\zeta \left(s\right)}}.$$

(3.3)

Proof. 

Logarithmic differentiation of the Euler product for $\zeta \left(s\right)$. □

Proposition 3.4 

(Grid of prime powers). Let $1_{\mathrm{pp}}\left(n\right)=1$ if $n=p^m$ for some $p\in \mathbb{P}$ and $m\ge 1$, and 0 otherwise. For $\Re \left(s\right)>1$,

$$\sum _{n\ge 1}{\displaystyle \frac{1_{\mathrm{pp}}\left(n\right)}{n^s}}=\sum _{p\in \mathbb{P}}{\displaystyle \frac{p^{-s}}{1-p^{-s}}}.$$

(3.4)

Proof. 

Sum of independent geometric series over primes p. □

Lemma 3.5 

(Convolution of grids). Let $A,B\subset \mathbb{N}$ and $c\left(n\right)=(1_A*1_B)\left(n\right)={\sum }_{ab=n}{1}_A\left(a\right)1_B\left(b\right)$. For $\Re \left(s\right)>1$,

$$\sum _{n\ge 1}{\displaystyle \frac{c\left(n\right)}{n^s}}=\left(\sum _{a\in A}{a}^{-s}\right)\left(\sum _{b\in B}{b}^{-s}\right).$$

(3.5)

Proof. 

This is a standard identity for Dirichlet convolutions. □

4. Double Grid and Primorial Mollifier

We define the double grid as

$$\mathcal{D}=\{p^{m}n:p\in \mathbb{P},m\ge 1,n\in \mathbb{N}\}.$$

(4.1)

Proposition 4.1 

(Generator of the double grid). Let $w_{p,m}$ be a weight independent of n. For $\Re \left(s\right)>1$,

$$\sum _{p\in \mathbb{P}}\sum _{m\ge 1}\sum _{n\ge 1}{\displaystyle \frac{w_{p,m}}{{\left(p^{m}n\right)}^s}}=\left(\sum _{p\in \mathbb{P}}\sum _{m\ge 1}{\displaystyle \frac{w_{p,m}}{p^{ms}}}\right)\zeta \left(s\right).$$

(4.2)

Proof. 

Factor out the sum ${\sum }_{n\ge 1}{n}^{-s}=\zeta \left(s\right)$. □

Theorem 4.2 

(Primorial as a finite Dirichlet polynomial).  For a prime P and $\Re \left(s\right)>0$,

$$M_P\left(s\right):=\prod _{p\le P}(1-p^{-s})=\sum _{\begin{array}{c}n\ge 1\\ {\mu }^2\left(n\right)=1\\ P^{+}\left(n\right)\le P\end{array}}{\displaystyle \frac{\mu \left(n\right)}{n^s}},$$

(4.3)

where μ is the Möbius function and $P^{+}\left(n\right)$ is the largest prime factor of n.

Proof. 

Expansion of the finite product; only square-free integers n with $P^{+}\left(n\right)\le P$ appear. □

Lemma 4.3 

(Elementary bounds for the mollifier). For $\sigma =\Re \left(s\right)>1$,

$$log|M_P\left(s\right)|\le \sum _{p\le P}log(1+p^{-\sigma })\le \sum _{p\le P}{p}^{-\sigma },log\left|M_P\left(s\right)\right|\ge -\sum _{p\le P}{\displaystyle \frac{p^{-\sigma }}{1-p^{-\sigma }}}.$$

(4.4)

Proof. 

Use $log|1-z|\le log(1+|z\left|\right)$ and $log(1-x)\ge -x/(1-x)$ for $x\in (0,1)$. □

5. Application: A Mean-Square Formula

The following identity quantifies (in the $L^2$ norm on vertical lines) the contribution of layers constructed by sum over divisors.

Theorem 5.1 

(Mean-square for series with sum over divisors). Let $\sigma >1$ and $a:\mathbb{N}\to \mathbb{C}$ with finite support (or absolutely summable with weight $d^{-2\sigma }$). Define $b\left(n\right)={\sum }_{d\mid n}a\left(d\right)$ and

$$F\left(s\right)=\sum _{n\ge 1}{\displaystyle \frac{b\left(n\right)}{n^s}},\Re \left(s\right)=\sigma .$$

Then

$$\underset{T\to \infty }{lim}{\displaystyle \frac{1}{2T}}{\int }_{-T}^T|F(\sigma +it){|}^{2}dt=\zeta \left(2\sigma \right)\sum _{d,e\ge 1}{\displaystyle \frac{a\left(d\right)\overline{a\left(e\right)}}{lcm{(d,e)}^{2\sigma }}}.$$

(5.1)

Proof. 

By absolute convergence, we can interchange the sum and the integral. Expanding the square and using

$${\displaystyle \frac{1}{2T}}{\int }_{-T}^T{e}^{-it(logm-logn)}dt\to \left\{\begin{array}{cc}1,\hfill & m=n,\hfill \\ 0,\hfill & m\ne n,\hfill \end{array}\right.$$

we obtain the orthogonality limit. Now, $b\left(n\right)={\sum }_{d\mid n}a\left(d\right)$ implies

$$\sum _{n\ge 1}{\displaystyle \frac{{\left|b\left(n\right)\right|}^2}{n^{2\sigma }}}=\sum _{d,e\ge 1}a\left(d\right)\overline{a\left(e\right)}\sum _{\begin{array}{c}n\ge 1\\ d\mid n,e\mid n\end{array}}{\displaystyle \frac{1}{n^{2\sigma }}}=\sum _{d,e\ge 1}a\left(d\right)\overline{a\left(e\right)}\sum _{k\ge 1}{\displaystyle \frac{1}{{(lcm(d,e)k)}^{2\sigma }}},$$

which is equal to $\zeta \left(2\sigma \right){\sum }_{d,e}a\left(d\right)\overline{a\left(e\right)}lcm{(d,e)}^{-2\sigma }$. □

