Zeros of \(L(s)+L(2s)+\cdots + L(Ns)\) in the region of absolute convergence (Q2079503)

This paper studies zeros of particular linear combinations of \(L\)-functions in the region of absolute convergence. Although the abstract would suggest that most of the paper focuses on Dirichlet \(L\)-functions, this is not the case. The authors consider \(L\)-functions in a broad class similar to the standard Selberg class. Their primary result is Theorem 1, which states that for one such \(L\)-function \(L(s)\), the linear combination \[ L(s) + L(2s) + \cdots + L(Ns) \] has infinitely many zeros with \(\mathrm{Re}(s) > 1\) if \(N\) is sufficiently large, or if \(N = 2\). If further the coefficients of the \(L\)-function satisfy \(\lvert a(n) \rvert \leq n^{1/4}\), then they show that this linear combination has infinitely many zeros with \(\mathrm{Re}(s) > 1\) for every \(N \geq 2\). More generally, we should expect that a general linear combination of \(L\)-functions won't have an Euler product and will have zeros in the region of absolute convergence. But proving that this actually occurs is nontrivial. To prove this result, the authors prove the following (Theorem 2): If \(f(s)\) is an absolutely convergent Dirichlet series for \(\mathrm{Re}(s) \geq 1\) with uniformly bounded logarithm in this region and \(L(s)\) as before, then \(L(s) + f(s)\) has infinitely many zeros in the half-plane \(\mathrm{Re}(s) > 1\). The majority of the work in the paper is to prove Theorem 2. As in the typical theory of almost periodic functions, finding a single zero in the region of absolute convergence guarantees that there are infinitely many zeros in the region of absolute convergence. Thus it suffices to find a single zero. To do this, the authors adapt an argument due to \textit{E. Saias} and \textit{A. Weingartner} [Acta Arith. 140, No. 4, 335--344 (2009; Zbl 1205.11101)]. Finally, the authors also show that for any \(k \geq 9\), the function \[ \zeta^k(2s) + \zeta^k(3s) \] also has infinitely many zeros in the half-plane \(\mathrm{Re}(s) > 1\), using totally different methods.