On the Uniform Distribution (mod 1) of the Farey Sequence, quadratic Farey and Riemann sums with a remark on local integrals of $\zeta(s)$

TL;DR

This paper investigates the distribution of Farey sequences mod 1 for functions with weak regularity, providing sharp estimates for Farey and Riemann sums, and explores their connection to local integrals of the Riemann zeta function.

Contribution

It offers new sharp estimates for Farey sums and Riemann sums under minimal regularity assumptions and relates these sums to local integrals of the Riemann zeta function.

Findings

Sharp bounds for Farey sums for functions with Dini-type regularity.

Estimates for Riemann quadratic sums related to Farey sequences.

Connection established between these sums and local zeta function integrals.

Abstract

For $1$-periodic functions $f$ satisfying only a weak local regularity assumption of Dini's type at rational points of $]0,1[$, we study the Farey sums $$F_n(f)= \sum_{\frac{\k}{\l}\in \F_n} f\big(\frac{\k}{\l}\big),\qq F_{n,\s}(f)= \sum_{\frac{\k}{\l}\in \F_n} \frac{1}{\k^\s\l^\s}f\big(\frac{\k}{\l}\big),\qq 1/2\le \s<1 , $$ where $\F_n$ is the Farey series of order $n\ge 1$. We obtain sharp estimates of $F_{n,\s}(f)$, for all $0< \s\le1$. We prove similar results for the corresponding Riemann quadratic sums $$ S_{n,\s}(f) \ =\ \sum_{1\le k\le \ell \le n}\frac{1}{(k\ell)^{\s }}\, f\big( \frac{k}{\ell}\big). $$ These sums are related to local integrals of the Riemann zeta-function over bounded intervals $I$, which are considered in the last part of the paper.

Equations

$$F_{n}(f)=\sum_{\frac{{\kappa}}{{\lambda}}\in{\mathcal{F}}_{n}}f\big{(}\frac{{\kappa}}{{\lambda}}\big{)},{\qquad}F_{n,{\sigma}}(f)=\sum_{\frac{{\kappa}}{{\lambda}}\in{\mathcal{F}}_{n}}\frac{1}{{\kappa}^{\sigma}{\lambda}^{\sigma}}f\big{(}\frac{{\kappa}}{{\lambda}}\big{)},{\qquad}1/2\leq{\sigma}<1,$$

$$S_{n,{\sigma}}(f)\ =\ \sum_{1\leq k\leq\ell\leq n}\frac{1}{(k\ell)^{{\sigma}}}\,f\big{(}\frac{k}{\ell}\big{)}.$$

$$F_{n}(f)=\sum_{\frac{{\kappa}}{{\lambda}}\in{\mathcal{F}}_{n}}f\big{(}\frac{{\kappa}}{{\lambda}}\big{)},{\qquad}F_{n,{\sigma}}(f)=\sum_{\frac{{\kappa}}{{\lambda}}\in{\mathcal{F}}_{n}}\frac{1}{{\kappa}^{\sigma}{\lambda}^{\sigma}}f\big{(}\frac{{\kappa}}{{\lambda}}\big{)}{\qquad}n=1,2,\ldots.$$

$$\displaystyle S_{n,{\sigma}}(f)\ =\ \sum_{1\leq k\leq\ell\leq n}\frac{1}{(k\ell)^{{\sigma}}}\,f\Big{(}\frac{k}{\ell}\Big{)}$$

$${\int }_n^{n+1}|\zeta (\sigma +it){|}^2\mathrm{d}t=Quadratic\; Riemann\; sum+Error\; term.$$

$$F_n(f)\sim \Phi (n){\int }_0^{1}f(x)\mathrm{d}x,n\to \infty .$$

$$E_n(f)=F_n(f)-\Phi (n){\int }_0^{1}f(x)\mathrm{d}x,$$

$$F_{n}(f)=\sum_{d\leq n}D_{f}(d)\,M\big{(}\frac{n}{d}\big{)}=\sum_{d\leq n}{\mu}*D_{f}(d),{\qquad}D_{f}(\ell)\,=\,\sum_{1\leq k\leq\ell}f\Big{(}\frac{k}{\ell}\Big{)},$$

$$M(n)=\sum _{\frac{\kappa }{\lambda }\in {\mathcal{F}}_n}\cos 2\pi (\frac{\kappa }{\lambda }),$$

$$M(x)=\mathcal{O}_{\varepsilon}\big{(}x^{\frac{1}{2}+{\varepsilon}}\big{)}.$$

$$R_f(\ell )={\int }_0^{1}f(t)\mathrm{d}t+\mathcal{O}(q^{\ell }).$$

$$F_{n,{\sigma}}\,=\,\frac{n^{2(1-{\sigma})}}{2(1-{\sigma})\zeta(2)}\int_{0}^{1}\frac{\cos 2\pi t}{t^{\sigma}}{\rm d}t\,+\mathcal{O}_{\sigma}\big{(}n^{1-{\sigma}}\big{)}{\qquad}\quad(0<{\sigma}<1).$$

$$F_{n,{\sigma}}(f)\ =\ \sum_{d=1}^{n}\frac{1}{d^{2{\sigma}}}\,{\mu}*D_{f_{\sigma}}(d)\ =\ \sum_{d=1}^{n}\frac{D_{f_{\sigma}}(d)}{d^{2{\sigma}}}\,\Big{(}\,\sum_{{\lambda}\leq\frac{n}{d}}\,\frac{{\mu}({\lambda})}{{\lambda}^{2{\sigma}}}\,\Big{)}.$$

$$\sum_{{\lambda}\leq n}\,\frac{{\mu}({\lambda})}{{\lambda}^{\tau}}=\mathcal{O}_{\varepsilon}\big{(}n^{\frac{1}{2}+{\varepsilon}}\big{)}.$$

$$\frac{1}{\zeta (s)}\sum _{k=1}^{\infty }\frac{D_{f_{\sigma }}(k)}{k^s}.$$

$$F_{n}(f)=\mathcal{O}_{\varepsilon}\Big{(}n^{\frac{1}{2}+{\varepsilon}}\sum_{d\leq n}\frac{|D_{f}(d)|}{d^{\frac{1}{2}+{\varepsilon}}}\Big{)},$$

$$F_n(f)\ge n^{\frac{1}{2}}\sum _{d\le n}\frac{|D_f(d)|}{d^{\frac{1}{2}}}.$$

$$M(x)={\Omega}\,\big{(}x^{\frac{1}{2}}\big{)},$$

$$g(a,x)=\frac{\sin a\log x}{\log x},\; 0<x<\mathrm{1,}g(a,1)=a,g(a,0)=\mathrm{0,}(a\ne 0)\; .$$

$$g(x)=g(1,x).$$

$$f(x-0)=f(x+0)$$

$$\displaystyle\mathcal{C}_{1}=\big{\{}f:[0,1]\to{\mathbb{R}}:\hbox{$f(x)$ is derivable for each $x\in{\mathbb{Q}}\cap]0,1[$\big{\}}}$$

$$\displaystyle\mathcal{C}_{2}=\big{\{}f:[0,1]\to{\mathbb{R}}:\hbox{${f}(x)$ satisfies a Lipschitz condition of order ${\alpha}>0$},$$

$$\displaystyle{\qquad}{\qquad}{\qquad}\hbox{in the neighbourhood of each $x\in{\mathbb{Q}}\cap]0,1[$}\big{\}}.$$

$${\sigma }_f(x)=\sum _{\nu \in \mathbb{Z}}{c}_f(\nu )e_{\nu }(x),(e_{\nu }(x)=e^{2\mathrm{i}\pi \nu x})$$

$$f(x)={\int }_0^{1}u(2\pi (x-t))v(2\pi t)\mathrm{d}t,$$

$$\Vert u\ast v{\Vert }_r\le \Vert u{\Vert }_p\Vert v{\Vert }_q,1\le p,q,r\le \infty ,\frac{1}{r}=\frac{1}{p}+\frac{1}{q}-1,$$

