# ``Quasi''-norm of an arithmetical convolution operator and the order of the Riemann zeta function

@inproceedings{Hilberdink2013QuasinormOA, title={``Quasi''-norm of an arithmetical convolution operator and the order of the Riemann zeta function}, author={Titus Hilberdink}, year={2013} }

In this paper we study Dirichlet convolution with a given arithmetical function f as a linear mapping 'f that sends a sequence (an) to (bn) where bn = Pdjn f(d)an=d.
We investigate when this is a bounded operator on l2 and ¯nd the operator norm. Of particular interest is the case f(n) = ni® for its connection to the Riemann zeta
function on the line 1, 'f is bounded with k'f k = ³(®). For the unbounded case, we show that 'f : M2 ! M2 where M2 is the subset of l2 of multiplicative sequences…

## 6 Citations

### The boundedness and spectral properties of multiplicative Toeplitz operators

- Mathematics
- 2019

The aim of this thesis is to study the properties of multiplicative Toeplitz operators
with an emphasis on boundedness and spectral points. In particular, we consider these
operators acting on the…

### Maximal order of a class of multiplicative functions

- Mathematics
- 2014

In this paper we obtain the maximal order of the multiplicative function given at the prime powers by f(p) = exp{h(k)l(p)} where h(·) and l(·) are increasing and decreasing functions respectively…

### Boundedness and Spectrum of Multiplicative Convolution Operators Induced by Arithmetic Functions

- MathematicsActa Mathematica Sinica, English Series
- 2019

In this paper, we consider a multiplicative convolution operator $${{\cal M}_f}$$ℳf acting on a Hilbert spaces ℓ2(ℕ,ω). In particular, we focus on the operators $${{\cal M}_1}$$ℳ1 and $${{\cal M}_\mu…

### Bounded multiplicative Toeplitz operators on sequence spaces

- Mathematics
- 2018

In this paper, we study the linear mapping which sends the sequence…

### MAXIMAL ORDER OF A CLASS

- Mathematics
- 2014

In this paper we obtain the maximal order of the multiplica- tive function given at the prime powers by f(p k ) = expfh(k)l(p)g where h( ) and l( ) are increasing and decreasing functions…

### Boundedness and Spectrum of Multiplicative Convolution Operators Induced by Arithmetic Functions

- Materials ScienceActa Mathematica Sinica, English Series
- 2019

In this paper, we consider a multiplicative convolution operator ℳf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb}…

## References

SHOWING 1-10 OF 19 REFERENCES

### An arithmetical mapping and applications to Ω-results for the Riemann zeta function

- Mathematics
- 2009

In this paper we study the linear mapping that sends a sequence (an) to (bn) where bn = ∑ d|n d −αan/d. We investigate for which values of α this is a bounded operator from l to l and show the…

### Finite Euler products and the Riemann hypothesis

- Mathematics
- 2007

We show that if the Riemann Hypothesis is true, then in a region containing most of the right-half of the critical strip, the Riemann zeta-function is well approximated by short truncations of its…

### Extreme values of $|ζ(1+it)|$

- Mathematics
- 2005

Improving on a result of J.E. Littlewood, N. Levinson [3] showed that there are arbitrarily large t for which |ζ(1 + it)| ≥ e log2 t + O(1). (Throughout ζ(s) is the Riemann-zeta function, and logj…

### GCD sums from Poisson integrals and systems of dilated functions

- Mathematics
- 2012

Upper bounds for GCD sums of the form [\sum_{k,{\ell}=1}^N\frac{(\gcd(n_k,n_{\ell}))^{2\alpha}}{(n_k n_{\ell})^\alpha}] are proved, where $(n_k)_{1 \leq k \leq N}$ is any sequence of distinct…

### On the distribution of extreme values of zeta and $L$-functions in the strip $1/2<\sigma<1$

- Mathematics
- 2010

We study the distribution of large (and small) values of several families of $L$-functions on a line $\text{Re(s)}=\sigma$ where $1/2<\sigma<1$. We consider the Riemann zeta function $\zeta(s)$ in…

### Real and complex analysis

- Mathematics
- 1966

Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures…

### Determinants of Multiplicative Toeplitz Matrices

- Mathematics
- 2006

In this paper we study matrices A = (aij) whose (i;j) th -entry is a function of i=j; that is, aij = f(i=j) for some f :Q + !C. We obtain a formula for the truncated determinants in the case where f…

### Mean values of finite Euler products

- Mathematics
- 2010

We prove several theorems concerning mean values of the modulus squared of finite Euler products in right half‐planes of the complex plane. We are particularly interested in knowing when the mean of…

### Extreme values of zeta and L-functions

- Mathematics
- 2007

We introduce a resonance method to produce large values of the Riemann zeta-function on the critical line, and large and small central values of L-functions.