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<title>L-function</title>

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<span id="openzim-page-title" class="mw-page-title-main"><i>L</i>-function</span>

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<p>In mathematics, an <b><i>L</i>-function</b> is a <a href="Meromorphic" class="mw-redirect" title="Meromorphic">meromorphic</a> <a href="Function_(mathematics)" title="Function (mathematics)">function</a> on the <a href="Complex_plane" title="Complex plane">complex plane</a>, associated to one out of several categories of <a href="Mathematical_object" title="Mathematical object">mathematical objects</a>. An <b><i>L</i>-series</b> is a <a href="Dirichlet_series" title="Dirichlet series">Dirichlet series</a>, usually <a href="Convergence_(mathematics)" class="mw-redirect" title="Convergence (mathematics)">convergent</a> on a <a href="Half-plane" class="mw-redirect" title="Half-plane">half-plane</a>, that may give rise to an <i>L</i>-function via <a href="Analytic_continuation" title="Analytic continuation">analytic continuation</a>. The <a href="Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> is an example of an <i>L</i>-function, and some important conjectures involving <i>L</i>-functions are the <a href="Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a> and its <a href="Generalized_Riemann_hypothesis" title="Generalized Riemann hypothesis">generalizations</a>.

</p><p>The theory of <i>L</i>-functions has become a very substantial, and still largely <a href="Conjectural" class="mw-redirect" title="Conjectural">conjectural</a>, part of contemporary <a href="Analytic_number_theory" title="Analytic number theory">analytic number theory</a>. In it, broad generalisations of the Riemann zeta function and the <a href="Dirichlet_L-function" title="Dirichlet L-function"><i>L</i>-series</a> for a <a href="Dirichlet_character" title="Dirichlet character">Dirichlet character</a> are constructed, and their general properties, in most cases still out of reach of proof, are set out in a systematic way. Because of the <a href="Euler_product_formula" class="mw-redirect" title="Euler product formula">Euler product formula</a> there is a deep connection between <i>L</i>-functions and the theory of <a href="Prime_number" title="Prime number">prime numbers</a>.

</p><p>The mathematical field that studies <i>L</i>-functions is sometimes called <b>analytic theory of <i>L</i>-functions</b>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Construction">Construction</h2></div>

<p>We distinguish at the outset between the <b><i>L</i>-series</b>, an <a href="Infinite_set" title="Infinite set">infinite</a> series representation (for example the <a href="Dirichlet_series" title="Dirichlet series">Dirichlet series</a> for the <a href="Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>), and the <b><i>L</i>-function</b>, the function in the complex plane that is its <a href="Analytic_continuation" title="Analytic continuation">analytic continuation</a>. The general constructions start with an <i>L</i>-series, defined first as a <a href="Dirichlet_series" title="Dirichlet series">Dirichlet series</a>, and then by an expansion as an <a href="Euler_product" title="Euler product">Euler product</a> indexed by prime numbers. Estimates are required to prove that this converges in some right half-plane of the <a href="Complex_number" title="Complex number">complex numbers</a>. Then one asks whether the function so defined can be analytically continued to the rest of the complex plane (perhaps with some <a href="Pole_(complex_analysis)" class="mw-redirect" title="Pole (complex analysis)">poles</a>).

</p><p>It is this (conjectural) <a href="Meromorphic" class="mw-redirect" title="Meromorphic">meromorphic</a> continuation to the complex plane which is called an <i>L</i>-function. In the classical cases, already, one knows that useful information is contained in the values and behaviour of the <i>L</i>-function at points where the series representation does not converge. The general term <i>L</i>-function here includes many known types of zeta functions. The <a href="Selberg_class" title="Selberg class">Selberg class</a> is an attempt to capture the core properties of <i>L</i>-functions in a set of axioms, thus encouraging the study of the properties of the class rather than of individual functions.

