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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
69:, is a useful starting point for translations, but translators must revise errors as necessary and confirm that the translation is accurate, rather than simply copy-pasting machine-translated text into the English Knowledge.
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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
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Do not translate text that appears unreliable or low-quality. If possible, verify the text with references provided in the foreign-language article.
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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
332:. Then one asks whether the function so defined can be analytically continued to the rest of the complex plane (perhaps with some
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488:, and the problem it attempts to solve is the prediction of the rank of the elliptic curve over the rational numbers (or another
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of 1.9. This rather large fractal dimension is found over zeros covering at least fifteen orders of magnitude for the
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Content in this edit is translated from the existing German
Knowledge article at ]; see its history for attribution.
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indexed by prime numbers. Estimates are required to prove that this converges in some right half-plane of the
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O. Shanker (2006). "Random matrices, generalized zeta functions and self-similarity of zero distributions".
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analysis. The general case now unifies at a conceptual level a number of different research programs.
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by a few years, and can be regarded as complementary to it: Langlands' work relates largely to
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are also of interest. The fractal structure of the distributions has been studied using
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of the zero distribution is quite remarkable, and is characterized by a large
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One of the influential examples, both for the history of the more general
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to the source of your translation. A model attribution edit summary is
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Articles about a breakthrough third degree transcendental L-function
240:. In it, broad generalisations of the Riemann zeta function and the
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355:-function here includes many known types of zeta functions. The
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Gradually it became clearer in what sense the construction of
232:-functions has become a very substantial, and still largely
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interesting values at integers related to quantities from
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to this template: there are already 1,880 articles in the
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continuation to the complex plane which is called an
794:. American Institute of Mathematics. March 13, 2008.
213:-function, and some important conjectures involving
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375:-functions that one would wish to see generalized:
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a machine-translated version of the German article.
371:One can list characteristics of known examples of
481:in the early part of the 1960s. It applies to an
174:, associated to one out of several categories of
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359:is an attempt to capture the core properties of
312:, the function in the complex plane that is its
528:, were defined several decades earlier, and to
455:-functions of different orders and conductors.
104:accompanying your translation by providing an
49:Click for important translation instructions.
36:expand this article with text translated from
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708:Grundlehren der mathematischen Wissenschaften
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148:can be thought of as the archetype for all
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316:. The general constructions start with an
286:We distinguish at the outset between the
16:Meromorphic function on the complex plane
831:. University of Bristol. August 6, 2008.
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788:"Glimpses of a new (mathematical) world"
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543:might be made to work to provide valid
297:series representation (for example the
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1042:Birch and Swinnerton-Dyer conjecture
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266:The mathematical field that studies
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255:there is a deep connection between
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635:"An Introduction to the Theory of
451:, and also for the zeros of other
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1087:Main conjecture of Iwasawa theory
799:Rehmeyer, Julie (April 2, 2008).
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604:Explicit formulae for L-functions
324:, and then by an expansion as an
825:"Hunting the elusive L-function"
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532:-functions attached to general
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270:-functions is sometimes called
1021:Ramanujan–Petersson conjecture
1011:Generalized Riemann hypothesis
907:-functions of Hecke characters
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562:Generalized Riemann hypothesis
508:This development preceded the
114:You may also add the template
1:
980:Analytic class number formula
619:
259:-functions and the theory of
985:Riemann–von Mangoldt formula
633:Steuding, Jörn (June 2005).
379:location of zeros and poles;
320:-series, defined first as a
116:{{Translated|de|L-Funktion}}
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768:Encyclopedia of Mathematics
678:10.1088/0305-4470/39/45/008
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534:automorphic representations
197:, that may give rise to an
86:will aid in categorization.
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541:Hasse–Weil zeta functions
420:, which describe certain
339:It is this (conjectural)
38:the corresponding article
801:"Creeping Up on Riemann"
937:Dedekind zeta functions
367:Conjectural information
236:, part of contemporary
125:For more guidance, see
761:Lavrik, A.F. (2001) .
707:
238:analytic number theory
153:
1057:Bloch–Kato conjecture
1052:Beilinson conjectures
1035:Algebraic conjectures
890:Riemann zeta function
658:J. Phys. A: Math. Gen
479:Peter Swinnerton-Dyer
449:Riemann zeta function
314:analytic continuation
303:Riemann zeta function
253:Euler product formula
207:Riemann zeta function
203:analytic continuation
146:Riemann zeta function
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127:Knowledge:Translation
98:copyright attribution
1118:Zeta and L-functions
1062:Langlands conjecture
1047:Deligne's conjecture
999:Analytic conjectures
811:on February 16, 2012
209:is an example of an
176:mathematical objects
1016:Lindelöf hypothesis
670:2006JPhA...3913983S
664:(45): 13983–13997.
383:functional equation
272:analytic theory of
249:Dirichlet character
217:-functions are the
156:In mathematics, an
1006:Riemann hypothesis
930:Algebraic examples
595:Special values of
585:Modularity theorem
219:Riemann hypothesis
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106:interlanguage link
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883:Analytic examples
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510:Langlands program
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407:Bernoulli numbers
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951:Hasse–Weil
829:Physorg.com
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639:-functions"
549:automorphic
475:Bryan Birch
431:theory and
341:meromorphic
305:), and the
234:conjectural
165:meromorphic
152:-functions.
1079:-functions
964:-functions
955:-functions
946:-functions
916:-functions
899:-functions
895:Dirichlet
872:-functions
739:0956.11021
620:References
599:-functions
567:Dirichlet
526:-functions
518:-functions
418:-functions
394:algebraic
276:-functions
195:half-plane
191:convergent
189:, usually
815:August 5,
773:EMS Press
613:-function
580:-function
571:-function
310:-function
161:-function
120:talk page
72:Consider
40:in German
1112:Category
973:Theorems
960:Motivic
701:(1999).
686:54958644
643:Preprint
609:Shimizu
555:See also
301:for the
295:infinite
221:and its
168:function
96:provide
731:1697859
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398:-theory
291:-series
245:-series
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170:on the
118:to the
100:in the
42:.
1075:-adic
942:Artin
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522:Hecke
514:Artin
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247:for a
205:. The
682:S2CID
334:poles
293:, an
193:on a
185:is a
178:. An
163:is a
63:DeepL
817:2008
717:ISBN
477:and
144:The
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92:You
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