Extremal bounds for Dirichlet polynomials with random multiplicative coefficients. (English) Zbl 1540.11106
A Steinhaus random multiplicative function \(f\) is defined as follows: Over the primes \(p\), \((f(p))_p\) is a sequence of i.i.d. Steinhaus random variables, that is, \(f(2)\) is uniformly distributed over the unitary complex circle. The definition is extended over the positive integers \(n\) by following the multiplicative rule, that is, \(f\) is defined to be completely multiplicative.
In the paper under review, the authors investigate the supremum of a Dirichlet polynomial of the form \[ D_N(t):=\frac{1}{\sqrt{N}}\sum_{n\leq N}f(n)n^{it}. \] They prove with probability \(1-o(1)\) an upper and a lower bound for the quantity \[ \mathcal{S}(N,C):=\sup_{|t|\leq N^{C(N)}}|D_N(t)|, \] where \(C(N)\) is assumed to satisfy a growth condition. These upper and lower bound are interesting to compare with the case where \((f(n))_n\) are i.i.d, where we see a much smaller related quantity to \(\mathcal{S}(N,C)\).
In the paper under review, the authors investigate the supremum of a Dirichlet polynomial of the form \[ D_N(t):=\frac{1}{\sqrt{N}}\sum_{n\leq N}f(n)n^{it}. \] They prove with probability \(1-o(1)\) an upper and a lower bound for the quantity \[ \mathcal{S}(N,C):=\sup_{|t|\leq N^{C(N)}}|D_N(t)|, \] where \(C(N)\) is assumed to satisfy a growth condition. These upper and lower bound are interesting to compare with the case where \((f(n))_n\) are i.i.d, where we see a much smaller related quantity to \(\mathcal{S}(N,C)\).
Reviewer: Marco Aymone (Belo Horizonte)
MSC:
| 11K65 | Arithmetic functions in probabilistic number theory |
| 11N56 | Rate of growth of arithmetic functions |
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