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<div class="mw-heading mw-heading2"><h2 id="Conjectural_information">Conjectural information</h2></div>

<p>One can list characteristics of known examples of <i>L</i>-functions that one would wish to see generalized:

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<ul><li>location of zeros and poles;</li>

<li><a href="Functional_equation_(L-function)" title="Functional equation (L-function)">functional equation</a>, with respect to some vertical line Re(<i>s</i>) = constant;</li>

<li>interesting values at integers related to quantities from <a href="Algebraic_K-theory" title="Algebraic K-theory">algebraic <i>K</i>-theory</a>.</li></ul>

<p>Detailed work has produced a large body of plausible conjectures, for example about the exact type of functional equation that should apply. Since the Riemann zeta function connects through its values at positive even integers (and negative odd integers) to the <a href="Bernoulli_numbers" class="mw-redirect" title="Bernoulli numbers">Bernoulli numbers</a>, one looks for an appropriate generalisation of that phenomenon. In that case results have been obtained for <a href="P-adic_L-function" title="P-adic L-function"><i>p</i>-adic <i>L</i>-functions</a>, which describe certain <a href="Galois_module" class="mw-redirect" title="Galois module">Galois modules</a>.

</p><p>The statistics of the zero distributions are of interest because of their connection to problems like the generalized Riemann hypothesis, distribution of prime numbers, etc. The connections with <a href="Random_matrix" title="Random matrix">random matrix</a> theory and <a href="Quantum_chaos" title="Quantum chaos">quantum chaos</a> are also of interest. The fractal structure of the distributions has been studied using <a href="Rescaled_range" title="Rescaled range">rescaled range</a> analysis.<sup id="cite_ref-Shanker_2-0" class="reference"><a href="#cite_note-Shanker-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The <a href="Self-similarity" title="Self-similarity">self-similarity</a> of the zero distribution is quite remarkable, and is characterized by a large <a href="Fractal_dimension" title="Fractal dimension">fractal dimension</a> of 1.9. This rather large fractal dimension is found over zeros covering at least fifteen orders of magnitude for the <a href="Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>, and also for the zeros of other <i>L</i>-functions of different orders and conductors.

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<div class="mw-heading mw-heading2"><h2 id="Birch_and_Swinnerton-Dyer_conjecture">Birch and Swinnerton-Dyer conjecture</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Birch_and_Swinnerton-Dyer_conjecture" title="Birch and Swinnerton-Dyer conjecture">Birch and Swinnerton-Dyer conjecture</a></div>

<p>One of the influential examples, both for the history of the more general <i>L</i>-functions and as a still-open research problem, is the conjecture developed by <a href="Bryan_Birch" class="mw-redirect" title="Bryan Birch">Bryan Birch</a> and <a href="Peter_Swinnerton-Dyer" title="Peter Swinnerton-Dyer">Peter Swinnerton-Dyer</a> in the early part of the 1960s. It applies to an <a href="Elliptic_curve" title="Elliptic curve">elliptic curve</a> <i>E</i>, and the problem it attempts to solve is the prediction of the rank of the elliptic curve over the rational numbers (or another <a href="Global_field" title="Global field">global field</a>): i.e. the number of free generators of its group of rational points. Much previous work in the area began to be unified around a better knowledge of <i>L</i>-functions. This was something like a paradigm example of the nascent theory of <i>L</i>-functions.

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<div class="mw-heading mw-heading2"><h2 id="Rise_of_the_general_theory">Rise of the general theory</h2></div>

<p>This development preceded the <a href="Langlands_program" title="Langlands program">Langlands program</a> by a few years, and can be regarded as complementary to it: Langlands' work relates largely to <a href="Artin_L-function" title="Artin L-function">Artin <i>L</i>-functions</a>, which, like <a href="Hecke_L-function_(disambiguation)" class="mw-redirect mw-disambig" title="Hecke L-function (disambiguation)">Hecke <i>L</i>-functions</a>, were defined several decades earlier, and to <i>L</i>-functions attached to general <a href="Automorphic_representation" class="mw-redirect" title="Automorphic representation">automorphic representations</a>.

