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# A Note on Moments of Random Multiplicative Functions and Truncated Characteristic Polynomials

David Naccache, Ofer Yifrach-Stav

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## To cite this version:

David Naccache, Ofer Yifrach-Stav. A Note on Moments of Random Multiplicative Functions and Truncated Characteristic Polynomials. 2023. hal-04089572

HAL Id: hal-04089572

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Preprint submitted on 4 May 2023

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# A Note on Moments of Random Multiplicative Functions and Truncated Characteristic Polynomials

David Naccache¹,² and Ofer Yifrach-Stav¹

¹ ÉNS (DI), Information Security Group, CNRS, PSL Research University, Paris, France

david.naccache@ens.fr, ofer.friedman@ens.fr

² SnT & University of Luxembourg, Luxembourg.

**Abstract.** In [2] Heap and Lindqvist give an asymptotic formula for the 2k-th moment of a sum of multiplicative Steinhaus variables. This provided an asymptotic equivalence with the moments of Steinhaus variables. In [2] a conjecture is formulated on an expectation related to a set of i.i.d. random variables {$X_p$}, equidistributed on the unit circle with variance 1 for prime $p$. This note improves the bound given in [2].

Note that it is now known [1] that $E|S_x| \asymp \frac{\sqrt{x}}{\log \log x}$.

# 1 Introduction

## 1.1 Introduction

At section 7 of [2], Heap and Lindqvist provide a conjectured bound:

$$E(|S_x|) \le (1 + o(1)) \cdot 0.9036020036 \dots \sqrt{x}$$

To avoid unnecessarily reproducing the context we refer the reader to [2] and include here only novel ideas improving the above bound. This note can hence be read as a continuation of [2].

We denote by $p_i$ the i-th prime.

Note that it is now known [1] that $E|S_x| \asymp \frac{\sqrt{x}}{\log \log x}$ which reduces the interest of this note to exploring the limits of the Cauchy-Schwarz strategy.

# 2 The improvement strategy

For understanding the improvement strategy we remark the following:

Let $P(X_{\ell(1)}, \dots, X_{\ell(t)})$ be a multivariate polynomial.

Consider the operator Replace taking $P$ and substituting all occurrences of $X_{\ell(i)}^2$ by 1 and all occurrences of $X_{\ell(i)}$ by $1/\ell(i)$.

Form the rational fraction $F$ in the variables $v_{\ell(1)}, \dots, v_{\ell(t)}$:
$$F(v_{\ell(1)}, \dots, v_{\ell(t)}) = \frac{\text{Replace} \left( \prod_{i=1}^{t} (1 - v_{\ell(i)} X_{\ell(i)})^2 \right)}{\prod_{i=1}^{t} (1 - v_{\ell(i)}^2)}$$

The minimum of $F$ over $[0, 1]^t$ is reached for:

$$\bar{v}_{\ell(i)} = \ell(i) - \sqrt{\ell(i)^2 - 1}$$

To see why, develop first

$$\begin{align*}
\Delta &= \text{Replace} \left( \prod_{i=1}^{t} (1 - v_{\ell(i)} X_{\ell(i)})^2 \right) \\
&= \text{Replace} \left( \prod_{i=1}^{t} (1 - 2v_{\ell(i)} X_{\ell(i)} + v_{\ell(i)}^2 X_{\ell(i)}^2) \right) \\
&= \text{Replace} \left( \prod_{i=1}^{t} (1 - 2v_{\ell(i)} X_{\ell(i)} + v_{\ell(i)}^2) \right) \\
&= \prod_{i=1}^{t} (1 - 2 \frac{v_{\ell(i)}}{\ell(i)} + v_{\ell(i)}^2)
\end{align*}$$

Which gives:

$$F(v_{\ell(1)}, \dots, v_{\ell(t)}) = \prod_{i=1}^{t} \frac{1 - 2 \frac{v_{\ell(i)}}{\ell(i)} + v_{\ell(i)}^2}{1 - v_{\ell(i)}^2}$$

