arXiv subjects

Kristian Seip

Publications and source records attributed to Kristian Seip.

At least 37 records · Page 2Linked to original sources

arXiv2019

A footnote to a theorem of Halász

We study multiplicative functions $f$ satisfying $|f(n)|\le 1$ for all $n$, the associated Dirichlet series $F(s):=\sum_{n=1}^{\infty} f(n) n^{-s}$, and the summatory function $S_f(x):=\sum_{n\le x} f(n)$. Up to a possible trivial contribution from the numbers $f(2^k)$, $F(s)$ may have at most one zero or one pole on the one-line, in a sense made precise by Halász. We estimate $\log F(s)$ away from any such point and show that if $F(s)$ has a zero on the one-line in the sense of Halász, then $|S_f(x)|\le (x/\log x) \exp\big(c\sqrt{\log \log x}\big)$ for all $c>0$ when $x$ is large enough. This bound is best possible.

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arXiv2019

Universality and distribution of zeros and poles of some zeta functions

This paper studies zeta functions of the form $\sum_{n=1}^{\infty} χ(n) n^{-s}$, with $χ$ a completely multiplicative function taking only unimodular values. We denote by $σ(χ)$ the infimum of those $α$ such that the Dirichlet series $\sum_{n=1}^{\infty} χ(n) n^{-s}$ can be continued meromorphically to the half-plane $\operatorname{Re} s>α$, and denote by $ζ_χ(s)$ the corresponding meromorphic function in $\operatorname{Re} s>σ(χ)$. We construct $ζ_χ(s)$ that have $σ(χ)\le 1/2$ and are universal for zero-free analytic functions on the half-critical strip $1/2<\operatorname{Re} s <1$, with zeros and poles at any discrete multisets lying in a strip to the right of $\operatorname{Re} s =1/2$ and satisfying a density condition that is somewhat stricter than the density hypothesis for the zeros of the Riemann zeta function. On a conceivable version of Cramér's conjecture for gaps between primes, the density condition can be relaxed, and zeros and poles can also be placed at $β+i γ$ with $β\le 1-λ\log\log |γ|/\log |γ|$ when $λ>1$. Finally, we show that there exists $ζ_χ(s)$ with $σ(χ) \le 1/2$ and zeros at any discrete multiset in the strip $1/2<\operatorname{Re} s \le 39/40$ with no accumulation point in $\operatorname{Re} s >1/2$; on the Riemann hypothesis, this strip may be replaced by the half-critical strip $1/2 < \operatorname{Re} s < 1$.

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arXiv2019

Linear space properties of $H^p$ spaces of Dirichlet series

We study $H^p$ spaces of Dirichlet series, called $\mathcal{H}^p$, for the range $0<p< \infty$. We begin by showing that two natural ways to define $\mathcal{H}^p$ coincide. We then proceed to study some linear space properties of $\mathcal{H}^p$. More specifically, we study linear functionals generated by fractional primitives of the Riemann zeta function; our estimates rely on certain Hardy--Littlewood inequalities and display an interesting phenomenon, called contractive symmetry between $\mathcal{H}^p$ and $\mathcal{H}^{4/p}$, contrasting the usual $L^p$ duality. We next deduce general coefficient estimates, based on an interplay between the multiplicative structure of $\mathcal{H}^p$ and certain new one variable bounds. Finally, we deduce general estimates for the norm of the partial sum operator $\sum_{n=1}^\infty a_n n^{-s}\mapsto \sum_{n=1}^N a_n n^{-s}$ on $\mathcal{H}^p$ with $0< p \le 1$, supplementing a classical result of Helson for the range $1<p<\infty$. The results for the coefficient estimates and for the partial sum operator exhibit the traditional schism between the ranges $1\le p \le \infty$ and $0<p<1$.

math.FA↗

arXiv2018

On certain sums over ordinates of zeta-zeros II

Let $γ$ denote the imaginary parts of complex zeros $ρ= β+iγ$ of $ζ(s)$. The problem of analytic continuation of the function $G(s) := \sum\limits_{γ> 0}γ^{-s}$ to the left of the line $\Re s = -1$ is investigated, and its Laurent expansion at the pole $s=1$ is obtained. Estimates for the second moment on the critical line $\int_1^T|G(1/2+it)|^2\,dt$ are revisited. This paper is a continuation of work begun by the second author in 2001.

