View source for Dirichlet L-function

''Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d0328903.png" />-series, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d0328905.png" />-series''

A function of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d0328906.png" /> that is defined for any [[Dirichlet character|Dirichlet character]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d0328907.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d0328908.png" /> by the series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d0328909.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

As functions of a real variable these were introduced by P.G.L. Dirichlet [[#References|[1]]] in 1837 in the context of the proof that the number of primes in an arithmetic progression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289010.png" />, where the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289011.png" /> and the first term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289012.png" /> are relatively prime numbers, is infinite. They are a natural generalization of the Riemann [[Zeta-function|zeta-function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289013.png" /> to an arithmetic progression and are a powerful tool in analytic number theory [[#References|[2]]]–[[#References|[4]]].

The series (1), known as a [[Dirichlet series]], converges absolutely and uniformly in any bounded domain in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289014.png" />-plane for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289016.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289017.png" /> is a non-principal character, one has

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

Since the sum in the integrand is bounded, this formula gives an analytic continuation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289019.png" /> to a regular function in the half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289020.png" />.

For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289021.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289022.png" /> it is possible to represent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289023.png" /> as an Euler product over prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289024.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

Hence, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289026.png" /> is a [[principal character]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289027.png" />, one has, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289028.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289029.png" /></td> </tr></table>

and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289030.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289031.png" /></td> </tr></table>

For this reason the properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289032.png" /> in the entire complex plane are mainly determined by the properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289033.png" />. In particular, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289034.png" /> is regular for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289035.png" />, except for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289036.png" /> where it has a simple pole with residue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289037.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289038.png" /> is Euler's function. If, on the other hand, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289039.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289040.png" /> is the primitive character inducing the character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289041.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289042.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289043.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

Thus, it is no essential restriction to consider only Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289044.png" />-functions for primitive characters. This property of Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289045.png" />-functions is important, since many results concerning <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289046.png" /> have a simple form for primitive characters only. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289047.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289048.png" /> is primitive, the analytic continuation to the entire plane and the functional equation for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289049.png" /> are obtained by direct generalization of Riemann's method for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289050.png" />. Putting

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289051.png" /></td> </tr></table>

the result has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289052.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289053.png" /> is the gamma-function, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289056.png" /> is a [[Gauss sum|Gauss sum]], and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289057.png" /> is the complex conjugate character to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289058.png" />. This equation is known as the functional equation of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289060.png" />. It follows from this formula and from formulas (2) and (4) that the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289062.png" /> are entire functions for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289063.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289065.png" /> only at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289067.png" /> and at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289068.png" /> where the product in (4) vanishes; these points are known as the trivial zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289070.png" />. The remaining zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289071.png" /> are said to be the non-trivial zeros. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289073.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289074.png" />. Ch.J. de la Vallée-Poussin showed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289075.png" />, so that all non-trivial zeros of a Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289076.png" />-function lie in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289077.png" />, which is known as the critical strip.

The distribution of the non-trivial zeros, and of the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289079.png" /> in the critical strip in general, is the most important problem in the theory of Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289080.png" />-functions, and is of fundamental importance in number theory.

That each function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289081.png" /> has infinitely many non-trivial zeros, and that the laws governing the distribution of primes in arithmetic progressions directly depend on the distribution of these zeros, is shown by the corresponding analogues of Riemann's formulas. In fact, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289082.png" /> be the number of zeros of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289083.png" /> with a primitive character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289084.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289085.png" /> in the rectangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289087.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289088.png" />. Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289089.png" /></td> </tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289090.png" /> be the [[Mangoldt function|von Mangoldt function]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289092.png" />, and let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289093.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289094.png" /></td> </tr></table>

Then it follows from the orthogonality property of the characters that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289095.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

where the summation is extended over all characters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289096.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289097.png" />. Moreover, for a primitive character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289098.png" /> and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d03289099.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890100.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890101.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890102.png" /> runs through the non-trivial zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890103.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890104.png" /> is the derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890105.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890106.png" />.

Approximate formulas for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890107.png" /> are more useful in practice: For arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890108.png" /> and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890109.png" /> one has

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890110.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890111.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890112.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>

The quantity in (8) is the principal term of the sum in (6).

According to the so-called extended Riemann hypothesis, all non-trivial zeros of a Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890113.png" />-function lie on the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890114.png" />. If this hypothesis is valid, one has, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890115.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890116.png" /></td> </tr></table>

and many other important problems in number theory would have their final solution. However, problems concerning the distribution of the non-trivial zeros of a Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890117.png" />-function are exceptionally difficult, and relatively little is yet (1988) known on the subject. Stronger results were obtained for complex rather than for real characters.

