2556:, who had many papers named (approximately) "(Some) Old and new problems and results about...". Indeed, the 1989 Amalfi conference was quite surprising in that both Selberg and ErdÅs were present, with the story being that Selberg did not know that ErdÅs was to attend.
1244:
787:
50:. Although the exact nature of the class is conjectural, the hope is that the definition of the class will lead to a classification of its contents and an elucidation of its properties, including insight into their relationship to
2206:
2371:
369:
2088:
589:
1354:
1650:
159:
1018:
263:
446:
1113:
1742:
856:
2932:
510:
2273:
2832:
1422:
671:
968:
1826:
2750:
2714:
2119:
2779:
1135:
2278:
1981:
203:
682:
2872:
2852:
2692:
3189:
1918:
can be written as a product of primitive functions. Selberg's conjectures, described below, imply that the factorization into primitive functions is unique.
276:
3386:
521:
1276:
3143:
3371:
1544:
3233:
3350:
894:
87:
3416:
3314:
3182:
268:
2221:
976:
221:
3355:
3340:
2459:
400:
3447:
3376:
3123:
1023:
42:
which obey four axioms that seem to capture the essential properties satisfied by most functions that are commonly called
3280:
1680:
1461:
The best results on the structure of the
Selberg class are due to Kaczorowski and Perelli, who show that the Dirichlet
3175:
2479:
1940:
1934:
798:
2877:
205:
has a meromorphic continuation to the entire complex plane, with the only possible pole (if any) when s equals 1.
2600:
458:
2784:
1362:
615:
1937:
1126:
903:
3345:
3266:
3241:
2533:
1436:
1429:
1772:
3381:
3219:
2509:) = 1. As mentioned above, conjectures 1 and 2 imply the unique factorization of functions in
2483:
1259:
3309:
3224:
2494:
1922:
1267:
208:
2719:
2697:
1239:{\displaystyle F_{p}(s)=\sum _{n=0}^{\infty }{\frac {a_{p^{n}}}{p^{ns}}}{\text{ for Re}}(s)>0.}
3335:
3289:
3094:
3068:
3013:
2475:
880:
55:
1465:-functions (including the Riemann zeta-function) are the only examples with degree less than 2.
2972:
Selberg, Atle (1992), "Old and new conjectures and results about a class of
Dirichlet series",
2755:
3401:
3391:
3139:
173:
3319:
3271:
3157:
3131:
3102:
3078:
3039:
3023:
2985:
2615:
2463:
79:
51:
39:
3153:
3090:
3035:
2981:
782:{\displaystyle F_{p}(s)=\exp \left(\sum _{n=1}^{\infty }{\frac {b_{p^{n}}}{p^{ns}}}\right)}
3161:
3149:
3127:
3106:
3086:
3043:
3031:
2989:
2977:
179:
2857:
2837:
2677:
379:
3441:
3205:
2553:
595:
47:
3082:
2674:
A celebrated conjecture of
Dedekind asserts that for any finite algebraic extension
2201:{\displaystyle \sum _{p\leq x}{\frac {a_{p}{\overline {a_{p}^{\prime }}}}{p}}=O(1).}
168:) > 1 that satisfy four axioms (or assumptions as Selberg calls them):
3426:
3421:
3115:
3052:
3001:
2471:
2451:
59:
3098:
3027:
897:
holds, and have a functional equation, but do not satisfy the
Riemann hypothesis.
3004:; Ghosh, Amit (1993), "On the Selberg class of Dirichlet series: small degrees",
900:
The condition that θ < 1/2 is important, as the θ = 1 case includes
2620:
2366:{\displaystyle F^{\chi }(s)=\sum _{n=1}^{\infty }{\frac {\chi (n)a_{n}}{n^{s}}}}
20:
3167:
66:), who preferred not to use the word "axiom" that later authors have employed.
3198:
3135:
890:
364:{\displaystyle \gamma (s)=Q^{s}\prod _{i=1}^{k}\Gamma (\omega _{i}s+\mu _{i})}
32:
2974:
Proceedings of the Amalfi
Conference on Analytic Number Theory (Maiori, 1989)
2083:{\displaystyle \sum _{p\leq x}{\frac {|a_{p}|^{2}}{p}}=n_{F}\log \log x+O(1)}
2513:
into primitive functions. Another consequence is that the primitivity of
584:{\displaystyle \Phi (s)=\alpha \,{\overline {\Phi (1-{\overline {s}})}};}
3018:
1349:{\displaystyle L(s,\Delta )=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}}
394:
complex with non-negative real part, as well as a so-called root number
1929:
of primitive
Dirichlet characters. Assuming conjectures 1 and 2 below,
1921:
Examples of primitive functions include the
Riemann zeta function and
3073:
1645:{\displaystyle N_{F}(T)=d_{F}{\frac {T\log(T+C)}{2\pi }}+O(\log T).}
28:
154:{\displaystyle F(s)=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}}
3171:
2634:
The zeroes on the boundary are counted with half-multiplicity.
2448:) is entire. In particular, they imply Dedekind's conjecture.
3126:, Readings in Mathematics, vol. 206 (Second ed.),
2388:(and consequently, they form the primitive factorization of
2164:
2834:
is entire. More generally, Dedekind conjectures that if
2653:, Selberg's result shows that their sum is well-defined.
1955:), Selberg made conjectures concerning the functions in
1013:{\displaystyle a_{n}\ll _{\varepsilon }n^{\varepsilon }}
258:{\displaystyle a_{n}\ll _{\varepsilon }n^{\varepsilon }}
2275:Ļ is a primitive Dirichlet character, and the function
1766:
are in the
Selberg class, then so is their product and
893:
associated with exceptional eigenvalues, for which the
2880:
2860:
2840:
2787:
2758:
2722:
2700:
2680:
2281:
2224:
2122:
1984:
1943:
that satisfy the
Ramanujan conjecture are primitive.
