# Multifractal Analysis of the Riemann's Series

Stéphane Jaffard, Quentin Rible, Guillaume Saës

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Stéphane Jaffard, Quentin Rible, Guillaume Saës. Multifractal Analysis of the Riemann's Series. Harmonic and Multifractal Analyses: from Mathematics to Quantitative Neuroscience, Jul 2023, Montréal (Canada (QC)), Canada. . hal-04255431v2

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Distributed under a Creative Commons Attribution 4.0 International License
# Multifractal Analysis of the Riemann's Series

Stéphane Jaffard¹, **Quentin Rible**¹, Guillaume Saës¹

¹Univ Paris Est Creteil, LAMA, CNRS (UMR 8050), F-94010 Creteil, France

jaffard@u-pec.fr, quentin.rible@u-pec.fr, guillaume.saes@u-pec.fr

<figure><img src="image_2.png" /></figure>

## 1. Introduction

The wavelet analysis is a widely used mathematical tool. First introduced in the 80's, it is now used to study temporal data such as functions, processes or signals and their regularity.

A well known example of continuous function which regularity changes from point to point is the Riemann's series. So we consider a slightly generalised type of function defined from the Riemann's series

$$R_s(x) = \sum_{n=1}^{+\infty} \frac{\sin(\pi n^2 x)}{n^s}, \quad x \in \mathbb{R}, s > 1/2.$$

<figure><img src="image_3.png" /><figcaption>Figure 1: The usual Riemann's series $R_2$ on $[0, 1]$</figcaption></figure>

## 3. Wavelets and p-leaders

**Definition 1.1.** A wavelet is a function $\psi$ which is compactly supported, oscillating, sufficiently regular and with several vanishing moments, such that the family $(2^{j/2}\psi_{j,k}(x) = 2^{j/2}\psi(2^j x - k))_{j,k \in \mathbb{Z}}$ form an orthonormal basis of $L^2(\mathbb{R})$. Thus

$$\forall f \in L^2(\mathbb{R}), f = \sum_{j \in \mathbb{Z}} \sum_{k \in \mathbb{Z}} c_{j,k} \psi_{j,k}$$

$$\text{with } c_{j,k} = 2^j \int_{\mathbb{R}} f(x) \psi(2^j x - k) dx.$$

Let's denote $\lambda_{j,k} = [k2^{-j}, (k+1)2^{-j}]$ the dyadic interval.

The exponent of uniform regularity of $f$, $h_f^{\min}$, is given (through $\log - \log$ regression) by

$$\sup_k |c_{j,k}| \sim 2^{-jh_f^{\min}}.$$

**Theorem 1.2.** If $h_f^{\min} > 0$, then $f \in L_{\text{loc}}^{\infty}(\mathbb{R})$. The leaders of $f$ are

$$l_{j,k} = \sup_{\lambda' \subset 3\lambda_{j,k}} |c_{\lambda'}|.$$

We have the characterization for $x \in \lambda_{j,k}$: $l_{j,k} \sim 2^{-jh_f(x)}$.

If $h_f^{\min} < 0$, we consider $p > 0$ such that $\eta(p) > 0$ for $\eta$ given by $2^{-j} \sum_k |c_{j,k}|^p \sim 2^{-j\eta(p)}$. If $0 < p \le 1$, then $f \in H^p(\mathbb{R})$ and if $p > 1$, $f \in L_{\text{loc}}^p(\mathbb{R})$.

The p-leaders are then

$$l_{j,k}^{(p)} = \left( \sum_{\lambda' \subset 3\lambda_{j,k}} |c_{\lambda'}|^p 2^{2j-j'} \right)^{\frac{1}{p}}.$$

We have the characterization for $x \in \lambda_{j,k}$: $l_{j,k}^{(p)} \sim 2^{-jh_f^{(p)}(x)}$.

## 5. Multifractal spectra

With the Meyer wavelet, the wavelet coefficients of $R_s$ are given by

$$c_{j,k} = \sum_{n=1}^{+\infty} \frac{1}{n^s} \exp \left( -in^2 \frac{2k-1}{2^{j+1}} \right) \sqrt{B_3 \left( \frac{n^2}{2^j} \right)}$$

with $B_3$ the 3rd-order B-spline.

Wavelet analysis is therefore ideal for studying regularity.

In figure 2, we estimate $\mathcal{L}_{R_2}$ numerically.

<figure><img src="image_4.png" /><figcaption>Figure 2: Multifractal spectrum $\mathcal{D}_{R_2}$ in red and estimated Legendre spectrum $\mathcal{L}_{R_2}$ in black. For almost any $x \in \mathbb{R}$, $h_{R_2}(x) = 3/4$ since $c_{R_2}^1 = 3/4$.</figcaption></figure>

But the same analysis can be performed for $R_{0.7}$ with p-spectra (since $R_{0.7} \notin L_{\text{loc}}^{\infty}(\mathbb{R})$).