$$\sum _{\ell \in \mathbb{Z}}|c_{g_{\sigma }}(\ell )|=\infty .$$

$$\displaystyle F_{n,{\sigma}}(f)=\$$

$$\displaystyle{\qquad}+\sum_{\ell\in{\mathbb{Z}}_{*}}c_{f_{\sigma}}(\ell)\Big{[}\sum_{d\leq n}\frac{1}{d^{2{\sigma}}}{\varepsilon}_{\ell}(d)\Big{(}\sum_{{\lambda}\leq\frac{n}{d}}\frac{{\mu}({\lambda})}{{\lambda}^{2{\sigma}}}\Big{)}\Big{]},$$

$${\epsilon }_{\ell }(d)=\{\begin{array}{ll}d-1& \mathrm{if}d|\ell ,\\ -1& \mathrm{if}d\nmid \ell .\end{array}$$

$$\displaystyle E_{n}(f)\,=\,f(1)-\int_{0}^{1}f(x){\rm d}x+\sum_{\ell\in{\mathbb{Z}}_{*}}c_{f}(\ell)\Big{[}\sum_{d\leq n}{\varepsilon}_{\ell}(d)M\big{(}\frac{n}{d}\big{)}\Big{]}.$$

Full text

[TABLE]

where ${\mathcal{F}}_{n}$ is the Farey series of order $n\geq 1$. We obtain sharp estimates of $F_{n,{\sigma}}(f)$, for all $0<{\sigma}\leq 1$. We prove similar results for the corresponding Riemann quadratic sums

[TABLE]

These sums are related to local integrals of the Riemann zeta-function over bounded intervals $I$, which are considered in the last part of the paper.

1. Introduction

Let {\mathcal{F}}_{n}=\big{\{}\frac{j}{m},1\leq j\leq m\leq n,(j,m)=1\big{\}} be the Farey series of order $n\geq 1$. Let also $f:{\mathbb{Q}}\cap[0,1]\to{\mathbb{C}}$ arbitrary and ${\sigma}\geq 0$. In this work we study the Farey sums

[TABLE]

As $F_{n,0}(f)=F_{n}(f)$, the second sums generalize the first ones. We also study the corresponding Riemann quadratic sums

[TABLE]

where $0\leq{\sigma}<1$ and $f\in L^{0}([0,1])$. These sums have a simpler structure as being weightings of the Riemann sums of $f_{\sigma}(x)=f(x)/x^{{\sigma}}$, and are connected with $\zeta$-local integrals, where $\zeta$ is the Riemann-zeta function, since for instance (see section 6)

[TABLE]

Some preliminary considerations are necessary. It is well-known that the Farey fractions are uniformly distributed (mod 1), see Mikolás [15], whence by the Weyl criterion, for any Riemann integrable function on $[0,1]$,

[TABLE]

Here $\Phi(n)=\#({\mathcal{F}}_{n})$ and $\Phi(n)=\sum_{d=1}^{n}{\varphi}(d)\sim\frac{3}{\pi^{2}}n^{2}$, ${\varphi}(d)$ being Euler’s totient function. The problem of estimating the error term

[TABLE]

is connected with the Riemann Hypothesis, and was studied by Mikolás [15] and by several authors, notably Codecà and Perelli [5], see references therein, and Yoshimoto [27]. Farey sums and Riemann sums are linked by the formula ([15, Lemma 2]),

[TABLE]

where $*$ stands for the Dirichlet convolution product. One notes that $\Phi(n)=\sum_{d=1}^{n}dM(\frac{n}{d})$ where $M(x)=\sum_{{\lambda}\leq x}{\mu}({\lambda})$, ${\mu}(n)$ being the Möbius function, and that $M(n)$ is the Farey sum

[TABLE]

which easily follows from (3.2). By a result of Littlewood [13], the Riemann Hypothesis is equivalent to the assertion

[TABLE]

The simplest example of a smooth periodic function $f(x)=\cos 2\pi x$, thus shows that the problem of estimating $E_{n}(f)$ ($=F_{n}(f)$ here) is out of reach, advances in this domain are therefore difficult. Farey sums much differ at this regard from Riemann sums $R_{f}(\ell)=\frac{1}{\ell}D_{f}(\ell)$, since by a result of Wintner [26, § 12], a continuous $1$-periodic function is analytic if and only if there exists $q$, $0<q<1$, such that

[TABLE]

A rate of convergence can be assigned, and the convergence of Riemann sums turns up the more rapid, the smoother $f$ is. If $f$ is only Lebesgue integrable, the corresponding convergence problems of Riemann sums, and by extension Riemann equidistant sums, Farey sums, are another attracting and difficult matter. We refer to Ch. XI of our book [25]. For the case considered above ($h(x)=\cos 2\pi x$, $F_{n,{\sigma}}=F_{n,{\sigma}}(h)$), we will prove that

[TABLE]

The analogous formula to (1.5) for the Farey sums $F_{n,{\sigma}}(f)$ is

[TABLE]

See Lemma 3.3. In comparison with (1.6) one knows (Mikolás [15, Lemma 8]) that for $\tau\geq 1/2$, RH is also equivalent to

[TABLE]

Further the Dirichlet series associated with ${\mu}*D_{f_{\sigma}}$ being the product

[TABLE]

$F_{n,{\sigma}}(f)$ can be precisely estimated by using Perron’s formula, once estimates of $D_{f_{\sigma}}(k)$ are at disposal. This was used in Mikolás [15] and Codecà-Perelli [5].

The first formula in (1.5) together with (1.6) imply that

[TABLE]

for all $f\colon[0,1]\to{\mathbb{C}}$, assuming the validity of the RH. Conversely we prove the following unconditional result, of very close order of magnitude.

Theorem 1.1**.**

For for infinitely many $n$, there exists $f\colon[0,1]\to{\mathbb{C}}$ such that

[TABLE]

The proof is a combination of a theorem of Pintz [18], which in particular implies that

[TABLE]

and infinite Möbius inversion formula. The work made in [5] concerns absolutely continuous functions on $[0,1]$, or equivalently, continuous functions $f$ with a derivative $f^{\prime}$ almost everywhere and $f^{\prime}$ is Lebesgue integrable. It is further imposed that $f^{\prime}\in L^{p}[0,1]$ for some $p\in(1,2]$. Let $\mathcal{C}_{\rm cp}$ denotes this class of functions. The main results obtained being of conditional nature, are by definition ineffective. However these results are nearly optimal with respect to the class $\mathcal{C}_{\rm cp}$, and [5] is one of the central papers in the theory with Mikolás [15, 16], notably by the ideas implemented. Some of these results were slightly extended in Yoshimoto [27], who notably much investigated some specific remarkable classes of functions related to Riemann sums. Let $\zeta(s)$ denotes the Riemann zeta function. More precisely, it is assumed that a weaker form of Riemann Hypothesis, noted RH(${\alpha}$) and meaning that: ${\sigma}>{\alpha}\ \Rightarrow\ \zeta({\sigma}+it)\neq 0$, where ${\alpha}\in[1/2,1)$, holds true. One notes that then $M(x)=\mathcal{O}_{\varepsilon}(x^{{\alpha}+{\varepsilon}})$. For $f\in\mathcal{C}_{\rm cp}$, there is a close link between the deviation of $R_{f}(n)$ from $\int_{0}^{1}f(t){\rm d}t$, and Fourier coefficients of $f^{\prime}$, which is the basis of the approach in [5].

Remark 1.2*.*

The following class of continuous functions $f\colon(0,1)\to{\mathbb{C}}$ satisfying Kubert identities f(x)=m^{s-1}\sum_{k=0}^{m-1}f\big{(}\frac{x+k}{m}\big{)} for every $m\in{\mathbb{N}}$ and every $x\in(0,1)$, was studied in [27], and more recently in other papers. We study this class of functions in a separate work, as well as some variant in Carlitz [4].