</p><p>Gradually it became clearer in what sense the construction of <a href="Hasse%E2%80%93Weil_zeta_function" title="Hasse–Weil zeta function">Hasse–Weil zeta functions</a> might be made to work to provide valid <i>L</i>-functions, in the analytic sense: there should be some input from analysis, which meant <i>automorphic</i> analysis. The general case now unifies at a conceptual level a number of different research programs.

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<ul><li><a href="Generalized_Riemann_hypothesis" title="Generalized Riemann hypothesis">Generalized Riemann hypothesis</a></li>

<li><a href="Dirichlet_L-function" title="Dirichlet L-function">Dirichlet <i>L</i>-function</a></li>

<li><a href="Automorphic_L-function" title="Automorphic L-function">Automorphic <i>L</i>-function</a></li>

<li><a href="Modularity_theorem" title="Modularity theorem">Modularity theorem</a></li>

<li><a href="Artin_conjecture_(L-functions)" class="mw-redirect" title="Artin conjecture (L-functions)">Artin conjecture</a></li>

<li><a href="Special_values_of_L-functions" title="Special values of L-functions">Special values of <i>L</i>-functions</a></li>

<li><a href="Explicit_formulae_for_L-functions" title="Explicit formulae for L-functions">Explicit formulae for L-functions</a></li>

<li><a href="Shimizu_L-function" title="Shimizu L-function">Shimizu <i>L</i>-function</a></li></ul>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFSteuding2005" class="citation web cs1">Steuding, Jörn (June 2005). <a rel="nofollow" class="external text" href="https://www.scribd.com/document/230217684/An-Introduction-to-the-Theory-of-L-Functions">"An Introduction to the Theory of <i>L</i>-functions"</a>. <i>Preprint</i>.</cite></span>

</li>

<li id="cite_note-Shanker-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Shanker_2-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFO._Shanker2006" class="citation journal cs1">O. Shanker (2006). "Random matrices, generalized zeta functions and self-similarity of zero distributions". <i>J. Phys. A: Math. Gen</i>. <b>39</b> (45): <span class="nowrap">13983–</span>13997. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006JPhA...3913983S">2006JPhA...3913983S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0305-4470%2F39%2F45%2F008">10.1088/0305-4470/39/45/008</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:54958644">54958644</a>.</cite></span>

</li>

</ol></div></div>

<ul><li><cite id="CITEREFNeukirch1999" class="citation book cs1"><a href="J%C3%BCrgen_Neukirch" title="Jürgen Neukirch">Neukirch, Jürgen</a> (1999). <i>Algebraische Zahlentheorie</i>. <span title="German-language text"><i lang="de">Grundlehren der mathematischen Wissenschaften</i></span>. Vol. 322. Berlin: <a href="Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-65399-8</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1697859">1697859</a>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0956.11021">0956.11021</a>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.lmfdb.org">"LMFDB, the database of L-functions, modular forms, and related objects"</a>.</cite></li>

<li><cite id="CITEREFLavrik2001" class="citation cs1">Lavrik, A.F. (2001) [1994]. <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=L-function">"L-function"</a>. <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>. <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>.</cite></li></ul>

<dl><dt>Articles about a breakthrough third degree transcendental L-function</dt>

<dd>

<ul><li><cite class="citation news cs1"><a rel="nofollow" class="external text" href="http://www.physorg.com/news124636003.html">"Glimpses of a new (mathematical) world"</a>. Mathematics. <i>Physorg.com</i>. American Institute of Mathematics. March 13, 2008.</cite></li>