<figure><img src="image_5.png" /><figcaption>Figure 3: Multifractal 2-spectrum $\mathcal{D}_{R_{0.7}}^{(2)}$ in red and estimated p-Legendre spectra $\mathcal{L}_{R_{0.7}}^{(p)}$ for multiple value of p. For almost any $x \in \mathbb{R}$, $h_{R_{0.7}}(x) = 1/10$ since $c_{R_{0.7}}^1 = 0.1$.</figcaption></figure>

## 6. Oscillating singularities and fractional integration

The fractional integral of order $\gamma$ of a function $f$, denoted $I^\gamma f$, is defined via its Fourier transform by the formula

$$(I^\gamma f)(\xi) = (1 + |\xi|^2)^{-\gamma/2} \hat{f}(\xi).$$

We know $\mathcal{D}_{I^\gamma f}^{(p)}$ for any $\gamma$ such that $I^\gamma f \in L_{\text{loc}}^p$ therefore

* for all fractional integrations ($\gamma > 0$),

* for some fractional derivatives ($\gamma < 0$).

If a point $x$ displays **no oscillating singularity**, $\mathcal{D}_{I^\gamma f}^{(p)}(h(x) + \gamma) = \mathcal{D}_f^{(p)}(h(x))$. However, for points displaying **oscillating singularities** like the **chirp** in figure 4, we have $\mathcal{D}_{I^\gamma f}^{(p)}(h(x) + K \cdot \gamma) = \mathcal{D}_f^{(p)}(h(x))$ with $K > 1$.

<figure><img src="image_6.png" /><figcaption>Figure 6: Characteristic values of the spectra of $I^\gamma R_2$</figcaption></figure>

<caption>Figure 7: 2-Legendre spectra of fractional integrates of $R_{0.7}$</caption>

**Observation:** The distance $h_{R_s}^{\min} - c_{R_s}^1$ remains constant, while the distance $c_{R_s}^1 - h_{R_s}^{\max}$ increases linearly with integration orders.

**Conclusion:** We can conjecture that the functions $R_s$ display oscillating singularities for all values of $s$.

## References

[1] A. Cohen. *Numerical Analysis of Wavelet Methods*, volume Studies in Mathematics and its Applications. JAI Press, 2003.

[2] S. Jaffard. The spectrum of singularities of riemann's function. *Rev. Mat. Iber.*, 12(2):441-460, 1996.

[3] S. Seuret and A. Ubis. Local $l^2$-regularity of Riemann's Fourier series. *Ann. de l'Ins. Four.*, 67(5):2237-2264, 2017.

## 2. Definition of p-Exponents

Let $p \in [1, +\infty)$, $f \in L_{\text{loc}}^p(\mathbb{R})$, $t_0 \in \mathbb{R}$ and $\alpha \ge -1/p$. The function **f** belongs to $T_{\alpha}^p(t_0)$ when there exists a polynomial $P_{t_0}$, $K, R > 0$ such that for all $0 < r \le R$,

$$(\frac{1}{r} \int_{t_0-r}^{t_0+r} |f(t) - P_{t_0}(t-t_0)|^p dt)^{\frac{1}{p}} \le K r^{\alpha}.$$

The **p-exponent** of $f$ is

$$h_f^{(p)}(t_0) = \sup\{\alpha : f \in T_{\alpha}^p(t_0)\}.$$

When $p = +\infty$, we have the **Hölder's exponent** $h_f(t_0) = h_f^{(+\infty)}(t_0)$.

## 4. p-Spectre

The multifractal **p-spectrum** of $f \in L_{\text{loc}}^p(\mathbb{R})$, is

$$D_f^{(p)} : H \to \dim_H \left( \left\{ x \in \mathbb{R} : h_f^{(p)}(x) = H \right\} \right),$$

with $\dim_H$ the Hausdorff dimension.

Let $S_f^{(p)}(j, q) = 2^{-j} \sum_k (l_{j,k}^{(p)})^q$. The **p-scaling** function $\zeta_f^{(p)}(q)$ is defined by the relation $S_p(j, q) \sim 2^{-j\zeta_f^{(p)}(q)}$.

**Theorem 1.3.** $D_f^{(p)}(H) \le L_f^{(p)}(H)$ where $\mathcal{L}_f^{(p)}$ is the Legendre transform of $\zeta_f^{(p)}$.

$$\forall H \in \mathbb{R}, \quad \mathcal{L}_f^{(p)}(H) = \inf_{q \in \mathbb{R}} \{1 + qH - \zeta_f^{(p)}(q)\}.$$

<figure><img src="image_7.png" /><figcaption>Figure 4: Singularity of type cusp and chirp in $x_0 = 0$.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 5: Oscillating singularity of $R_2$ in $x_0 = 1$.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 7: 2-Legendre spectra of fractional integrates of $R_{0.7}$</figcaption></figure>