In this paper we are interested in the study of Farey sums (1.1) under minimal conditions, and our results are effective and will depend on the Fourier coefficients of $f$. Recall that every function defined almost everywhere in $[0,1]$, in particular every integrable function, has its Fourier series, see Zygmund [28, p. 9]. One motivation is that the limit (1.3) actually holds true for any function $f$ defined on rational points of $[0,1]$, whose Riemann sums are converging, see [15, Th. 2]. Another motivation lies in the fact that there are important classes of functions $f$ having a (non integrable) derivative on $]0,1[$, so $f$ is not absolutely continuous. The following specific functions, familiar in Fourier analysis and relevant in section 6, are typical cases,

[TABLE]

They are not absolutely continuous on $(0,1]$, for otherwise they would have an absolutely continuous extension on $[0,1]$, thereby continuous on $[0,1]$, which is not. Let further $g_{\sigma}(a,x)=g(a,x)/x^{\sigma}$, $0<{\sigma}<1$. Then $g_{\sigma}(a,.)\in L^{p}[0,1]$ if $p<1/{\sigma}$, $g_{\sigma}(a,.)^{\prime}$ is not integrable, and as $g_{\sigma}(a,.)$ is unbounded in the neighbourhood of [math], its Fourier series does not converge absolutely.

For the problem studied, considering the restriction of $g_{\sigma}(a,.)$ on $[{\varepsilon},1]$, ${\varepsilon}>0$ small, and applying Euler-McLaurin’s formula + Parseval’s formula is inoperant.

We will use the simplified notation (section 5)

[TABLE]

We consider in this work $1$-periodic functions $f$ satisfying a weak local regularity assumption of Dini’s type at rational points of $]0,1[$, which is in accordance with the fact that Farey sums $F_{n}(f)$, $F_{n,{\sigma}}(f)$ are determined by the values taken by $f$ on rational numbers. More precisely, introduce the class $\mathcal{C}$ of functions $f:[0,1]\to{\mathbb{R}}$ such that

[TABLE]

This defines a fairly wide setting, one has the following obvious inclusions: $\mathcal{C}_{1}\subset\mathcal{C}_{2}\subset\mathcal{C}$, where

[TABLE]

These functions being not necessarily absolutely continuous, Euler-McLaurin’s formula, which is the pivot of the approach in [5], does not apply. As $x=0$ is excluded in definition (1.16), we note that $g_{\sigma}(a,.)\in\mathcal{C}$. We clarify that the approach used and most of the results obtained extend with no difficulty to classes of functions subject to sharper types of criteria such as the one of Jordan, Young, de la Vallée-Poussin, Lebesgue, see [2, Vol. I]. These ones being more elaborated we chosed to develop the present work in this simpler setting.

The paper is organized as follows. In the two next sections we respectively state and give the proofs of results concerning Farey sums $F_{n}(f)$, $F_{n,{\sigma}}(f)$, and further comment and discuss our assumptions, comparing them notably with those in [5]. Section 3 contains preparatory results which are interesting on their own. In section 4, the proofs of Theorems 1.1 and 2.8 are given. Our results concerning quadratic Riemann sums $S_{n,{\sigma}}(f)$ are stated and proved in section 5. In section 6, we discuss some questions related to the previous sections and concerning local integrals of $\zeta({\sigma}+it)$.

The investigation of a related non trivial question concerning the unboundedness of the $\mathcal{S}^{2}$-Stepanov’s norm of the Riemann zeta function $\zeta({\sigma}+it)$ for $1/2<{\sigma}<1$ is concluding the paper, the case ${\sigma}=1/2$ being trivial.

2. Farey sums and quadratic Farey sums.

We first provide general explicit formulas of $F_{n,{\sigma}}(f)$ or $F_{n}(f)$, valid for all $f$ such that ${f_{\sigma}}\in\mathcal{C}$, with no additional condition. We next study these sums, mainly under two type of conditions on the complex Fourier coefficients of $f_{\sigma}$ where $f_{\sigma}(x)=\frac{f(x)}{x^{\sigma}}$, ${\sigma}\geq 0$. We assume that: Either (i) the series $\sum_{\ell\in{\mathbb{Z}}}c_{f_{\sigma}}(\ell)$ converges, or (ii) the series $\sum_{\ell\in{\mathbb{Z}}}|c_{f_{\sigma}}(\ell)|$, converges. Recall some classical facts. The first assumption of course holds if (for instance) ${f_{\sigma}}(x)$ is derivable at $x=0$. By Bernstein’s theorem [2, Vol. I, p. 216], the second assumption holds if $f\in{\rm Lip}({\alpha})$ for ${\alpha}>1/2$. It is also implied by the absolute convergence of the Fourier series

[TABLE]

at a single point, and implies the absolute and uniform convergence of the series ${\sigma}_{f}(x)$ for all $x$. It also implies that ${\sigma}_{f}(x)$ converges to $f(x)$ at almost every $x$, and if $f$ is continuous on $[0,1]$, that convergence holds for every $x$. Further, if $f$ is absolutely continuous and $|f^{\prime}|\ln^{+}|f^{\prime}|\in L^{1}([0,1]$, then ${\sigma}(f)$ converges absolutely. Also there exists an absolutely continuous function $f$ such that ${\sigma}(f)$ does not possess a single point of absolute convergence. See [2, Vol. II, p. 162]. Furthermore the second assumption implies that $f$ is bounded, since using for instance Riesz’s criterion [2, Vol. II], p. 184, as the series ${\sigma}_{f}$ converges absolutely, $f$ can be represented in the form

[TABLE]

$u,v\in L^{2}[0,1]$. By applying Young’s inequality

[TABLE]

with $p=q=2$, $r=\infty$, we deduce that $\|f\|_{\infty}<\infty$. In particular for $g_{\sigma}$ defined in (1.15), we note that

[TABLE]

Consider first the case when $f$ is such that ${f_{\sigma}}\in\mathcal{C}$, and no additional condition is imposed.

Proposition 2.1**.**

(i)* Let $f:[0,1]\to{\mathbb{R}}$ and assume that ${f_{\sigma}}\in\mathcal{C}$. Then we have,*

[TABLE]

where

[TABLE]

(ii)* Let $f:[0,1]\to{\mathbb{R}}$ be such that ${f}\in\mathcal{C}$. Then we have,*

[TABLE]

In the next theorems, we derive precise estimates of $F_{n,{\sigma}}(f)$ under the afore mentionned assumptions.

Theorem 2.2** ($0<{\sigma}<1$).**

Let $f:[0,1]\to{\mathbb{R}}$ be such that ${f_{\sigma}}\in\mathcal{C}$. (i) Assume that the series $\sum_{\ell\in{\mathbb{Z}}}c_{f_{\sigma}}(\ell)$ is convergent. Then

[TABLE]

[TABLE]

and the constant $A(f_{\sigma})$ is defined as follows,

[TABLE]

(ii)* Assume that the series $\sum_{\ell\in{\mathbb{Z}}_{*}}|c_{f_{\sigma}}(\ell)|$ is convergent. Then*

[TABLE]

where

[TABLE]

In the series above, the summand of order $\ell$ not only depends on estimates of Möbius sums $\sum_{{\lambda}\leq x}\frac{{\mu}({\lambda})}{{\lambda}^{2{\sigma}}}$, for instance conditionally to RH(${\alpha}$), or using (3.4), but also on the divisors of $\ell$ which are less than $n$, namely on the arithmetical structure of the support of the Fourier coefficient sequence. In Theorem 2.2, the series

[TABLE]

cannot be estimated in general. The term in parenthesis can be close to $n^{2(1-{\sigma})}$ (for those $\ell$ such that $d\leq n\,\Rightarrow\,d|\ell$), or close to $2$ when the Fourier coefficients are supported by a sequence of numbers having few divisors, typically a sequence of primes. In this case we get for instance if ${\sigma}=\frac{1}{2}$,

[TABLE]

if $\sum_{\ell\in{\mathbb{Z}}_{*}}|c_{f_{1/2}}(\ell)|$ converges.

