Dirichlet polynomial

Let $\sigma + i t$ be a complex variable. A finite sum

\begin{equation*} S _ { M } ( s ) = \sum _ { m \in M } a _ { m } e ^ { - \lambda_{m} s }, \end{equation*}

where $M$ is a finite set of natural numbers, is called the Dirichlet polynomial with coefficients $a _ { m }$ (complex numbers) and exponents $\lambda _ { m }$ ($\{ \lambda _ { m } \}$ is an increasing sequence of positive real numbers). In particular, Dirichlet polynomials are partial sums of corresponding Dirichlet series.

Dirichlet polynomials are extensively used and studied in analytic and multiplicative number theory (cf. also Analytic number theory). Most zeta-functions (cf. also Zeta-function) and $L$-functions (cf. also Dirichlet $L$-function), as well as their powers, can be approximated by Dirichlet polynomials, mostly with $\lambda _ { m } = \operatorname { log } m$. For example, uniformly for $\sigma \geq \sigma _ { 0 } > 0$, $| t | \leq \pi x$, the equality

\begin{equation*} \zeta ( s ) = \sum _ { m \leq x } m ^ { - s } + \frac { x ^ { 1 - s } } { s - 1 } + O ( x ^ { - \sigma } ) \end{equation*}

is valid for the Riemann zeta-function [a6]. Dirichlet polynomials also occur in approximate functional equations of zeta-functions [a2], [a6], and have a great influence on their analytic properties. A sufficient condition [a6] for the Riemann hypothesis (cf. Riemann hypotheses) is that the Dirichlet polynomial $\sum _ { { m } = 1 } ^ { { n } } m ^ { - s }$ should have no zeros in $\sigma > 1$.

There exist inversion formulas for Dirichlet series (see, for example, [a2]), which give an integral expression of the Dirichlet polynomial $p _ { n } ( s ) = \sum _ { m = 1 } ^ { n } a _ { m } m ^ { - s }$ by a sum of corresponding Dirichlet series.

In applications, mean-value theorems for Dirichlet polynomials are very useful. The Montgomery–Vaughan theorem [a5] is the best of them, and has, for $p _ { n } ( s )$, the form

\begin{equation*} \int _ { 0 } ^ { 1 } | p _ { n } ( i t ) | ^ { 2 } d t = \sum _ { m = 1 } ^ { n } | a _ { m } | ^ { 2 } ( T + O ( m ) ). \end{equation*}

Transformation formulas for special Dirichlet polynomials were obtained by M. Jutila [a3].

Dirichlet polynomials have a limit distribution in the sense of weak convergence of probability measures. For example, let $G$ be a region on the complex plane, let $H ( G )$ denote the space of analytic functions on $G$ equipped with the topology of uniform convergence on compacta, let $\mathcal{B} ( H ( G ) )$ stand for the class of Borel sets of $H ( G )$ (cf. also Borel set), and let $\operatorname{meas} \, \{ A \}$ be the Lebesgue measure of the set $A$. Then [a4] there exists a probability measure $P$ on $( H ( G ) , \mathcal{B} ( H ( G ) ) )$ such that the measure

\begin{equation*} \frac { 1 } { T } \text { meas } \{ \tau \in [ 0 , T ] : p _ { n } ( s + i \tau ) \in A \}, \end{equation*}

$A \in \mathcal{B} ( H ( G ) )$, converges weakly to $P$ as $T \rightarrow \infty$.

Dirichlet polynomials $S _ { M } ( i t )$ (with arbitrary real numbers $\lambda _ { m }$) play an important role in the theory of almost-periodic functions (cf. also Almost-periodic function) [a1].

References

[a1]	A. Besicovitch, "Almost periodic functions" , Cambridge Univ. Press (1932)
[a2]	A. Ivič, "The Riemann zeta-function" , Wiley–Interscience (1985)
[a3]	M. Jutila, "Transformation formulae for Dirichlet polynomials" J. Number Th. , 18 : 2 (1984) pp. 135–156
[a4]	A. Laurinčikas, "Limit theorems for the Riemann zeta-function" , Kluwer Acad. Publ. (1996)
[a5]	H.L. Montgomery, R.C. Vaughan, "Hilbert's inequality" J. London Math. Soc. , 8 : 2 (1974) pp. 73–82
[a6]	E.C. Titchmarsh, "The theory of the Riemann zeta-function" , Clarendon Press (1986) (Edition: Second)