On the proximity of multiplicative functions to the function counting prime factors with multiplicity

TL;DR

This paper investigates the similarity between multiplicative functions and additive functions, providing bounds on the number of integers where they agree, thus advancing understanding of their structural relationship.

Contribution

It establishes a lower bound for the count of integers where a multiplicative function matches a given additive function, highlighting their proximity.

Findings

Lower bound for E(Ω,g,x) established

Quantitative measure of similarity between multiplicative and additive functions

Insights into the structural relationship between these classes of functions

Abstract

We examine how closely a multiplicative function resembles an additive function. Given a multiplicative function $g$ and an additive function $f$, we examine the size of the quantity $E(f,g;x)=\# \{n\leq x:f(n)=g(n)\}$. We establish a lower bound for $E(\Omega,g,x)$.

Figures

Equations

$$E(\omega ,g;x_n)\gg \frac{x_n}{(\log \log x_n{)}^{\frac{1}{2}+ϵ}},$$

$$\phi (x)={\phi }_f(x)=\frac{B(x)}{A(x)}⟶0$$

$$\displaystyle A(x):=\sum\limits_{p^{\alpha}\leq x}f(p^{\alpha})\bigg{(}1-\frac{1}{p}\bigg{)}\quad\text{~{}and~{}}\quad B(x):=\sum\limits_{p^{\alpha}\leq x}\frac{|f(p^{\alpha})|^{2}}{p^{\alpha}},$$

$$\displaystyle\max_{z\in\mathbb{R}}\#\{n\leq x:f(n)=z\}=O\bigg{(}\frac{x}{K(x)}\bigg{)},$$

$$E(f,g;x)=o(x)$$

$$\#\{n\le x:|\omega (n)-\log \log n|>(\log \log x{)}^{1+\delta }\}=o(x)(x⟶\infty ).$$

$$E(\omega ,g;r_j)\gg \frac{x}{(\log \log r_j{)}^{\frac{1}{2}+ϵ}}.$$

$$\max \{s_{j-1},j^{1+\delta }\},$$

$$\#\left\{m\le \frac{r_j}{s_j}:s_k\nmid m\text{ for each }{s}_k\in \mathcal{S},\omega (m)=z_j-1\right\},$$

$$g(p)=\{\begin{array}{ll}{z}_j& \text{ if }p=s_j\in \mathcal{S}\\ 1& \text{ if }p\notin \mathcal{S}.\end{array}$$

$$E(\omega ,g;r_j)$$

$$\ge \#\{n\le r_j:s_j|n,s_k\nmid m\text{for }k\ne j,\omega (m)=z_j\}$$

$$\ge \#\{m\le \frac{r_j}{s_j}:s_k\nmid m\text{ for }{s}_k\in \mathcal{S},\omega (m)=z_j-1\}.$$

$$\#\{m\le \frac{r_j}{s_j}:s_k\nmid m\text{ for }{s}_k\in \mathcal{S}\}=\#\{m\le \frac{r_j}{s_j}:s_k\nmid m\text{ for }{s}_k\in S,\omega (m)\notin I_j\}$$

$$+\#\{m\le \frac{r_j}{s_j}:s_k\nmid m,\omega (m)\in I_j\}.$$

$$\displaystyle\#\{m\leq\frac{r_{j}}{s_{j}}:s_{k}\not|m\text{~{}for~{}}s_{k}\in S,~{}\omega(m)\notin I_{j}\}=o\bigg{(}\frac{r_{j}}{s_{j}}\bigg{)}~{}~{}(j\longrightarrow\infty).$$

$$\#\{m\le \frac{r_j}{s_j}:s_k\nmid m,\omega (m)\in I_j\}$$

$$\le 2(\log \log r_j{)}^{\frac{1}{2}+ϵ}\#\{m\le \frac{r_j}{s_j}:\text{ for }{s}_k\in S,\omega (m)=z_j-1\}.$$

$$\#\{m\le \frac{r_j}{s_j}:s_k\nmid m\text{ for }{s}_k\in S,\omega (m)=z_j-1\}\ge \frac{\#\{m\le \frac{r_j}{s_j}:s_k\nmid m\text{ for }{s}_k\in S\}}{2(\log \log r_j{)}^{\frac{1}{2}+ϵ}}.$$