A generalization of the method proposed in 1899 by de la Vallée-Poussin for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890118.png" /> yields a bound on the non-trivial zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890119.png" />: For a complex character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890120.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890121.png" /> there exists an absolute constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890122.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890123.png" /> has no zeros in the domain

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890124.png" /></td> </tr></table>

However, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890125.png" /> is a real non-principal character modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890126.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890127.png" /> may have in this domain at most one simple real (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890128.png" />) zero, known as the exceptional zero of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890130.png" />. The following inequality was deduced for the exceptional zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890131.png" /> from the analytic class number formula for quadratic fields:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890132.png" /></td> </tr></table>

A well-known best (pre 1975) bound for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890133.png" /> was obtained in 1935 by C.L. Siegel: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890134.png" /> there exists a positive number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890135.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890136.png" /></td> </tr></table>

However, this estimate has the major drawback of being ineffective in the sense that the knowledge of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890137.png" /> is insufficient to make an estimate for the numerical constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890138.png" />. This is also the disadvantage of the number-theoretic results based on Siegel's estimate.

From the above bounds for the non-trivial zeros of Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890139.png" />-functions and formulas (6)–(8), the following asymptotic law for the distribution of prime numbers can be derived:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890140.png" /></td> </tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890141.png" /> is an effectively computable constant for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890142.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890143.png" />. Otherwise, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890144.png" /> ineffectively, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890145.png" /> is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890146.png" />.

These results are the best results available in the problem of uniform distribution of prime numbers in arithmetic progressions with increasing difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890147.png" />. A little more is known in the case where the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890148.png" /> is fixed. In such a case the theory of Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890149.png" />-functions for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890150.png" /> resembles in many respects the theory of the Riemann zeta-function [[#References|[5]]], and the most recent bound on the zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890151.png" />, obtained by the [[Vinogradov method|Vinogradov method]] for estimating trigonometric sums, has the form:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890152.png" /></td> </tr></table>

for

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890153.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890154.png" /> is a positive constant depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890155.png" />.

To this bound for the non-trivial zeros of Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890156.png" />-functions modulo a fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890157.png" /> corresponds the best (1977) remainder term in the asymptotic formula for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890158.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890159.png" /></td> </tr></table>

All formulas concerning the asymptotics of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890160.png" /> have analogues for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890161.png" />, viz. for the number of primes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890162.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890163.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890164.png" />), with principal term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890165.png" /> instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890166.png" /> and a residual term which is smaller by a factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890167.png" />.

A major subject in modern studies on the theory of Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890168.png" />-functions is research on the density of the distribution of the non-trivial zeros of such functions. This research is concerned with giving estimates for the quantities

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890169.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890170.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890171.png" /> denotes the number of zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890172.png" /> in the rectangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890173.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890174.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890175.png" /> is a primitive character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890176.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.G.L. Dirichlet,   "Vorlesungen über Zahlentheorie" , Vieweg  (1894)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Davenport,   "Multiplicative number theory" , Springer  (1980)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K. Prachar,   "Primzahlverteilung" , Springer  (1957)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.G. Chudakov,   "Introductions to the theory of Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890177.png" />-functions" , Moscow-Leningrad  (1947)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Walfisz,   "Weylsche Exponentialsummen in der neueren Zahlentheorie" , Deutsch. Verlag Wissenschaft.  (1963)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  H. Montgomery,   "Topics in multiplicative number theory" , Springer  (1971)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.F. Lavrik,   "Development of the method of density of zeros of Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890178.png" />-functions"  ''Math. Notes'' , '''17''' :  5  (1975)  pp. 483–488  ''Mat. Zametki'' , '''17''' :  5  (1975)  pp. 809–817</TD></TR></table>



====Comments====
The effective bound

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890179.png" /></td> </tr></table>

for the exceptional zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890180.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890181.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890182.png" /> is a real non-principal character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890183.png" />, was improved by D. Goldfeld and A. Schinzel [[#References|[a1]]] to

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890184.png" /></td> </tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890185.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890186.png" /></td> </tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890187.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890188.png" /> is an effectively computable constant. Using work of B.H. Gross and D. Zagier [[#References|[a2]]] the result for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890189.png" /> can be improved to

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890190.png" /></td> </tr></table>

for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890191.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890192.png" /> is an effective constant.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Goldfeld,   A. Schinzel,   "On Siegel's zero"  ''Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)'' , '''2'''  (1975)  pp. 571–583</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.H. Gross,   D. Zagier,   "Heegner points and derivatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032890/d032890193.png" />-series"  ''Invent. Math.'' , '''84'''  (1986)  pp. 225–320</TD></TR></table>