Corollary 5.2 

(Additive layer: multiples of a finite set of primes). Let $S\subset \mathbb{P}$ be a finite set and let $a\left(d\right)=1$ if $d\in S$, and 0 otherwise. Then $b\left(n\right)={\sum }_{p\in S}{1}_{p\mid n}$ and, for $\sigma >1$,

$$\underset{T\to \infty }{lim}{\displaystyle \frac{1}{2T}}{\int }_{-T}^T|\sum _{n\ge 1}{\displaystyle \frac{b\left(n\right)}{n^{\sigma +it}}}{|}^{2}dt=\zeta \left(2\sigma \right)\left(\sum _{p\in S}{\displaystyle \frac{1}{p^{2\sigma }}}+\sum _{\begin{array}{c}p,q\in S\\ p\ne q\end{array}}{\displaystyle \frac{1}{{\left(pq\right)}^{2\sigma }}}\right).$$

(5.2)

Proof. 

In (5.1), $lcm(p,q)=pq$ if $p\ne q$ and $lcm(p,p)=p$. □

Corollary 5.3 

(Prime powers with weights). Let $a\left(p^m\right)={\beta }_{m}logp$ (and $a\left(d\right)=0$ if d is not a prime power). Then $b\left(n\right)={\sum }_{p^m\mid n}{\beta }_{m}logp$ and

$$\underset{T\to \infty }{lim}{\displaystyle \frac{1}{2T}}{\int }_{-T}^T|\sum _{n\ge 1}{\displaystyle \frac{b\left(n\right)}{n^{\sigma +it}}}{|}^{2}dt=\zeta \left(2\sigma \right)\sum _{p,q\in \mathbb{P}}\sum _{m,\ell \ge 1}{\displaystyle \frac{{\beta }_m{\beta }_{\ell }logplogq}{lcm{(p^m,q^{\ell })}^{2\sigma }}}.$$

(5.3)

Remark 1. 

The preceding identities quantify, in the mean-square norm, the contribution of additive and multiplicative grids along the vertical line $\Re \left(s\right)=\sigma$. They provide, for instance, a basis for comparing mollifiers or filters constructed from such grids under an $L^2$ criterion.

6. Bounds for Projections of $Re\zeta$

Lemma 6.1 

(Universal bound). Let $R\subset \mathbb{N}$ and $w:\mathbb{N}\to [0,W]$. For $\sigma >1$ and all $t\in \mathbb{R}$,

$$\left|\sum _{n\in R}{\displaystyle \frac{w\left(n\right)cos(tlogn)}{n^{\sigma }}}\right|\le W\sum _{n\in R}{n}^{-\sigma }.$$

(6.1)

Proof. 

From $|cos(x\left)\right|\le 1$. □

Lemma 6.2 

(Multiplicative grids). If R is multiplicatively closed and generated by $S\subset \mathbb{P}$, then

$$\sum _{n\in R}{n}^{-\sigma }=\prod _{p\in S}{\displaystyle \frac{1}{1-p^{-\sigma }}},\left|\sum _{n\in R}{\displaystyle \frac{w\left(n\right)cos(tlogn)}{n^{\sigma }}}\right|\le W\prod _{p\in S}{\displaystyle \frac{1}{1-p^{-\sigma }}}.$$

(6.2)

Proof. 

Restrict the Euler product to S and apply Lemma 6.1. □

Lemma 6.3 

(Finite additive layer). Let $R=\{np:n\ge 1,p\in S\}$ with $S\subset \mathbb{P}$ being a finite set. Then

$$\sum _{np\in R}{\left(np\right)}^{-\sigma }=\zeta \left(\sigma \right)\sum _{p\in S}{p}^{-\sigma },\left|\sum _{np\in R}{\displaystyle \frac{w\left(np\right)cos(tlog(np\left)\right)}{{\left(np\right)}^{\sigma }}}\right|\le W\zeta \left(\sigma \right)\sum _{p\in S}{p}^{-\sigma }.$$

(6.3)

Proof. 

Factor out ${\sum }_n{n}^{-\sigma }$ and sum over $p\in S$; apply Lemma 6.1. □

7. Final Remarks

The results above provide a deterministic toolkit for isolating, quantifying, and bounding contributions from arithmetic grids to $Re\zeta (\sigma +it)$ for $\sigma >1$. In particular: (i) the additive layer is identified with $\omega \left(n\right)$ and the prime zeta function, (ii) the prime power layer is related to the multiplicative spine via $\Lambda$, and (iii) the double grid is factored exactly. The mean-square formula Theorem 5.1 and its corollaries Corollaries 5.2 and 5.3 offer an explicit $L^2$ criterion for evaluating filters and mollifiers based on these grids.

Perspectives. 

Extending these identities to $\Re \left(s\right)\ge 1$ requires tools from analytic continuation and control of truncation errors; we do not address this here. Another natural direction is to incorporate Dirichlet characters to separate parities and arithmetic progressions, which is straightforward: ${\sum }_{n\ge 1}\chi \left(n\right)n^{-s}=L(s,\chi )$ for $\Re \left(s\right)>1$.