An estimate for the sum of a Dirichlet series in terms of the minimum of its modulus on a vertical line segment (Q2880051)

Let \(f\) be a transcendental entire function. Set NEWLINE\[NEWLINE M(r, f)=\max_{|z|=r}|f(z)|\quad\text{and}\quad m(r, f)=\min_{|z|=r}|f(z)|. NEWLINE\]NEWLINE We know that the behaviour of a transcendental entire in terms of the minimum of its modulus is a topical problem in function theory. When we consider the asymptotic values of an entire function NEWLINE\[NEWLINE f(z)=\sum_{n=1}^{\infty}a_{n}z^{p_{n}}, \quad z=x+i y, NEWLINE\]NEWLINE where \(p_{n}\) is an increasing sequence of positive numbers, how can we find conditions on the sequence \(\{p_{n}\}\) ensuring that for any curve \(\gamma\) going to infinity there exists a sequence \(\{\xi_{n}\}\), \(\xi_{n}\in \gamma\), such that NEWLINE\[NEWLINE \ln M(|\xi_{n}, f|)=(1+o(1))\ln|f(\xi_{n})|,\quad \xi_{n}\rightarrow \infty. NEWLINE\]NEWLINE This problem is called the Pólya problem in the literature.NEWLINENEWLINEIn this paper, the authors mainly deal with this problem, and obtain results on the behaviour of the sum of an entire Dirichlet series in terms of the minimum of its modulus on a system of vertical line segments.