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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 series]].

Dirichlet polynomials are extensively used and studied in analytic and multiplicative number theory (cf. also [[Analytic number theory|Analytic number theory]]). Most zeta-functions (cf. also [[Zeta-function|Zeta-function]]) and $L$-functions (cf. also [[Dirichlet L-function|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 $\sigma \geq \sigma _ { 0 } &gt; 0$, $| t | \leq \pi x$, 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|Riemann zeta-function]] [[#References|[a6]]]. Dirichlet polynomials also occur in approximate functional equations of zeta-functions [[#References|[a2]]], [[#References|[a6]]], and have a great influence on their analytic properties. A sufficient condition [[#References|[a6]]] for the Riemann hypothesis (cf. [[Riemann hypotheses|Riemann hypotheses]]) is that the Dirichlet polynomial $\sum _ {  { m } = 1 } ^ {  { n } } m ^ { - s }$ should have no zeros in $\sigma &gt; 1$.

There exist inversion formulas for Dirichlet series (see, for example, [[#References|[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 [[#References|[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 [[#References|[a3]]].

Dirichlet polynomials have a limit distribution in the sense of [[Weak convergence of probability measures|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|uniform convergence]] on compacta, let $\mathcal{B} ( H ( G ) )$ stand for the class of Borel sets of $H ( G )$ (cf. also [[Borel set|Borel set]]), and let $\operatorname{meas} \, \{ A \}$ be the [[Lebesgue measure|Lebesgue measure]] of the set $A$. Then [[#References|[a4]]] there exists a [[Probability measure|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|Almost-periodic function]]) [[#References|[a1]]].

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