View source for Dirichlet polynomial

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d1202001.png" /> be a complex variable. A finite sum

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d1202003.png" /> is a finite set of natural numbers, is called the Dirichlet polynomial with coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d1202004.png" /> (complex numbers) and exponents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d1202005.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d1202006.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d1202007.png" />-functions (cf. also [[Dirichlet-L-function|Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d1202008.png" />-function]]), as well as their powers, can be approximated by Dirichlet polynomials, mostly with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d1202009.png" />. For example, uniformly for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d12020010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d12020011.png" />, the equality

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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d12020013.png" /> should have no zeros in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d12020014.png" />.

There exist inversion formulas for Dirichlet series (see, for example, [[#References|[a2]]]), which give an integral expression of the Dirichlet polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d12020015.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d12020016.png" />, the form

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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d12020018.png" /> be a region on the complex plane, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d12020019.png" /> denote the space of analytic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d12020020.png" /> equipped with the topology of [[Uniform convergence|uniform convergence]] on compacta, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d12020021.png" /> stand for the class of Borel sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d12020022.png" /> (cf. also [[Borel set|Borel set]]), and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d12020023.png" /> be the [[Lebesgue measure|Lebesgue measure]] of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d12020024.png" />. Then [[#References|[a4]]] there exists a [[Probability measure|probability measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d12020025.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d12020026.png" /> such that the measure

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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d12020028.png" />, converges weakly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d12020029.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d12020030.png" />.

Dirichlet polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d12020031.png" /> (with arbitrary real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120200/d12020032.png" />) 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>