<li><cite id="CITEREFRehmeyer2008" class="citation news cs1">Rehmeyer, Julie (April 2, 2008). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120216201232/http://www.sciencenews.org/view/generic/id/9542/title/Math_Trek__Creeping_Up_on_Riemann">"Creeping Up on Riemann"</a>. <i>Science News</i>. Archived from <a rel="nofollow" class="external text" href="http://www.sciencenews.org/view/generic/id/9542/title/Math_Trek__Creeping_Up_on_Riemann">the original</a> on February 16, 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">August 5,</span> 2008</span>.</cite></li>

<li><cite class="citation news cs1"><a rel="nofollow" class="external text" href="http://www.physorg.com/news137248087.html">"Hunting the elusive L-function"</a>. Mathematics. <i>Physorg.com</i>. University of Bristol. August 6, 2008.</cite></li></ul></dd></dl>

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</style><div id="L-functions_in_number_theory183" style="font-size:114%;margin:0 4em"> in <a href="Number_theory" title="Number theory">number theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Analytic examples</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a></li>

<li><a href="Dirichlet_L-function" title="Dirichlet L-function">Dirichlet <i>L</i>-functions</a></li>

<li><a href="L-function_with_Gr%C3%B6ssencharakter" class="mw-redirect" title="L-function with Grössencharakter"><i>L</i>-functions of Hecke characters</a></li>

<li><a href="Automorphic_L-function" title="Automorphic L-function">Automorphic <i>L</i>-functions</a></li>

<li><a href="Selberg_class" title="Selberg class">Selberg class</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algebraic examples</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dedekind_zeta_function" title="Dedekind zeta function">Dedekind zeta functions</a></li>

<li><a href="Artin_L-function" title="Artin L-function">Artin <i>L</i>-functions</a></li>

<li><a href="Hasse%E2%80%93Weil_zeta_function" title="Hasse–Weil zeta function">Hasse–Weil <i>L</i>-functions</a></li>

<li><a href="Motivic_L-function" title="Motivic L-function">Motivic <i>L</i>-functions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Class_number_formula" title="Class number formula">Analytic class number formula</a></li>

<li><a href="Riemann%E2%80%93von_Mangoldt_formula" title="Riemann–von Mangoldt formula">Riemann–von Mangoldt formula</a></li>

<li><a href="Weil_conjectures" title="Weil conjectures">Weil conjectures</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Analytic conjectures</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a></li>

<li><a href="Generalized_Riemann_hypothesis" title="Generalized Riemann hypothesis">Generalized Riemann hypothesis</a></li>

<li><a href="Lindel%C3%B6f_hypothesis" title="Lindelöf hypothesis">Lindelöf hypothesis</a></li>

<li><a href="Ramanujan%E2%80%93Petersson_conjecture" title="Ramanujan–Petersson conjecture">Ramanujan–Petersson conjecture</a></li>

<li><a href="Artin_conjecture_(L-functions)" class="mw-redirect" title="Artin conjecture (L-functions)">Artin conjecture</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algebraic conjectures</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Birch_and_Swinnerton-Dyer_conjecture" title="Birch and Swinnerton-Dyer conjecture">Birch and Swinnerton-Dyer conjecture</a></li>

<li><a href="Special_values_of_L-functions" title="Special values of L-functions">Deligne's conjecture</a></li>

<li><a href="Beilinson_conjectures" class="mw-redirect" title="Beilinson conjectures">Beilinson conjectures</a></li>

<li><a href="Bloch%E2%80%93Kato_conjecture_(L-functions)" class="mw-redirect" title="Bloch–Kato conjecture (L-functions)">Bloch–Kato conjecture</a></li>

<li><a href="Langlands_program" title="Langlands program">Langlands conjecture</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="P-adic_L-function" title="P-adic L-function"><i>p</i>-adic <i>L</i>-functions</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Main_conjecture_of_Iwasawa_theory" title="Main conjecture of Iwasawa theory">Main conjecture of Iwasawa theory</a></li>

<li><a href="Selmer_group" title="Selmer group">Selmer group</a></li>

<li><a href="Euler_system" title="Euler system">Euler system</a></li></ul>

</div></td></tr></tbody></table></div>

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