Theorem 2.3** (${\sigma}=1$).**

Let $f:[0,1]\to{\mathbb{R}}$ be such that ${f_{1}}\in\mathcal{C}$. (i) Assume that the series $\sum_{\ell\in{\mathbb{Z}}}c_{f_{1}}(\ell)$ is convergent. Then,

[TABLE]

(ii)* If the series $\sum_{\ell\in{\mathbb{Z}}}|c_{f_{1}}(\ell)|$ is convergent, then*

[TABLE]

In particular, if the series $\sum_{\ell\in{\mathbb{Z}}}|c_{f_{1}}(\ell)|{\sigma}_{-1}(\ell)$ is convergent, then

[TABLE]

Remark 2.4*.*

We recall that by [9], ${\sigma}_{-1}(\ell)=\mathcal{O}(\log\log\ell)$, thus the above estimate holds true under the mild condition

[TABLE]

The following Theorem concerns Farey sums $F_{n}(f)$ (case ${\sigma}=0$) and provides a simple formula for the error term $E_{n}(f)$ under minimal assumption, as well as a new estimate.

Theorem 2.5**.**

Let ${f}\in\mathcal{C}$. (i) If the series $\sum_{\ell\in{\mathbb{Z}}}c_{f}(\ell)$ is convergent, then

[TABLE]

(ii)* If the series $\sum_{\ell\in{\mathbb{Z}}_{*}}|c_{f}(\ell)|d(\ell)$ is convergent, where $d(\ell)$ is the divisor function, then*

[TABLE]

In addition if $f(1)=\sum_{\ell\in{\mathbb{Z}}}c_{f}(\ell)$, then

[TABLE]

Remarks 2.6**.**

According to a classical estimate, $E_{n}(f)\,=\,o(n)$ if $\sum_{\ell\in{\mathbb{Z}}_{*}}|c_{f}(\ell)|e^{c\log n/\log\log n}$ is convergent for some $c>\log 2$. Codecà and Perelli [5] showed using Euler-McLaurin’s formula and a lemma due to Féjer, that if $f$ is absolutely continuous, then $E_{n}(f)\,=\,o(n)$. See also Corollary p. 105 in Mikolas [15]. This can in fact be improved if in addition $f^{\prime}\in L^{p}[0,1]$ for some $p\in(1,2]$, by using Vinogradov-Korobov’s estimate. They further showed, under assumption RH(${\alpha}$), and using example 2.9 below, that for every ${\varepsilon}>0$, there exists an absolutely continuous function $f_{\varepsilon}:[0,1]\to{\mathbb{R}}$ such that $E_{n}(f)\,=\,{\Omega}_{\varepsilon}(n^{1-{\varepsilon}})$.

Corollary 2.7**.**

Assume that RH(${\alpha}$) holds. Then for any ${f}\in\mathcal{C}$ such that the series

[TABLE]

converges for some ${\alpha}^{\prime}>{\alpha}$, we have

[TABLE]

Proof.

As RH(${\alpha}$) implies $M(x)=\mathcal{O}_{\varepsilon}(x^{{\alpha}+{\varepsilon}})$, we have

[TABLE]

Thus

[TABLE]

which by Theorem 2.5 implies that $E_{n}(f)\,=\,\mathcal{O}_{{\alpha}^{\prime}}(n^{{\alpha}^{\prime}})$. ∎

Discussion: We compare our assumptions with the ones made in [5]. First note, as a consequence of (1.5) and of the fact that \sum_{d\leq n}M\big{(}\frac{n}{d}\big{)}=1 (see (3.5)), that

[TABLE]

In [5, Th. 1&2], assuming the validity of RH(${\alpha}$), it was proved that $E_{n}(f)=\mathcal{O}_{\varepsilon}(n^{({\alpha}\vee{1}/{p})-{\varepsilon}})$ for all $f\in\mathcal{C}_{\rm cp}$, and the estimate is nearly optimal ([5, Th. 1&2]). The authors underlined the link between $\tilde{D}_{f}(n)$ and Fourier coefficients $(c_{f^{\prime}}(m))_{m}$ of $f^{\prime}$, which follows from Euler-MacLaurin sum formula (for absolutely continuous $f$, [5, p. 417]),

[TABLE]

Note, however (Bary [2, Vol. I] p. 78) that for absolutely continuous $f$, $c_{f^{\prime}_{\sigma}}(\ell)=i\ell c_{f_{\sigma}}(\ell)$, $\ell\in{\mathbb{Z}}_{*}$; thus the link with the Fourier coefficients of $f$ is direct. We further observe that assumption $f\in\mathcal{C}_{\rm cp}$ implies that

[TABLE]

which is much stronger than our assumptions. Indeed, by Hausdorff-Young’s theorem,

[TABLE]

In the next Theorem we provide a sharp estimate of the quadratic Farey sum $F_{n,{\sigma}}=F_{n,{\sigma}}(h)$, $h(x)=\cos 2\pi x$, recalling that $F_{n}(h)=M(n)$.

Theorem 2.8**.**

Let $0<{\sigma}<1$. Then

[TABLE]

(The integral term cannot be expressed elementarily.)

Before passing to the proofs, let us give one more example of Farey sums, linked to Euler’s generalized totient function.

Example 2.9**.**

Let $\zeta_{h}(n)=n^{h}$ with summatory function $M_{\zeta_{h}}(x)$. Let $J_{s}(n)=\zeta_{s}*{\mu}(n)=\sum_{d|n}d^{s}{\mu}(n/d)$ be Euler’s generalized totient function. Let $f(x)=c(a)\sum_{\ell\in{\mathbb{Z}}^{*}}|\ell|^{-a}e_{\ell}(x)$, $1<a<2$, where c(a)=\big{(}\sum_{\ell\in{\mathbb{Z}}^{*}}|\ell|^{-a}\big{)}^{-1}. Then

[TABLE]

the summatory function of $J_{1-a}(n)$ therefore writes as a Farey sum. First note that $f(1)=f(0)=1$, and so by Theorem 2.5,

[TABLE]

By well-known formula for partial sums of a Dirichlet product ([1, Th. 3.10]),

[TABLE]

whence (2.7).

3. Proofs of Proposition 2.1 and Theorems 2.2, 2.3, 2.5.

We first establish some auxiliary lemmas and intermediate results.

3.1. Preliminary results.

Proposition 3.1**.**

Let $f\in\mathcal{C}$. Then,

[TABLE]

*recalling that ${\varepsilon}_{\ell}(d)=d-1$ if $d|\ell$, and ${\varepsilon}_{\ell}(d)=-1$ if $d\not|\ell$, by (2.3).  (ii) If the series $\sum_{\ell\in{\mathbb{Z}}}c_{f}(\ell)$ converges, then *

[TABLE]

Proof.

Since $f\in\mathcal{C}$, by Dini’s test (Bary [2, Vol. I] p. 113, see also p. 114) the Fourier series of ${f}$

[TABLE]

converges to ${f}(x)$ (i.e. the partials sums are converging to ${f}(x)$) for any $x\in{\mathbb{Q}}\cap]0,1[$. Thus

[TABLE]

whence

[TABLE]

permutation between finitely many convergent series being permitted. As $\sum_{{\kappa}=1}^{d-1}e^{2{\rm i}\pi\ell\frac{{\kappa}}{d}}={\varepsilon}_{\ell}(d)$, we get

[TABLE]

Further if the series $\sum_{\ell\in{\mathbb{Z}}}c_{f}(\ell)$ converges, we can write

[TABLE]

which completes the proof. ∎

Corollary 3.2**.**

Let $f\in\mathcal{C}$. Assume that the series $A=\sum_{\ell\in{\mathbb{Z}}}|c_{f}(\ell)|d(\ell)$ converges. Then,

[TABLE]

Proof.

