arXiv subjects

Johan Andersson

Publications and source records attributed to Johan Andersson.

28 records · Page 2Linked to original sources

arXiv2012

On a problem of Ramachandra and approximation of functions by Dirichlet polynomials with bounded coefficients

We prove effective results on when a function can be approximated by a Dirichlet polynomial with bounded coefficients. Assuming that Φ(n) is an increasing function we prove that the set of polynomials {\sum_{n=2}^N a_n n^{it-1}: N \geq 2, |a_n| \leq Φ(n)}, is dense in L^2(0,H) if and only if \sum_{n=2}^\infty \frac{\log Φ(n)} {n \log^2 n} = \infty. We also prove variants of this result for generalized Dirichlet polynomials. The main tools are theorems of Paley and Wiener related to quasianalyticity and the Pechersky rearrangement theorem. We use this result to give precise conditions on when a conjecture of Ramachandra is true and when it is false. We prove that whenever Φ(n) is a positive increasing function then \lim_{N \to \infty} \min_{|a_n| \leq Φ(n)} \int_0^H \abs{1+\sum_{n=2}^N a_n n^{it-1}}^2 dt =0, if and only if the above sum is divergent. This has applications on lower bounds for moments of the Riemann zeta-functions in short intervals close to Re(s)=1, and to questions of Universality for zeta-functions on and close to their abscissa of convergence.

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arXiv2012

On the zeta function on the line Re(s) = 1

We show the estimates \inf_T \int_T^{T+δ} |ζ(1+it)|^{-1} dt =e^{-γ}/4 δ^2+ O(δ^4) and \inf_T \int_T^{T+δ} |ζ(1+it)| dt =e^{-γ} π^2/24 δ^2+ O(δ^4) as well as corresponding results for sup-norm, L^p-norm and other zeta-functions such as the Dirichlet L-functions and certain Rankin-Selberg L-functions. This improves on previous work of Balasubramanian and Ramachandra for small values of δand we remark that it implies that the zeta-function is not universal on the line Re(s)=1. We also use recent results of Holowinsky (for Maass wave forms) and Taylor et al. (Sato-Tate for holomorphic cusp forms) to prove lower bounds for the corresponding integral with the Riemann zeta-function replaced with Hecke L-functions and with δ^2 replaced by δ^{11/12+ε} and δ^{8/(3 π)+ε} respectively.

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arXiv2010

Lavrentiev's approximation theorem with nonvanishing polynomials and universality of zeta-functions

We prove a variant of the Lavrentiev's approximation theorem that allows us to approximate a continuous function on a compact set K in C without interior points and with connected complement, with polynomial functions that are nonvanishing on K. We use this result to obtain a version of the Voronin universality theorem for compact sets K, without interior points and with connected complement where it is sufficient that the function is continuous on K and the condition that it is nonvanishing can be removed. This implies a special case of a criterion of Bagchi, which in the general case has been proven to be equivalent to the Riemann hypothesis.

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arXiv2007

On some power sum problems of Montgomery and Turan

We use an estimate for character sums over finite fields of Katz to solve open problems of Montgomery and Turan. Let h=>2 be an integer. We prove that inf_{|z_k| => 1} max_{v=1,...,n^h} |sum_{k=1}^n z_k^v| <= (h-1+o(1)) sqrt n. This gives the right order of magnitude for the quantity and improves on a bound of Erdos-Renyi by a factor of the order sqrt log n.

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arXiv2007

Turan's problem 10 revisited

In this paper we prove that inf_{|z_k| => 1} max_{v=1,...,n^2} |sum_{k=1}^n z_k^v| = sqrt n+O(n^{0.2625+epsilon}). This improves on the bound O(sqrt (n log n)) of Erdos and Renyi. In the special case of $n+1$ being a prime we have previously proved the much sharper result that the quantity lies in the interval [sqrt(n),sqrt(n+1)] The method of proof combines a general lower bound (of Andersson), explicit arithmetical constructions (of Montgomery, Fabrykowski or Andersson), moments (probabilistic methods) and estimates for the difference of consecutive primes (of Baker, Harman and Pintz). We also prove some (conditional and unconditional) related results.

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arXiv2007

Lower bounds in some power sum problems

We study the power sum problem max_{v=1,...,m} | sum_{k=1}^n z_k^v | and by using features of Fejer kernels we give new lower bounds in the case of unimodular complex numbers z_k and m cn^2 for constants c>1.

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arXiv2007

Explicit solutions to certain inf max problems from Turan power sum theory

Let s_v denote the pure power sum \sum_{k=1}^n z_k^v. In a previous paper we proved that \sqrt n <= \inf_{|z_k| => 1} \max_{v=1,...,n^2} |s_v| <= \sqrt{n+1} when n+1 is prime. In this paper we prove that \inf_{|z_k| = 1} \max_{v=1,...,n^2-n} |s_v| = \sqrt{n-1} when n-1 is a prime power, and if 2 <= i <= n-1 and n => 3 is a prime power then \inf_{|z_k| => 1} \max_{v=1,...,n^2-i} |s_v| =\sqrt n. We give explicit constructions of n-tuples (z_1,...,z_n) which we prove are global minima for these problems. These are two of the few times in Turan power sum theory where solutions in the inf max problem can be explicitly calculated.

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arXiv2006

On the solutions to a power sum problem

In a recent paper we proved that if (*)=\inf_{|z_k|=1}\max_{v=1,...,n^2-n} |\sum_{k=1}^n z_k^v|, then (*)=\sqrt{n-1} if n-1 is a prime power. We proved that a construction of Fabrykowski gives minimal systems (z_1,...,z_n) to this problem. The construction depends on the existence of perfect difference sets of order n-1. As an open problem we asked whether all solutions would arise from this construction. In this paper we show that this is true and in fact if there exist no perfect difference set of order n-1 (which by the prime power conjecture is true if n-1 is not a prime power), then we have the strict inequality (*)>\sqrt{n-1}.

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arXiv2006

On the number of plane partitions and non isomorphic subgroup towers of abelian groups

We study the number of $k \times r$ plane partitions, weighted on the sum of the first row. Using Erhart reciprocity, we prove an identity for the generating function. For the special case $k=1$ this result follows from the classical theory of partitions, and for $k=2$ it was proved in Andersson-Bhowmik with another method. We give an explicit formula in terms of Young tableaux, and study the corresponding zeta-function. We give an application on the average orders of towers of abelian groups. In particular we prove that the number of isomorphism classes of ``subgroups of subgroups of ... ($k-1$ times) ... of abelian groups'' of order at most $N$ is asymptotic to $c_k N (\log N)^{k-1}$. This generalises results from Erd{\H o}s-Szekeres and Andersson-Bhowmik where the corresponding result was proved for $k=1$ and $k=2$.

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arXiv2006

On Factorisations of Matrices and Abelian Groups

We establish correspondances between factorisations of finite abelian groups (direct factors, unitary factors, non isomorphic subgroup classes) and factorisations of integer matrices. We then study counting functions associated to these factorisations and find average orders.

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