On the distribution of Dirichlet sums
Assuming \(| t_ r|\leq T\), \(| t_ r-t_ s|\geq 1\) and \(N<T<N^ 2\), \(R,N\in\mathbb{N}\), \(a_ n\in\mathbb{C}\) the author defines functions \(\alpha(N)\), \(\beta(N)\) such that \[ \sum_{r=1}^ R \Biggl| \sum_{n=1}^ N a_ n n^{it_ r} \Biggr|^ 2\leq (\alpha(N)\cdot N+ \beta(N)\cdot R)\cdot N\cdot \max_{n\leq N} | a_ n|^ 2. \tag{1} \] Then he shows that for each \(0<\rho<1\), there is an \(\varepsilon(\rho)>0\) such that (1) may only be valid if either \(\alpha(N)> \exp(\log N)^ \rho\) or \(\beta(N)> N^{\varepsilon(\rho)}\). At least in case of \(\chi_ r=1\) this statement disproves some well- known conjectures about the distribution of Dirichlet sums, esp. \[ \sum_{r=1}^ R \Biggl| \sum_{n=1}^ N a_ n n^{-{1\over 2}+it_ r} \chi_ r(n)\Biggr|^ 2\leq (N+RD^ \varepsilon)\cdot \max_{n\leq N} | a_ n|^ 2, \] where \(\chi_ r\) is a primitive Dirichlet character with a modulus \(q_ r\leq Q\), \(r=1,\dots,R\) and \(D:= {{Q^ 2T} \over {q_ 0}}, q_ 0\mid q_ r\). So the natural reformulation is the conjecture \[ \sum_{r=1}^ R \Biggl| \sum_{n=1}^ N a_ n n^{it_ r} \Biggr|^ 2\leq N^{1+\varepsilon} (N+R)\cdot \max_{n\leq N} | a_ n|^ 2, \] letting both \(\alpha(N), \beta(N)= N^ \varepsilon\). Rather unexpected this inequality has implications on the Hausdorff dimension of Besicovitch sets in higher dimensions.
distribution of Dirichlet character sums
scientific article; zbMATH DE number 528685