By Proposition 3.1-(ii), since $c_{f}(0)=\int_{0}^{1}f(x){\rm d}x$,

[TABLE]

By (2.4),

[TABLE]

Thus

[TABLE]

as claimed.∎

Lemma 3.3**.**

Let $f:{\mathbb{Q}}\to{\mathbb{R}}$ be arbitrary and ${\sigma}\in{\mathbb{C}}$. Then,

[TABLE]

Proof of Lemma 3.3.

We recall that

[TABLE]

Then

[TABLE]

We write $k={\kappa}u$ with ${\kappa}\leq\frac{\ell}{u}$, and get

[TABLE]

Now we write the divisors $u$ of $\ell$ under the form $u=\frac{\ell}{d}$, $d$ running along all divisors of $\ell$, and continue as follows

[TABLE]

Writing $\ell={\lambda}d$ with ${\lambda}\leq\frac{n}{d}$ in the last sum, finally gives

[TABLE]

∎

We also need the following lemma.

Lemma 3.4**.**

We have the following estimates. (a) ($\frac{1}{2}<{\sigma}<1$)

[TABLE]

recalling that \zeta(s)=\lim_{x\to\infty}\big{(}\sum_{n\leq x}\frac{1}{n^{s}}-\frac{x^{1-s}}{1-s}\big{)}, $0<s<1$. (b) ($0<{\sigma}<\frac{1}{2}$)

[TABLE]

(c)* If ${\sigma}=\frac{1}{2}$, then*

[TABLE]

where $B$ is an absolute constant. (d) If ${\sigma}=1$, then

[TABLE]

Proof.

(i) We quote the estimates ([1, Th. 3.2]),

[TABLE]

where ${\gamma}$ is Euler’s constant

(a) We get with (3.3),

[TABLE]

Further,

[TABLE]

(b) For ${\rm(b\!\cdot\!1)}$ one gets similarly

[TABLE]

Moreover,

[TABLE]

As [3, Th. 8.17] $M(x)=\mathcal{O}(x\,e^{-c\sqrt{\log x}})$ for some positive number $c$, it follows by using Abel summation that

[TABLE]

We deduce

[TABLE]

(c) For ${\rm(c\!\cdot\!1)}$, see Exercise 4-(b) in [1, Ch. 3]. As to ${\rm(c\!\hskip-1.0pt\cdot\!\hskip-1.0pt2)}$, this is Rubel’s estimate ([3, Prob. 8.20]). (d) We have

[TABLE]

and obviously,

[TABLE]

∎

Proof of Proposition 2.1.

(i) Using Lemma 3.3 we have

[TABLE]

where we isolated the term related to ${f_{\sigma}}(1)$. As $f_{\sigma}\in\mathcal{C}$, by Proposition 3.1,

[TABLE]

Thus,

[TABLE]

since only a finite number of convergent series is involved. (ii) It is easy to observe with Proposition 2.1 (applied with ${\sigma}=0$ and thus ${f_{\sigma}}=f$) that we also have (recalling that \Phi(n)=\sum_{d\leq n}dM\big{(}\frac{n}{d}\big{)}),

[TABLE]

where ${\varepsilon}_{\ell}(d)$ is defined in (2.3). By Theorem 3.12 in [1], we have

[TABLE]

So that

[TABLE]

Applying this to $\bar{f}=f-\int_{0}^{1}f(x){\rm d}x$ in place of $f$ gives ($c_{\bar{f}}(\ell)=c_{f}(\ell)$, $\ell\neq 0$ and $c_{\bar{f}}(0)=0$).

[TABLE]

∎

3.2. Proof of Theorem 2.2

By Proposition 2.1,

[TABLE]

Since the series $\sum_{\ell\in{\mathbb{Z}}}c_{f_{\sigma}}(\ell)$ is convergent, we note that

[TABLE]

So that

[TABLE]

(a) ($\frac{1}{2}<{\sigma}<1$) (i) By applying Lemma 3.4, we obtain

[TABLE]

recalling that A={f_{\sigma}}(1)+\big{(}\int_{0}^{1}f_{\sigma}(x){\rm d}x\big{)}\big{(}\frac{\zeta(2{\sigma}-1)}{\zeta(2{\sigma})}-1\big{)}-\sum_{\ell\in{\mathbb{Z}}_{*}}c_{f_{\sigma}}(\ell). (ii) If the series $\sum_{\ell\in{\mathbb{Z}}}|c_{f_{\sigma}}(\ell)|$ is convergent, then

[TABLE]

Thus

[TABLE]

(b) ($0<{\sigma}<\frac{1}{2}$) (i) From (3.6) and Lemma 3.4 follows that

[TABLE]

(ii) If the series $\sum_{\ell\in{\mathbb{Z}}}|c_{f_{\sigma}}(\ell)|$ is convergent, then by (3.4),

[TABLE]

And so,

[TABLE]

(c) (${\sigma}=\frac{1}{2}$) (i) Assume that the series $\sum_{\ell\in{\mathbb{Z}}}c_{f_{1/2}}(\ell)$ is convergent. By (3.6),

[TABLE]

By Lemma 3.4 we first get

[TABLE]

next

[TABLE]

(ii) By Theorem 3.13 in [1],

[TABLE]

if $x\geq 1$. Thus

[TABLE]

the series $\sum_{\ell\in{\mathbb{Z}}_{*}}|c_{f_{1/2}}(\ell)|$ converging by assumption. We therefore get

[TABLE]

As $c_{f_{1/2}}(0)=\int_{0}^{1}f_{1/2}(x){\rm d}x$, this achieves the proof.

3.3. Proof of Theorem 2.3

If the series $\sum_{\ell\in{\mathbb{Z}}}c_{f}(\ell)$ converges, it follows from (3.6) that

[TABLE]

And if the series $\sum_{\ell\in{\mathbb{Z}}}|c_{f}(\ell)|$ converges, then

[TABLE]

3.4. Proof of Theorem 2.5

(i) Since

[TABLE]

and by assumption the series $\sum_{\ell\in{\mathbb{Z}}}c_{f}(\ell)$ is convergent, we get using Proposition 2.1-(ii) and (3.5),

[TABLE]

(ii) As $M(x)=o(x)$, we deduce from (i) and the assumption made that

[TABLE]

Now if further $f(1)=\sum_{\ell\in{\mathbb{Z}}}c_{f}(\ell)$, Corollary 3.2 reads

[TABLE]

recalling that $\tilde{D}_{f}(d)=D_{f}(d)-d\int_{0}^{1}f(x){\rm d}x$. By Theorem 8.17 of [3], there exists a positive number $c$ such that

[TABLE]

By (2.4), E_{n}(f)\ =\ \sum_{d\leq n}\tilde{D}_{f}(d)\,M\big{(}\frac{n}{d}\big{)}. Let $\varpi(n)$ be non-decreasing. Then,

[TABLE]

Taking $\varpi(v)=v^{2}$ we get,

[TABLE]

by (3.16).

4. Proof of Theorems 1.1 and 2.8.

Proof of Theorem 1.1..

Let $0<Y_{k}\uparrow\infty$ with $k$, be an integer sequence such that

[TABLE]

We use the following precise result of Pintz [18]: for $Y>c$ effective, there exists $x^{\prime},x^{\prime\prime}\in[Ye^{-5(\log\log Y)^{3/2}},Y]$ such that

[TABLE]

$L=136000$. Assume that $Y_{1}>c$. Let $n=Y_{k+1}$. There thus exist integers $m_{j}$, $1\leq j\leq k$, such that

[TABLE]

For each $1\leq j\leq k$ let $d_{j}$ integer be such that $m_{j}=\lfloor\frac{n}{d_{j}}\rfloor$. The numbers $d_{j}$ are mutually distinct since

[TABLE]

Next we use the infinite Möbius inversion formula which we recall.Infinite Möbius inversion formula. By Theorem 270 of [10], we have the other (infinite) Möbius inversion formula

[TABLE]

if for instance

[TABLE]

It also suffices that

[TABLE]

As no proof is given in [10], we provide it for sake of completeness. First, under this assumption $f$ is obviously well defined. Now

[TABLE]

where $R_{M}(x)={\displaystyle\sum_{n=1}^{\infty}}{\mu}(n){\displaystyle\sum_{m=1\atop nm>M}^{M}}g(mnx)$ verifies

[TABLE]

and thus is small if $M$ is large. Further,

[TABLE]

by (3.2).