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arXiv2018

Pseudomoments of the Riemann zeta function

The $2$kth pseudomoments of the Riemann zeta function $ζ(s)$ are, following Conrey and Gamburd, the $2k$th integral moments of the partial sums of $ζ(s)$ on the critical line. For fixed $k>1/2$, these moments are known to grow like $(\log N)^{k^2}$, where $N$ is the length of the partial sum, but the true order of magnitude remains unknown when $k\le 1/2$. We deduce new Hardy--Littlewood inequalities and apply one of them to improve on an earlier asymptotic estimate when $k\to\infty$. In the case $k<1/2$, we consider pseudomoments of $ζ^α(s)$ for $α>1$ and the question of whether the lower bound $(\log N)^{k^2α^2}$ known from earlier work yields the true growth rate. Using ideas from recent work of Harper, Nikeghbali, and Radziwi{łł} and some probabilistic estimates of Harper, we obtain the somewhat unexpected result that these pseudomements are bounded below by $\log N$ to a power larger than $k^2α^2$ when $k<1/e$ and $N$ is sufficiently large.

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arXiv2018

Contractive inequalities for Hardy spaces

We state and discuss several interrelated results, conjectures, and questions regarding contractive inequalities for classical $H^p$ spaces of the unit disc. We study both coefficient estimates in terms of weighted $\ell^2$ sums and the Riesz projection viewed as a map from $L^q$ to $H^p$ with $q\ge p$. Some numerical evidence is given that supports our conjectures.

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arXiv2018

Extreme values of the Riemann zeta function and its argument

We combine our version of the resonance method with certain convolution formulas for $ζ(s)$ and $\log\, ζ(s)$. This leads to a new $Ω$ result for $|ζ(1/2+it)|$: The maximum of $|ζ(1/2+it)|$ on the interval $1 \le t \le T$ is at least $\exp\left((1+o(1)) \sqrt{\log T \log\log\log T/\log\log T}\right)$. We also obtain conditional results for $S(t):=1/π$ times the argument of $ζ(1/2+it)$ and $S_1(t):=\int_0^t S(τ)dτ$. On the Riemann hypothesis, the maximum of $|S(t)|$ is at least $c \sqrt{\log T \log\log\log T/\log\log T}$ and the maximum of $S_1(t)$ is at least $c_1 \sqrt{\log T \log\log\log T/(\log\log T)^3}$ on the interval $T^β \le t \le T$ whenever $0\le β< 1$.

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arXiv2017

Note on the resonance method for the Riemann zeta function

We improve Montgomery's $Ω$-results for $|ζ(σ+it)|$ in the strip $1/2<σ<1$ and give in particular lower bounds for the maximum of $|ζ(σ+it)|$ on $\sqrt{T}\le t \le T$ that are uniform in $σ$. We give similar lower bounds for the maximum of $|\sum_{n\le x} n^{-1/2-it}|$ on intervals of length much larger than $x$. We rely on our recent work on lower bounds for maxima of $|ζ(1/2+it)|$ on long intervals, as well as work of Soundararajan, Gál, and others. The paper aims at displaying and clarifying the conceptually different combinatorial arguments that show up in various parts of the proofs.

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arXiv2016

Large GCD sums and extreme values of the Riemann zeta function

It is shown that the maximum of $|ζ(1/2+it)|$ on the interval $T^{1/2}\le t \le T$ is at least $\exp\left((1/\sqrt{2}+o(1)) \sqrt{\log T \log\log\log T/\log\log T}\right)$. Our proof uses Soundararajan's resonance method and a certain large GCD sum. The method of proof shows that the absolute constant $A$ in the inequality \[ \sup_{1\le n_1<\cdots < n_N} \sum_{k,{\ell}=1}^N\frac{\gcd(n_k,n_{\ell})}{\sqrt{n_k n_{\ell}}} \ll N \exp\left(A\sqrt{\frac{\log N \log\log\log N}{\log\log N}}\right), \] established in a recent paper of ours, cannot be taken smaller than $1$.

