By S. Mandelbrojt
It isn't really our purpose to offer a treatise on Dirichlet sequence. This a part of harmonic research is so big, so wealthy in guides and in 'theorems' that it sounds as if to us impossible and, to our brain, void of curiosity to gather something yet a constrained (but quite entire) department of the idea. we've not attempted to provide an account of the extremely important result of G. P6lya which hyperlink his proposal of utmost density to the analytic continuation of the sequence, nor the researches to which the names of A. Ostrowski and V. Bernstein are in detail hooked up. the superb publication of the latter, which used to be released within the assortment Borel greater than thirty years in the past, offers an account of them with the entire readability you can actually want for. however, a few scattered effects proved via those authors have chanced on their position one of the correct effects, in part via their statements, partially as a operating device. now we have followed a extra own perspective, in explaining the tools and the foundations (as the identify of the e-book exhibits) that originate in our learn paintings and supply a set of effects which we boost the following; now we have additionally integrated others, as a result of present-day authors, which permit us to shape a coherent whole.
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Additional resources for Dirichlet Series: Principles and Methods
Lp=m+A n• For every p~ 1 and sufficiently large al we can write: Ip = lim ~ J: lJ'(al + it) eUp(fJ 1 +tt> dt. Lpl-at 2 , O:::;t:::;1T, and the segments SJ,p, which are the parts of the segments -log S1 (s E -log S1=e- s E S1) contained between this arc and the line a=O (these segments are evidently parallel to the real axis t=O), where the ordinates of these segments belong to ]0,1T]. Lp 1 - all :::; a :::; 0, o< t, :::; 1T with
However, we shall see that these restrictions can be removed. So let us consider the series (A, A)=2:an e- llnS with Ua
A and let us set for every a Ms = max If(a It-tol .. a + it)l. The quantity . log log M,la) I1m sup = Ps -a (1-+-00 is called the order off in the strip S. We can now prove the following theorem, which plays an important role. Suppose that the step of A is positive and that the function f represented by (A, A) is entire. Then the order off in every strip S=S(to,1Ta) with a> IY is equal to the (R)-order off(in the plane). Let P be the (R)-order of f and let Ps be its order in a strip S(to,1Ta) with IY < a.