Note that (4.3) is clearly satisfied if $g$ is finitely supported on integers. Next choose $f:[0,1]\to{\mathbb{R}}$ so that

[TABLE]

where

[TABLE]

Then

[TABLE]

Now,

[TABLE]

∎

Remark 4.1*.*

Assume that $g$ is positive everywhere. Let $x$ be a real such that for every $c>0$, (weaker conditions are available)

[TABLE]

Then as a special case of Corollary 1.2 in [21], condition (4.3) holds as soon as

[TABLE]

In other words, it suffices that (4.3) holds with the mean value of $d(n)$ in place of $d(n)$, since $\sum_{k\leq n}d(k)\sim n\log n$.

Proof of Theorem 2.8..

By Lemma 3.3,

[TABLE]

for any ${\sigma}\in{\mathbb{C}}$. Let $0\leq{\sigma}<1$. We prove that

[TABLE]

As $h_{\sigma}^{\prime}(x)=(-2\pi\sin 2\pi x)x^{-{\sigma}}+\cos 2\pi x(-{\sigma}x^{-1-{\sigma}})$, we have $|h_{\sigma}^{\prime}(x)|\leq(2\pi+{\sigma})x^{-1-{\sigma}}$, for any $x\in]0,1]$. Let $1\leq k\leq\ell$ and $t\in]\frac{k-1}{\ell},\frac{k}{\ell}]$. Then,

[TABLE]

for some $\xi\in]\frac{k-1}{\ell},\frac{k}{\ell}[$. Thus

[TABLE]

if $k>1$, and if $k=1$, $\int_{0}^{\frac{1}{\ell}}|h_{\sigma}(t)|{\rm d}t\leq\frac{1}{(1-{\sigma})\ell^{1-{\sigma}}}$, and $\int_{0}^{\frac{1}{\ell}}|h_{\sigma}(\frac{1}{\ell})|{\rm d}t\leq\frac{1}{\ell^{1-{\sigma}}}$. Thus

[TABLE]

Now

[TABLE]

Thus

[TABLE]

Therefore \big{|}R_{h_{\sigma}}(\ell)-\int_{0}^{1}h_{\sigma}(t){\rm d}t\big{|}\leq{C_{\sigma}}/{\ell^{1-{\sigma}}} as claimed.

We deduce that

[TABLE]

Let $1/2<{\sigma}<1$. By Lemma 3.4,

[TABLE]

Consequently,

[TABLE]

Now let ${\sigma}=1/2$. By Lemma 3.3, next Lemma 3.4 and (3.11),

[TABLE]

Finally let $0<{\sigma}<1/2$. By Lemma 3.4,

[TABLE]

and by (3.4), $\sum_{{\lambda}\leq x}\frac{{\mu}({\lambda})}{{\lambda}^{2{\sigma}}}=\mathcal{O}_{\sigma}(x^{1-2{\sigma}})$. Thus

[TABLE]

By reporting these estimates in (4.11), we get

[TABLE]

∎

5. Quadratic Riemann sums.

We prove the following theorems.

Theorem 5.1**.**

Let $f:[0,1]\to{\mathbb{R}}$. Let $\frac{1}{2}<{\sigma}<1$ and assume that $f_{\sigma}\in\mathcal{C}$. Further assume that the series

[TABLE]

are convergent, where $c_{f_{\sigma}}(\ell)$, $\ell\in{\mathbb{Z}}$ are the Fourier coefficients of ${f_{\sigma}}$, and ${\sigma}_{1-2{\sigma}}(\nu)$ denotes the sum of the $(1-2{\sigma})$-th powers of the divisors of $\nu$. Then,

[TABLE]

as $n\to\infty$.

Theorem 5.1 will be deduced from the following preliminary result.

Theorem 5.2**.**

Let ${\sigma}$ be a real number. Let $f:[0,1]\to{\mathbb{R}}$ and assume that $f_{\sigma}\in\mathcal{C}$.

(1)* We have,*

[TABLE]

for each positive $n$, where ${\varepsilon}_{\ell}(d)$ is defined in (2.3). (2) Assume that the series $\sum_{\nu\in{\mathbb{Z}}}c_{f_{\sigma}}(\nu)$ is convergent. Then,

[TABLE]

for each positive $n$.

Without assuming $f_{\sigma}\in\mathcal{C}$, we have the following basic result.

Theorem 5.3**.**

Assume that $R_{f_{\sigma}}(\ell)$ converge to a finite limit $I(f_{\sigma})$, as $\ell\to\infty$. Then

[TABLE]

Consider $g(x)$ defined in (1.15),

[TABLE]

Theorem 5.4**.**

If $1/2<{\sigma}<1$, then

[TABLE]

Further,

[TABLE]

Remark 5.5*.*

The asymptotic size’s order of $S_{n,{\sigma}}(g)$ is thus the one given by the trivial bound $S_{n,{\sigma}}(g)=\mathcal{O}_{\sigma}(n^{2(1-{\sigma})})$.

As a corollary we get

Corollary 5.6**.**

Let $1/2<{\sigma}<1$. Then,

[TABLE]

as $n$ tends to infinity.

Remark 5.7*.*

It will be clear from the proofs given, that the previous Theorems extend with no difficulty to the modified Riemann quadratic sums

[TABLE]

where $0<{\alpha}<1$.

5.1. Proof of Theorem 5.3.

We note that

[TABLE]

where $R_{f_{\sigma}}(\ell)=D_{f_{\sigma}}(\ell)/\ell$. Thus

[TABLE]

where

[TABLE]

This reduces the problem to a matrix summation question. In the next lemma, we just add to well-known Toeplitz’s criterion a rate of convergence.

Lemma 5.8**.**

Let $\{{\vartheta}_{n,\ell},n\geq 1,1\leq\ell\leq n\}$ be a triangular array of complex numbers verifying the following conditions

[TABLE]

Let $\{x_{\ell},\ell\geq 1\}$ be a bounded sequence of reals and set $T_{n}=\sum_{\ell=1}^{n}{\vartheta}_{n,\ell}x_{\ell}$. Then for any $n\geq 1$ and any $D\geq 1$,

[TABLE]

In particular, if $\lim_{\ell\to\infty}x_{\ell}=0$, then $\lim_{n\to\infty}T_{n}=0$.

Proof.

Immediate since

[TABLE]

If $\lim_{\ell\to\infty}x_{\ell}=0$, given any positive real ${\varepsilon}$ and fixing $D=D({\varepsilon})$ sufficiently large so that $\sup_{\ell>D}|x_{\ell}|\leq{\varepsilon}$, we have for any $n\geq 1$,

[TABLE]

Whence \limsup_{n\to\infty}\big{|}T_{n}\big{|}\leq{\varepsilon}M, by (5.5)-(i). As ${\varepsilon}$ can be arbitrary small, this achieves the proof.∎

The triangular array (5.4) obviously verifies the conditions (5.5). Assume that the limit $I_{\sigma}(f)=\lim_{\ell\to\infty}R_{f_{\sigma}}(\ell)$ exists. By (5.3),

[TABLE]

Lemma 5.8 applied to $x_{\ell}=R_{f_{\sigma}}(\ell)-I(f_{\sigma})$, $\ell\geq 1$, together with (5.7) imply that

[TABLE]

which achieves the proof.

5.2. Proof of Theorem 5.4.

We first prove the convergence of $R_{g_{\sigma}}(\ell)$ to $\int_{0}^{1}g_{\sigma}(t){\rm d}t$, for any ${\sigma}\in[0,1[$, and provide a speed of convergence.