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arXiv2016

Volterra operators on Hardy spaces of Dirichlet series

For a Dirichlet series symbol $g(s) = \sum_{n \geq 1} b_n n^{-s}$, the associated Volterra operator $\mathbf{T}_g$ acting on a Dirichlet series $f(s)=\sum_{n\ge 1} a_n n^{-s}$ is defined by the integral $f\mapsto -\int_{s}^{+\infty} f(w)g'(w)\,dw$. We show that $\mathbf{T}_g$ is a bounded operator on the Hardy space $\mathcal{H}^p$ of Dirichlet series with $0 < p < \infty$ if and only if the symbol $g$ satisfies a Carleson measure condition. When appropriately restricted to one complex variable, our condition coincides with the standard Carleson measure characterization of ${\operatorname{BMOA}}(\mathbb{D})$. A further analogy with classical ${\operatorname{BMO}}$ is that $\exp(c|g|)$ is integrable (on the infinite polytorus) for some $c > 0$ whenever $\mathbf{T}_g$ is bounded. In particular, such $g$ belong to $\mathcal{H}^p$ for every $p < \infty$. We relate the boundedness of $\mathbf{T}_g$ to several other ${\operatorname{BMO}}$ type spaces: ${\operatorname{BMOA}}$ in half-planes, the dual of $\mathcal{H}^1$, and the space of symbols of bounded Hankel forms. Moreover, we study symbols whose coefficients enjoy a multiplicative structure and obtain coefficient estimates for $m$-homogeneous symbols as well as for general symbols. Finally, we consider the action of $\mathbf{T}_g$ on reproducing kernels for appropriate sequences of subspaces of $\mathcal{H}^2$. Our proofs employ function and operator theoretic techniques in one and several variables; a variety of number theoretic arguments are used throughout the paper in our study of special classes of symbols $g$.

math.FA↗

arXiv2016

The multiplicative Hilbert matrix

It is observed that the infinite matrix with entries $(\sqrt{mn}\log (mn))^{-1}$ for $m, n\ge 2$ appears as the matrix of the integral operator $\mathbf{H}f(s):=\int_{1/2}^{+\infty}f(w)(ζ(w+s)-1)dw$ with respect to the basis $(n^{-s})_{n\ge 2}$; here $ζ(s)$ is the Riemann zeta function and $H$ is defined on the Hilbert space ${\mathcal H}^2_0$ of Dirichlet series vanishing at $+\infty$ and with square-summable coefficients. This infinite matrix defines a multiplicative Hankel operator according to Helson's terminology or, alternatively, it can be viewed as a bona fide (small) Hankel operator on the infinite-dimensional torus ${\Bbb T}^{\infty}$. By analogy with the standard integral representation of the classical Hilbert matrix, this matrix is referred to as the multiplicative Hilbert matrix. It is shown that its norm equals $π$ and that it has a purely continuous spectrum which is the interval $[0,π]$; these results are in agreement with known facts about the classical Hilbert matrix. It is shown that the matrix $(m^{1/p} n^{(p-1)/p}\log (mn))^{-1}$ has norm $π/\sin(π/p)$ when acting on $\ell^p$ for $1<p<\infty$. However, the multiplicative Hilbert matrix fails to define a bounded operator on ${\mathcal H}^p_0$ for $p\neq 2$, where ${\mathcal H}^p_0$ are $H^p$ spaces of Dirichlet series. It remains an interesting problem to decide whether the analytic symbol $\sum_{n\ge 2} (\log n)^{-1} n^{-s-1/2}$ of the multiplicative Hilbert matrix arises as the Riesz projection of a bounded function on the infinite-dimensional torus ${\Bbb T}^\infty$.

math.FA↗

arXiv2016

Gál-type GCD sums beyond the critical line

We prove that \[ \sum_{k,{\ell}=1}^N\frac{(n_k,n_{\ell})^{2α}}{(n_k n_{\ell})^α} \ll N^{2-2α} (\log N)^{b(α)} \] holds for arbitrary integers $1\le n_1<\cdots < n_N$ and $0<α<1/2$ and show by an example that this bound is optimal, up to the precise value of the exponent $b(α)$. This estimate complements recent results for $1/2\le α\le 1$ and shows that there is no "trace" of the functional equation for the Riemann zeta function in estimates for such GCD sums when $0<α<1/2$.

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arXiv2016

Some open questions in analysis for Dirichlet series

We present some open problems and describe briefly some possible research directions in the emerging theory of Hardy spaces of Dirichlet series and their intimate counterparts, Hardy spaces on the infinite-dimensional torus. Links to number theory are emphasized throughout the paper.

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arXiv2015

Approximation numbers of composition operators on $H^p$ spaces of Dirichlet series

By a theorem of Bayart, $φ$ generates a bounded composition operator on the Hardy space $\Hp$of Dirichlet series ($1\le p<\infty$) only if $φ(s)=c_0 s+ψ(s)$, where $c_0$ is a nonnegative integer and $ψ$ a Dirichlet series with the following mapping properties: $ψ$ maps the right half-plane into the half-plane $\Real s >1/2$ if $c_0=0$ and is either identically zero or maps the right half-plane into itself if $c_0$ is positive. It is shown that the $n$th approximation numbers of bounded composition operators on $\Hp$ are bounded below by a constant times $r^n$ for some $0 0$ when $c_0$ is positive. Both results are best possible. Estimates rely on a combination of soft tools from Banach space theory ($s$-numbers, type and cotype of Banach spaces, Weyl inequalities, and Schauder bases) and a certain interpolation method for $\Ht$, developed in an earlier paper, using estimates of solutions of the $\overline{\partial}$ equation. A transference principle from $H^p$ of the unit disc is discussed, leading to explicit examples of compact composition operators on $\Ho$ with approximation numbers decaying at a variety of sub-exponential rates. Finally, a new Littlewood--Paley formula is established, yielding a sufficient condition for a composition operator on $\Hp$ to be compact.

