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.Comment: v1: 9 pages; v2: Minor changes. Fixed error in last three lines of proof of Theorem 2: v3: New title. Minor change

On questions of Cassels and Drungilas-Dubickas

We answer a question of Drungilas-Dubickas in the affirmative under the assumption of standard conjectures on smooth numbers in polynomial sequences. This gives evidence against the "Dubickas Conjecture", which Ka\v{c}inskait\.e and Laurin\v{c}ikas proved implies universality results for the Hurwitz zeta-function with certain algebraic irrational parameters. Under these standard conjectures we also prove some results that confirms observations of Worley relating to a problem of Cassels on the multiplicative dependence of algebraic numbers shifted by integers.Comment: 10 pages, A version of this manuscript was circulated in a smaller group in July 2011. When looking at the manuscript this year I decided that it might have some interest for a wider circle. The current manuscript is very similar to the manuscript from 5 years ago, although I have rewritten the proofs to hold for algebraic numbers rather than algebraic integer

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

We show the estimates \inf_T \int_T^{T+\delta} |\zeta(1+it)|^{-1} dt =e^{-\gamma}/4 \delta^2+ O(\delta^4) and \inf_T \int_T^{T+\delta} |\zeta(1+it)| dt =e^{-\gamma} \pi^2/24 \delta^2+ O(\delta^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 \delta and 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 \delta^2 replaced by \delta^{11/12+\epsilon} and \delta^{8/(3 \pi)+\epsilon} respectively.Comment: 33 page

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.Comment: 5 page

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.Comment: v1: 20 pages; v2: 22 pages. Misprints/minor errors fixed. Added some material and some references; v3: 21 pages. Minor errors fixed. Changed problem section and added reference to new paper where we solve some of the open problems of v

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}.Comment: 6 pages, v2: Minor changes. Typos fixe

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 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 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.Comment: v1: 8 pages; v2: 7 pages. Minor changes. Exposition improved. To appear in Indagationes Mathematica

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.Comment: 9 page

Bounded prime gaps in short intervals

We generalise Zhang's and Pintz recent results on bounded prime gaps to give a lower bound for the the number of prime pairs bounded by 6*10^7 in the short interval $[x,x+x (\log x)^{-A}]$. Our result follows only by analysing Zhang's proof of Theorem 1, but we also explain how a sharper variant of Zhang's Theorem 2 would imply the same result for shorter intervals.Comment: v2: 6 pages. Fixed minor errors and added reference to polymath projec

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 $k=2$.Comment: 20 pages, 2 figure