Lemma 5.9**.**

We have

[TABLE]

Further

[TABLE]

We need a lemma.

Lemma 5.10**.**

We have

[TABLE]

Proof.

As $g^{\prime}(x)={(\log x)\cos(\log x)-\sin(\log x)\over x(\log x)^{2}}$, we have

[TABLE]

Thus $|g_{\sigma}^{\prime}(x)|\leq\frac{2+{\sigma}}{x^{1+{\sigma}}|\log x|}$, for any $x\in]0,1]$.

Let $\ell$ and $k$ be two arbitrary integers such that $1\leq k\leq\ell$, and let $t\in]\frac{k-1}{\ell},\frac{k}{\ell}]$. We have

[TABLE]

for some $\xi\in]\frac{k-1}{\ell},\frac{k}{\ell}[$. Thus if $k>1$,

[TABLE]

If $k=1$, then $\int_{0}^{\frac{1}{\ell}}|g_{\sigma}(t)|{\rm d}t\leq\int_{0}^{\frac{1}{\ell}}\frac{1}{t^{\sigma}\log\frac{1}{t}}{\rm d}t\leq\frac{1}{(1-{\sigma})\ell^{1-{\sigma}}\log\ell}$, and $\int_{0}^{\frac{1}{\ell}}|g_{\sigma}(\frac{1}{\ell})|{\rm d}t\leq\frac{1}{\ell}\frac{\ell^{\sigma}}{\log\ell}=\frac{1}{\ell^{1-{\sigma}}\log\ell}$. So that the bound in (5.11) remains valid for $k=1$ either. Thus

[TABLE]

Now,

[TABLE]

If $\ell/2<k\leq\ell$, we write as in the proof of lemma 7.2 in [20], $k=\ell-r$ where $1\leq r<\ell/2$. Then \log\frac{\ell}{k}=-\log\big{(}1-\frac{r}{\ell})>\frac{r}{\ell}. Whence also,

[TABLE]

By combining we thus deduce

[TABLE]

By inserting (5.13) in (5.12), we therefore obtain

[TABLE]

∎

Proof of Lemma 5.9.

As for arbitrary positive integers $\ell$ and $k$,

[TABLE]

we have

[TABLE]

By Lemma 5.10,

[TABLE]

Thus

[TABLE]

Dividing both sides by $\ell^{1-{\sigma}}$, we get

[TABLE]

Whence the first part of the Lemma. As (Dwight [8, 863.4]),

[TABLE]

we get (here comes the restriction ${\sigma}<1$),

[TABLE]

∎

Proof of Theorem 5.4.

We get in view of Theorem 5.3 and Lemma 5.9,

[TABLE]

Now by (5.6), next by the first part of Lemma 5.9, for $1/2\leq{\sigma}<1$,

[TABLE]

Choosing $D\sim n^{2/3}$ gives

[TABLE]

As by (5.7)

[TABLE]

we get

[TABLE]

By (3.3), $\sum_{1\leq\ell\leq n}\ell^{1-2{\sigma}}=\frac{n^{2(1-{\sigma})}}{2(1-{\sigma})}+\zeta(2{\sigma}-1)+\mathcal{O}\left(n^{1-2{\sigma}}\right)$, so that

[TABLE]

We have obtained

[TABLE]

This achieves the proof.∎

Remark 5.11*.*

Let $a\geq 1$ and let $h(x)=\frac{\sin a\log x}{\log x}$. The same proof allows one to get

[TABLE]

Indeed, in this case

[TABLE]

Thus $|h_{\sigma}^{\prime}(x)|\leq\frac{2a+{\sigma}}{x^{1+{\sigma}}|\log x|}$, for any $x\in]0,1]$. So it suffices to substitute the new constant $2a+{\sigma}$ to $2+{\sigma}$ everywhere in the proof. In the critical case $a=n^{\beta}$, this is better than the classical bound $S_{n,{\sigma}}(g)=\mathcal{O}_{\sigma}(n^{2(1-{\sigma})}\log n)$ (cf. [20, Eq. (7.2.1)]), only if ${\beta}<{2(1-{\sigma})/3}$. As ${\sigma}>1/2$, ${2(1-{\sigma})/3}<1/3$; so ${\beta}$ must be less than $1/3$. See also Remark 5.5.

Problem 5.12**.**

How to improve the error term in (5.24) when $a=n^{\beta}$ with ${\beta}\geq 1/3$?

5.3. Proof of Corollary 5.6.

Follows from Theorem 5.4 since

[TABLE]

5.4. Proof of Theorem 5.2.

Here again we use Dini’s test (Bary [2, Vol. I]) which implies that the Fourier series of ${f_{\sigma}}$

[TABLE]

converges to ${f_{\sigma}}(x)$ for any real $x$, $0<x<1$. Thus

[TABLE]

Whence

[TABLE]

Therefore

[TABLE]

Whence the first assertion. Now if the series $\sum_{\ell\in{\mathbb{Z}}}c_{f_{\sigma}}(\ell)$ is convergent, we can write

[TABLE]

since $\sum_{k=1}^{\ell}e^{2{\rm i}\pi\nu\frac{k}{\ell}}$ equals to $\ell$ or [math] according to $\ell|\nu$ or not. Thus

[TABLE]

5.5. Proof of Theorem 5.1.

As $f$ is real-valued

[TABLE]

For each $N\geq 1$,

[TABLE]

From the assumptions made, by using a standard approximation argument, we deduce

[TABLE]

Using Theorem 5.2 and estimate (3.3) we get,

[TABLE]

as $n\to\infty$.

In the next Proposition we provide with (5.27) an $\ell^{1}$-type control of the error term

[TABLE]

Proposition 5.13**.**

Let $\bar{f}_{\sigma}=f_{\sigma}-\int_{0}^{1}f_{\sigma}(t){\rm d}t$. Let $\varpi(n)$ be positive reals such that the series $\sum_{n\geq 1}1/\varpi(n)$ converges, and let $\rho(d)=\sum_{n\geq d}1/\varpi(n)$. Assume that

[TABLE]

Then

[TABLE]

Proof.

By linearity, $R_{\bar{f}_{\sigma}}(\ell)=R_{f_{\sigma}}(\ell)-\int_{0}^{1}f_{\sigma}(t){\rm d}t$, next by (5.2),

[TABLE]

Thus

[TABLE]

by assumption. ∎

Remark 5.14*.*

In the case of example (1.15), it can be easily checked that Proposition 5.13 however provides a weaker estimate than (see in particular (5.20)) the one proven in Theorem 5.4.

6.

Concluding remarks: Local integrals of $\boldsymbol{\zeta(s)}$ and amalgams.

We discuss some questions related to the previous sections, in particular to Remark 5.11, and to local integrals of the $\zeta$-function. We notably consider three interesting related problems. Let $1\leq a<b<\infty$, $a\geq cb$ for some positive $c$. We first note that for ${\sigma}>\frac{1}{2}$,

[TABLE]

where

[TABLE]

and $g(a,b)(1)=\frac{1}{2}(b-a)$, $g(a,b)(0)=0$. Indeed (as $\sin a-\sin b=2\sin\frac{1}{2}(a-b)\cos\frac{1}{2}(a+b)$ ),

[TABLE]

By the classical approximation formula ([19], Theorem 3.5), given ${\sigma}_{0}>0$, $0<{\delta}<1$, we have uniformly for ${\sigma}\geq{\sigma}_{0}$, $x\geq 1$ , $0<|t|\leq(1-{\delta})2\pi x$,

[TABLE]

Thus

[TABLE]

Now assume $a=n$ and $b=n+1$. Then

[TABLE]

More precisely,

[TABLE]

where

[TABLE]

and $g_{n}(1)=\frac{1}{2}$, $g_{n}(0)=0$. Let also $g_{n,{\sigma}}(x)=g_{n}(x)/x^{\sigma}$, $n\geq 1$. This stresses if necessary, the importance of the quadratic sums $S_{n,{\sigma}}(g_{n})$. We could not find in the literature asymptotic estimates for these sums when $n\to\infty$, even partial ones, which is a bit surprising. The key quantities to be estimated are the Riemann sums $R_{g_{n,{\sigma}}}(d)$, $d\leq n$. However $g_{n,{\sigma}}(x)$ oscillates wildly near $0_{+}$, with peaks increasing to infinity with $n$, and it seems illusory to directly estimate them. The Fourier coefficients can however be computed. We first note by arguing as in (2.2) that

[TABLE]

Next Theorem 5.2 directly implies the following Proposition.