Hence:

$$\frac{\partial F}{\partial v_{\ell(u)}} = \frac{4\ell(u)v_{\ell(u)} - 2(1 + v_{\ell(u)}^2)}{\ell(u)(v_{\ell(u)}^2 - 1)^2} \prod_{i \neq u} \frac{1 - 2 \frac{v_{\ell(i)}}{\ell(i)} + v_{\ell(i)}^2}{1 - v_{\ell(i)}^2}$$

And:

$$\frac{\partial F}{\partial v_{\ell(u)}} = 0 \Rightarrow \bar{v}_{\ell(u)} = \ell(u) \pm \sqrt{\ell(u)^2 - 1}$$

Substituting back the second root $\ell(u) - \sqrt{\ell(u)^2 - 1}$ we get:

$$F(\bar{v}_{\ell(1)}, \dots, \bar{v}_{\ell(t)}) = \sqrt{\prod_{i=1}^{t} \frac{\ell(i)^2 - 1}{\ell(i)^2}} < 1$$

Hence, when $\ell(i) = p_i$ we have that:
$$ \lim_{t \to \infty} F(\bar{v}_{\ell(1)}, \dots, \bar{v}_{\ell(t)}) = \sqrt{\frac{1}{\zeta(2)}} = \frac{\sqrt{6}}{\pi} $$

The above improves the upper bound on the Steinhaus expectation for $k = 1/2$ given in [2].

To see why, let $0 \le v_{\ell(1)}, \dots, v_{\ell(t)} \le 1$ and let $S_x = \sum_{n \le x} X_n$ as in [2]. By the Cauchy-Schwarz inequality we have:

$$ \begin{aligned} \mathbb{E}[|S_x|]^2 &\le \mathbb{E}[\left|\prod_{i=1}^t (1 - v_{\ell(i)} X_{\ell(i)}) S_x\right|^2] \cdot \mathbb{E}[\left|\prod_{i=1}^t (1 - v_{\ell(i)} X_{\ell(i)})\right|^{-2}] \\ &= \frac{\mathbb{E}[\left|\prod_{i=1}^t (1 - v_{\ell(i)} X_{\ell(i)}) S_x\right|^2]}{\prod_{i=1}^t (1 - v_{\ell(i)}^2)} \\ &= F(v_{\ell(1)}, \dots, v_{\ell(t)}) \end{aligned} $$

whose minimum was given before.

<table><caption>**Table 1.** Convergence to $\sqrt{6}/\pi$</caption><thead><tr><th>t</th><th>bound</th><th>exact expression</th><th>comment</th></tr></thead><tbody><tr><td>2</td><td>0.816497</td><td>√2/3</td><td>bound of [2]</td></tr><tr><td>3</td><td>0.800000</td><td>4/5</td><td></td></tr><tr><td>4</td><td>0.791795</td><td>16√3/35</td><td></td></tr><tr><td>5</td><td>0.788516</td><td>96√2/5/77</td><td></td></tr><tr><td>6</td><td>0.786180</td><td>384√3/35/143</td><td></td></tr><tr><td>7</td><td>0.784818</td><td>4608√6/35/2431</td><td></td></tr><tr><td>8</td><td>0.783731</td><td>55296√3/7/46189</td><td></td></tr><tr><td>9</td><td>0.782989</td><td>663552/(96577√77)</td><td></td></tr><tr><td>10</td><td>0.782524</td><td>1327104√30/11/2800733</td><td></td></tr><tr><td>...</td><td>...</td><td>...</td><td></td></tr><tr><td>∞</td><td>0.779697</td><td>√6/π</td><td></td></tr></tbody></table>

## 3 Acknowledgment

We thank Winston Heap for his remarks concerning a preliminary version of this note.

## References

1. Harper, A.J.: Moments of random multiplicative functions, i: Low moments, better than square root cancellation, and multiplicative chaos. Forum of Mathematics, Pi 8, e1 (2020)
2. Heap, W., Lindqvist, S.: Moments of random multiplicative functions and truncated characteristic polynomials (2015)