$$\displaystyle\#\{m\leq\frac{r_{j}}{s_{j}}:s_{k}\not|m\text{~{}for~{}}s_{k}\in S\}=\bigg{(}1+o(1)\bigg{)}\frac{r_{j}}{s_{j}}C(\delta)~{}~{}(j\longrightarrow\infty).$$

$$\displaystyle\#\{m\leq\frac{r_{j}}{s_{j}}:s_{k}\not|m\text{~{}for~{}}s_{k}\in S,\omega(m)=z_{j}-1\}\gg\bigg{(}\frac{1}{2}+o(1)\bigg{)}\frac{r_{j}}{s_{j}}C(\delta)\cdot\frac{1}{(\log\log r_{j})^{\frac{1}{2}+\epsilon}}.$$

$$\displaystyle\bigg{(}\frac{1}{2}+o(1)\bigg{)}\frac{r_{j}}{s_{j}}C(\delta)\cdot\frac{1}{(\log\log r_{j})^{\frac{1}{2}+\epsilon}}\gg\bigg{(}\frac{1}{2}+o(1)\bigg{)}\frac{r_{j}}{(\log\log r_{j})^{\frac{1}{2}+2\epsilon}},$$

$$E(\omega ,g;r_j)\gg \frac{r_j}{(\log \log r_j{)}^{\frac{1}{2}+ϵ}},$$

$$\sum _{j=1}^{\infty }\frac{1}{s_j}<\infty .$$

$$E(\Omega ,g,x)\gg \frac{x}{(\log \log x{)}^{\frac{1}{2}+ϵ}}.$$

$$\#\{r\le \frac{x}{s_j}:s_i\nmid r\text{ for each }{s}_i\in \mathcal{S},\Omega (r)=z_j-2\},$$

$$\#\{r\le \frac{x}{s_j}:s_i\nmid r\text{ for each }{s}_i\in \mathcal{S},\Omega (r)=z_j\},$$

$$g(p)=\{\begin{array}{l}{z}_j-1\text{ if }p=s_j\equiv 1(\mathrm{mod}4),s_j\in \mathcal{S}\\ z_j+1\text{ if }p=s_j\equiv 3(\mathrm{mod}4),s_j\in \mathcal{S}\\ 1\text{ if }p\notin \mathcal{S}.\end{array}$$

$$E(\Omega ,g,x)$$

$$\ge \sum _{\alpha \ge 1}\#\{n\le x:s_j^{\alpha }|n,s_i\nmid n\text{ for }i\ne j,\Omega (n)=g(n)\}$$

$$\ge \sum _{\alpha \ge 1}\#\{r\le \frac{x}{s_j^{\alpha }}:\Omega (r)=g(s_j)-\alpha ,s_i\nmid r\text{ for each }{s}_i\in \mathcal{S},s_j\equiv 1(\mathrm{mod}4)\}$$

$$\ge \#\{r\le \frac{x}{s_j}:\Omega (r)=g(s_j)-1,s_i\nmid r\text{ for each }{s}_i\in \mathcal{S},s_j\equiv 1(\mathrm{mod}4)\}$$

$$\ge \#\{r\le \frac{x}{s_j}:\Omega (r)=z_j-2,s_i\nmid r\text{ for each }{s}_i\in \mathcal{S}\}.$$

$$I=[\log \log x-(\log \log x{)}^{\frac{1}{2}+ϵ},\log \log x+(\log \log x{)}^{\frac{1}{2}+ϵ}].$$

Full text

1. Introduction

Let us set $E(f,g;x):=\#\{n\leq x:f(n)=g(n)\}$, where $f$ and $g$ are arbitrary additive and multiplicative functions, respectively. One of the basic questions one can ever ask is, how large and how small can this quantity be. In 2014, Dekonick, Doyon and Letendre [1] proved that for some suitable choice of multiplicative function and some choice of sequence $(x_{n})$ of positive integers

[TABLE]

for any small $\epsilon>0$. Above all they were able to show that if $f$ is an integer-valued additive function such that

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as $x\longrightarrow\infty$, where

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and that

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where $K(x)\longrightarrow\infty$ as $x\longrightarrow\infty$. Then, for any multiplicative function $g$

[TABLE]

as $x\longrightarrow\infty$. In particular, $E(\omega,g,x)=o(x)$ as $x\longrightarrow\infty$.

111On the proximity of multiplicative functions to the function counting the number of prime divisors with multiplicity

.