Proposition 6.1**.**

for $\frac{1}{2}<{\sigma}<1$,

[TABLE]

as $n\to\infty$, recalling that ${\varepsilon}_{k}(\ell)$ is defined in (2.3). Further for any $k\in{\mathbb{Z}}*$,

[TABLE]

Proof.

By (6),

[TABLE]

Obviously $g_{n,{\sigma}}(x)$ is derivable for any real $x$, $0<x<1$. Thus by Theorem 5.2,

[TABLE]

Whence

[TABLE]

Note that $c_{g_{n,{\sigma}}}(0)$ is small, $c_{g_{n,{\sigma}}}(0)\sim\,{C_{\sigma}}/{n^{2}}$, since

[TABLE]

Thus

[TABLE]

Therefore

[TABLE]

as claimed. Now we compute the Fourier coefficients of $g_{n,{\sigma}}$. We have

[TABLE]

Thus

[TABLE]

∎

We conclude this paper with a remark on the $\mathcal{S}^{2}$-Stepanov norm of $\zeta(s)$. The Stepanov space $\mathcal{S}^{2}$ is defined as the sub-space of functions $f$ of $L^{2}_{\rm loc}({\mathbb{R}})$ verifying the following analogue of Bohr almost periodicity property: For all ${\varepsilon}>0$, there exists $K_{\varepsilon}>0$ such that for any $x_{0}\in{\mathbb{R}}$, there exists $\tau\in[x_{0},x_{0}+K_{\varepsilon}]$ such that $\|f(.+\tau)-f(.)\|_{\mathcal{S}^{2}}\leq{\varepsilon}$. The Stepanov norm in $\mathcal{S}^{2}$ is equivalent to the “amalgam” norm

[TABLE]

The almost everywhere convergence properties of almost periodic Fourier series in $\mathcal{S}^{2}$ and of corresponding series of dilates were recently studied with Cuny in [7]; and a new form of Carleson’s theorem for almost periodic Fourier series was proved. A natural question arising from this study concerns the Riemann zeta function $\zeta(s)$, $s={\sigma}+it$, and more precisely the evaluation of its Stepanov’s norm, namely the supremum over all $n$ of the local integrals $\int_{n}^{n+1}|\zeta({\sigma}+it)|^{2}{\rm d}t$. It is clear that

Lemma 6.2**.**

[TABLE]

Indeed otherwise it would imply that \int_{0}^{T}|\zeta({1\over 2}+it)|^{2}{\rm d}t\leq C\,\|\zeta\big{(}\frac{1}{2}+i\cdot\big{)}\|_{\mathcal{S}^{2}}T, which contradicts the classical mean value estimate $\int_{0}^{T}|\zeta({1\over 2}+it)|^{2}{\rm d}t\sim T\log T$, ([20] p. 176). Note that

[TABLE]

It is interesting to observe that the above $\Omega$-result is in a sense optimal when $t$ is modelled by a Cauchy random walk. The behavior of the Riemann zeta-function on the critical line, along the Cauchy random walk, was studied by Lifshits and Weber in [12]. Let $X_{1},X_{2},\ldots$ denote an infinite sequence of independent Cauchy distributed random variables (with characteristic function $\varphi(t)=e^{-|t|}$), and consider the partial sums $S_{n}=X_{1}+\ldots+X_{n}$. We shall prove the following precise result.

Proposition 6.3**.**

We have

[TABLE]

Moreover,

[TABLE]

Proof.

Put

[TABLE]

Put for any positive integer $n$

[TABLE]

By Theorem 1.1 in Lifshits and Weber [12], there exist explicit constants $C,C_{0}$ such that

[TABLE]

Therefore,

[TABLE]

By [12, (3.3)],

[TABLE]

By combining with (6.15)-(i) the first claim follows. As $|a-b|^{2}\leq 2(a^{2}+b^{2})$, we also get

[TABLE]

which proves the second claim.∎

For $\frac{1}{2}<{\sigma}<1$, the simple argument used in Lemma 6.2 no longer works since by Landau and Schnee’s result, $\int_{0}^{T}|\zeta({\sigma}+it)|^{2}{\rm d}t\sim\zeta(2{\sigma})T$ if ${\sigma}>\frac{1}{2}$; in fact ([14, p.13&16])

[TABLE]

if $\frac{1}{2}<{\sigma}<1$. Thus

[TABLE]

for infinitely many $n$, with $A<\zeta(2{\sigma})$, and the question arises whether $A$ can be arbitrary large, namely whether

Problem 6.4**.**

It is true that for $1/2<{\sigma}\leq 1$

[TABLE]

This question is of a different kind from that one of estimating the local integrals \int_{n}^{n+1}\big{|}\zeta({\sigma}+it)\big{|}^{2}{\rm d}t; both problems might rely on independent devices. Maybe the answer to Problem 6.4 is negative. In this direction, it seems that Proposition 6.3 can be extended to the range of values $1/2<{\sigma}<1$ as follows:

[TABLE]

This is examined in a separate work [22].

We also note the following equivalent reformulation: for any ${\sigma}>0$, any $b>0$,

[TABLE]

where $k_{b},K_{b}$ are positive finite constants depending on $b$ only. This follows from the Lemma below.

Lemma 6.5**.**

Let $b>0$. For any $f\in L^{2}_{{\rm loc}}({\mathbb{R}})$,

[TABLE]

Moreover,

[TABLE]

where $k_{b}=\frac{1}{b^{2}+1}$ and $K_{b}=2\sum_{w\geq 0}\frac{1}{b^{2}+w^{2}}+\frac{1}{b^{2}}$.

Proof.

On the one hand, for any real $u$,

[TABLE]

Taking supremum over $u$ in both sides yields

[TABLE]

On the other hand,

[TABLE]

∎

Concerning the integrals appearing in (6.21), we recall the classical formula [20, (2.1.5)],

[TABLE]

where $\{x\}=x-[x]$ denotes the fractional part of $x$. Thus by Parseval equality for Mellin’s transform,

[TABLE]

for $0<{\sigma}<1$, the quantity in brackets being negative. For the first equality, see [11, Cor. 1]. The second equality was proved by Coffey [6, Prop. 7], as well as [6, Cor. 8],

[TABLE]

and we note that by Lemma 6.2 and Corollary 5.6,

[TABLE]

Remark 6.6*.*

It is not clear to the author whether the integrals

[TABLE]

can be similarly estimated. We further could not find any reference in the literature.

It also appears that formula (6.15)-(i), which exactly means

[TABLE]

seems not be easily obtained by using complex integration.

The Stepanov space $\mathcal{S}^{2}$ is one instance of amalgam. We recall that the weighted amalgam $\ell^{q}(L^{p},w)$ consists of functions $f$ on ${\mathbb{R}}$ such that

[TABLE]

where $1<p,q<\infty$ and $w$ is a weight function. We end with the following question.

Problem 6.7**.**

In which weighted amalgams $\ell^{q}(L^{2},w)$ lies the Riemann zeta function?

Final Note. We recently solved Problem 6.4. This is the object of a separate writing [23].

Acknowledgments I thank Christophe Cuny for readings and comments on first attempts. I also thank Benjamin Enriquez for friendly discussions around Problem 6.4. I further thank Antanas Laurinčikas and Kohji Matsumoto for comments on this problem. Finally, I thank Jörn Steuding for exchanges around Remark 6.6.