Extremal bounds for Dirichlet polynomials with random multiplicative coefficients

In the first part of this expository paper, we present and discuss the interplay of Dirichlet polynomials in some classical problems of number theory, notably the Lindel\"of Hypothesis. We review some typical properties of their means and continue with some investigations concerning their supremum properties. Their random counterpart is considered in the last part of the paper, where a analysis of their supremum properties, based on methods of stochastic processes, is develop...

We study the supremum of some random Dirichlet polynomials with independent coefficients and obtain sharp upper and lower bounds for supremum expectation thus extending the results from our previous work (see http://arXiv.org/abs/math/0703691). Our approach in proving these results is entirely based on methods of stochastic processes, in particular the metric entropy method.

We determine the order of magnitude of $\mathbb{E}|\sum_{n \leq x} f(n)|^{2q}$ up to factors of size $e^{O(q^2)}$, where $f(n)$ is a Steinhaus or Rademacher random multiplicative function, for all real $1 \leq q \leq \frac{c\log x}{\log\log x}$. In the Steinhaus case, we show that $\mathbb{E}|\sum_{n \leq x} f(n)|^{2q} = e^{O(q^2)} x^q (\frac{\log x}{q\log(2q)})^{(q-1)^2}$ on this whole range. In the Rademacher case, we find a transition in the behaviour of the moments when...

We determine the order of magnitude of $\mathbb{E}|\sum_{n \leq x} f(n)|^{2q}$, where $f(n)$ is a Steinhaus or Rademacher random multiplicative function, and $0 \leq q \leq 1$. In the Steinhaus case, this is equivalent to determining the order of $\lim_{T \rightarrow \infty} \frac{1}{T} \int_{0}^{T} |\sum_{n \leq x} n^{-it}|^{2q} dt$. In particular, we find that $\mathbb{E}|\sum_{n \leq x} f(n)| \asymp \sqrt{x}/(\log\log x)^{1/4}$. This proves a conjecture of Helson that on...

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 $q>0$.

We study the supremum of some random Dirichlet polynomials and obtain sharp upper and lower bounds for supremum expectation that extend the optimal estimate of Hal\'asz-Queff\'elec and enable to cunstruct random polynomials with unusually small maxima. Our approach in proving these results is entirely based on methods of stochastic processes, in particular the metric entropy method.

Let $T>0$ and consider the random Dirichlet polynomial $S_T(t)=Re\, \sum_{n\leq T} X_n n^{-1/2-it}$, where $(X_n)_{n}$ are i.i.d. Gaussian random variables with mean $0$ and variance $1$. We prove that the expected number of roots of $S_T(t)$ in the dyadic interval $[T,2T]$, say $\mathbb{E} N(T)$, is approximately $2/\sqrt{3}$ times the number of zeros of the Riemann $\zeta$ function in the critical strip up to height $T$. Moreover, we also compute the expected number of zero...

The well-known result states that the square-free counting function up to $N$ is $N/\zeta(2)+O(N^{1/2})$. This corresponds to the identity polynomial $\text{Id}(x)$. It is expected that the error term in question is $O_\varepsilon(N^{\frac{1}{4}+\varepsilon})$ for arbitrarily small $\varepsilon>0$. Usually, it is more difficult to obtain a similar order of error term for a higher degree polynomial $f(x)$ in place of $\text{Id}(x)$. Under the Riemann hypothesis, we show that t...

In this paper we study the asymptotic behavior of the maximum magnitude of a complex random polynomial with i.i.d. uniformly distributed random roots on the unit circle. More specifically, let $\{n_k\}_{k=1}^{\infty}$ be an infinite sequence of positive integers and let $\{z_{k}\}_{k=1}^{\infty}$ be a sequence of i.i.d. uniform distributed random variables on the unit circle. The above pair of sequences determine a sequence of random polynomials $P_{N}(z) = \prod_{k=1}^{N}{(z...

We improve on previous upper bounds for the $q$th norm of the partial sums of the Riemann zeta function on the half line when $0<q\leqslant 1$. In particular, we show that the 1-norm is bounded above by $(\log N)^{1/4}(\log\log N)^{1/4}$.