2. Prelimary results

Lemma 2.1**.**

Let $\pi_{k}(x)=\#\{n\leq x:\omega(n)=k\}$ for each positive integer $k$. Then the maximum value of $\pi_{k}(x)$ is $\frac{x}{\sqrt{\log\log x}}(1+o(1))$ and the value of $k$ for which it occurs is $k=\log\log x+O(1)$

Proof.

This follows from a result of Balazard [2]. ∎

Lemma 2.2**.**

For all $x\geq 2$ and for every $\delta>0$

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Proof.

This follows from Theorem 8.12 in the book of Nathason [3]. ∎

Remark 2.3*.*

Now we present one of the results and the techniques of Dekoninck, Doyon and Letendre [1] employed in obtaining the lower bound for the quantity $E(\omega,g,x)$.

Theorem 2.4**.**

Let $\epsilon>0$ be very small. Then, there exist a multiplicative function $g$ and a sequence $(r_{j})$ of positive integers such that

[TABLE]

Proof.

Given $\epsilon>0$ very small, let $\mathcal{S}=\{s_{1},s_{2},\ldots\}$ be an infinite set of primes, with $s_{1}=2$ and $s_{j}$ to be the smallest prime number larger than

[TABLE]

for $j\geq 2$ and $\delta>0$ very small. Choose $r_{j}=e^{e^{2^{j}}}$ and let $(z_{j})$ be sequence of integers maximizing the quantity

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for each $j\geq 1$, which is well defined by Lemma 2.1. Define $g$ a strongly multiplicative function on the primes as

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To find a lower bound for $E(\omega,g;r_{j})$, it suffices to consider integers of the form $n=m\cdot s_{j}$ such that $s_{j}\not|m$ for $s_{j}\in\mathcal{S}.$

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Now, let $I_{j}=[\log\log r_{j}-(\log\log r_{j})^{\frac{1}{2}+\epsilon},\log\log r_{j}+(\log\log r_{j})^{\frac{1}{2}+\epsilon}]$. Then

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In relation to Lemma 2.2

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Also

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It follows from (2) and (2.2) that

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It is also clear that

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Plugging (2.5) into (2.4), we have that

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Then, for sufficiently large $j$ we see that , $s_{j}<j^{1+2\delta}\leq(2^{j})^{\epsilon}$, where $(2^{j})^{\epsilon}=(\log\log r_{j})^{\epsilon}$. Using this fact, we obtain

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and from (2.7) we have

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thus completing the proof. ∎

Remark 2.5*.*

It has to be said that this is a good lower bound, but it only works for a particular type of sequence and therefore is not uniform.

3. Main result

In this section we use the techniques employed by Dekonick, Doyon and Letendre [1] to obtain a uniform lower bound for $E(\Omega,g,x)$.

Theorem 3.1**.**

Let $\mathcal{S}=\{s_{1},s_{2},\ldots\}$ be an infinite set of the primes such that

[TABLE]

Then for any small $\epsilon>0$, there exists a strongly multiplicative function $g$ such that

[TABLE]

Proof.

Let $(z_{j})$ be sequence of positive integers maximizing the quantity

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for $s_{j}\equiv 1\pmod{4}$ for each $j\geq 1$, and

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for $s_{j}\equiv 3\pmod{4}$ for each $j\geq 1$, which is well defined by Lemma 2.1. Define $g$, a strongly multiplicative function, on the primes as

[TABLE]

To obtain a lower bound for $E(\Omega,g,x)$, it suffices to consider only integers of the form $n=r\cdot s_{j}^{\alpha}$ for $\alpha\geq 1$, $s_{j}\not|r$. Clearly

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Again consider the interval

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Let us consider

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We observe, in relation to Theorem 2.2, \#\{r\leq\frac{x}{s_{j}}:s_{i}\not|r\text{~{}for~{}each~{}}s_{i}\in\mathcal{S},~{}s_{j}\equiv 1\pmod{4},~{}\Omega(r)\notin I_{j}\}=o\bigg{(}\frac{x}{s_{j}}\bigg{)}, as $j\longrightarrow\infty$. On the other hand

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[TABLE]

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It follows from (3.1), that

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for some positive real number $\mathrm{K}$ and

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Carrying out the same process for the other residue class $s_{j}\equiv 3\pmod{4}$ and combining the result, we will obtain

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∎

4. Conclusion

The lower bound obtained in the original work of Dekoninck, Doyon and Letendre [1] can be made uniform by using a similar choice of multiplicative function in the main result; that is, if we let

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Then

[TABLE]

holds uniformly.