math.FA↗

arXiv2015

Helson's problem for sums of a random multiplicative function

We consider the random functions $S_N(z):=\sum_{n=1}^N z(n) $, where $z(n)$ is the completely multiplicative random function generated by independent Steinhaus variables $z(p)$. It is shown that ${\Bbb E} |S_N|\gg \sqrt{N}(\log N)^{-0.05616}$ and that $({\Bbb E} |S_N|^q)^{1/q}\gg_{q} \sqrt{N}(\log N)^{-0.07672}$ for all $q>0$.

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arXiv2015

An inequality of Hardy--Littlewood type for Dirichlet polynomials

The $L^q$ norm of a Dirichlet polynomial $F(s)=\sum_{n=1}^{N} a_n n^{-s}$ is defined as \[\| F\|_q:=(\lim_{T\to\infty}\frac{1}{T}\int_{0}^T |F(it)|^qdt)^{1/q}\] for $0<q<\infty$. It is shown that \[ (\sum_{n=1}^{N} |a_n|^2|μ(n)|[d(n)]^{\frac{\log q}{\log 2} -1})^{1/2}\le \| F\|_q \] when $0<q<2$; here $μ$ is the Möbius function and $d$ the divisor function. This result is used to prove that the $L^q$ norm of $D_N(s):=\sum_{n=1}^{N} n^{-1/2-s}$ satisfies $\|D_N\|_q\gg (\log N)^{q/4}$ for $0<q<\infty$. By Helson's generalization of the M. Riesz theorem on the conjugation operator, the reverse inequality $\|D_N\|_q \ll (\log N)^{q/4}$ is shown to be valid in the range $1<q<\infty$. Similar bounds are found for a fairly large class of Dirichlet series including, on one of Selberg's conjectures, the Selberg class of $L$-functions.

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arXiv2014

Approximation numbers of composition operators on the $H^2$ space of Dirichlet series

By a theorem of Gordon and Hedenmalm, $φ$ generates a bounded composition operator on the Hilbert space $\mathscr{H}^2$ of Dirichlet series $\sum_n b_n n^{-s}$ with square-summable coefficients $b_n$ if and only if $φ(s)=c_0 s+ψ(s)$, where $c_0$ is a nonnegative integer and $ψ$ a Dirichlet series with the following mapping properties: $ψ$ maps the right half-plane into the half-plane $\operatorname{Re} s >1/2$ if $c_0=0$ and is either identically zero or maps the right half-plane into itself if $c_0$ is positive. It is shown that the $n$th approximation numbers of bounded composition operators on $\mathscr{H}^2$ are bounded below by a constant times $r^n$ for some $0 0$ when $c_0$ is positive. Both results are best possible. The case when $c_0=0$, $ψ$ is bounded and smooth up to the boundary of the right half-plane, and $\sup \operatorname{Re} ψ=1/2$, is discussed in depth; it includes examples of non-compact operators as well as operators belonging to all Schatten classes $S_p$. For $φ(s)=c_1+\sum_{j=1}^d c_{q_j} q_j^{-s}$ with $q_j$ independent integers, it is shown that the $n$th approximation number behaves as $n^{-(d-1)/2}$, possibly up to a factor $(\log n)^{(d-1)/2}$. Estimates rely mainly on a general Hilbert space method involving finite linear combinations of reproducing kernels. A key role is played by a recently developed interpolation method for $\mathscr{H}^2$ using estimates of solutions of the $\bar{\partial}$ equation. Finally, by a transference principle from $H^2$ of the unit disc, explicit examples of compact composition operators with approximation numbers decaying at essentially any sub-exponential rate can be displayed.

math.FA↗

arXiv2014

Convergence of series of dilated functions and spectral norms of GCD matrices

We establish a connection between the $L^2$ norm of sums of dilated functions whose $j$th Fourier coefficients are $\mathcal{O}(j^{-α})$ for some $α\in (1/2,1)$, and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in $L^2$ and for the almost everywhere convergence of series of dilated functions.

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