1775:
1683:
1547:
1365:
1279:
1138:
1026:
979:
906:
801:
685:
618:
524:
461:
441:{\displaystyle \alpha \in \mathbb {C} ,\;|\alpha |=1}
403:
279:
224:
182:
90:
2552:
The title of
Selberg's paper is somewhat a spoof on
2470:
corresponding to irreducible representations of the
1108:{\displaystyle L(s+1/3,\chi _{4})L(s-1/3,\chi _{4})}
3400:
3364:
3328:
3302:
3259:
3212:
2926:
2866:
2846:
2826:
2773:
2744:
2708:
2686:
2365:
2267:
2200:
2082:
1820:
1737:{\displaystyle d_{F}=2\sum _{i=1}^{k}\omega _{i}.}
1736:
1644:
1416:
1348:
1238:
1107:
1012:
962:
850:
781:
665:
583:
504:
440:
363:
257:
197:
153:
2934:should be entire. This conjecture is still open.
2599:Jerzy Kaczorowski & Alberto Perelli (2011).
1485:that arise from the poles of the gamma factor Ī³(
1512:. Denoting the number of non-trivial zeroes of
1473:As with the Riemann zeta function, an element
16:Axiomatic definition of a class of L-functions
3183:
3061:Bulletin of the American Mathematical Society
8:
2976:, Salerno: Univ. Salerno, pp. 367ā385,
2601:"On the structure of the Selberg class, VII"
851:{\displaystyle b_{p^{n}}=O(p^{n\theta }).\,}
2955:
2927:{\displaystyle \zeta _{K}(s)/\zeta _{F}(s)}
1489:). The other zeroes are referred to as the
875:be non-negative is because there are known
610:) can be written as a product over primes:
3190:
3176:
3168:
2752:is divisible by the Riemann zeta function
1497:. These will all be located in some strip
1254:The prototypical example of an element in
970:whose zeros are not on the critical line.
869:The condition that the real part of μ
418:
3072:
3055:(1994), "Selberg's conjectures and Artin
3017:
2909:
2900:
2885:
2879:
2859:
2839:
2807:
2792:
2786:
2757:
2727:
2721:
2702:
2701:
2699:
2679:
2619:
2355:
2344:
2325:
2319:
2308:
2286:
2280:
2256:
2246:
2235:
2223:
2163:
2158:
2152:
2146:
2139:
2127:
2121:
2044:
2025:
2020:
2013:
2004:
2001:
1989:
1983:
1809:
1796:
1780:
1774:
1725:
1715:
1704:
1688:
1682:
1580:
1574:
1552:
1546:
1404:
1400:
1391:
1370:
1364:
1338:
1328:
1322:
1316:
1305:
1278:
1216:
1205:
1193:
1188:
1182:
1176:
1165:
1143:
1137:
1096:
1081:
1057:
1042:
1025:
1004:
994:
984:
978:
945:
920:
905:
847:
832:
811:
806:
800:
763:
751:
746:
740:
734:
723:
690:
684:
648:
638:
617:
559:
544:
543:
523:
505:{\displaystyle \Phi (s)=\gamma (s)F(s)\,}
501:
460:
427:
419:
411:
410:
402:
352:
336:
320:
309:
299:
278:
249:
239:
229:
223:
181:
143:
133:
127:
121:
110:
89:
2268:{\displaystyle F=\prod _{i=1}^{m}F_{i},}
2827:{\displaystyle \zeta _{F}(s)/\zeta (s)}
2545:
1952:
1751: = 1 is the only function in
1417:{\displaystyle a_{n}=\tau (n)/n^{11/2}}
1115:which violates the Riemann hypothesis.
666:{\displaystyle F(s)=\prod _{p}F_{p}(s)}
63:
271:: there is a gamma factor of the form
2943:
2662:
2587:
2576:
2565:
2458:) that conjectures 1 and 2 imply the
2455:
2104:Conjecture 2: For distinct primitive
963:{\displaystyle (1-2^{-s})(1-2^{1-s})}
889:is negative. Specifically, there are
7:
3372:Birch and Swinnerton-Dyer conjecture
2993:Reprinted in Collected Papers, vol
1821:{\displaystyle d_{FG}=d_{F}+d_{G}.}
1118:It is a consequence of 4. that the
879:-functions that do not satisfy the
74:The formal definition of the class
3120:Problems in analytic number theory
2424:Conjectures 1 and 2 imply that if
2320:
1317:
1292:
1177:
735:
547:
525:
462:
326:
122:
14:
3417:Main conjecture of Iwasawa theory
388:real and positive, and the μ
2997:, Springer-Verlag, Berlin (1991)
2493:also satisfy an analogue of the
38:. The members of the class are
3083:10.1090/s0273-0979-1994-00479-3
2505:) has no zeroes on the line Re(
2420:Consequences of the conjectures
3351:RamanujanāPetersson conjecture
3341:Generalized Riemann hypothesis
3237:-functions of Hecke characters
2921:
2915:
2897:
2891:
2821:
2815:
2804:
2798:
2768:
2762:
2739:
2733:
2337:
2331:
2298:
2292:
2192:
2186:
2077:
2071:
2021:
2005:
1636:
1624:
1604:
1592:
1564:
1558:
1388:
1382:
1295:
1283:
1227:
1221:
1155:
1149:
1102:
1069:
1063:
1030:
957:
932:
929:
907:
895:RamanujanāPeterssen conjecture
841:
825:
702:
696:
660:
654:
628:
622:
569:
550:
534:
528:
498:
492:
486:
480:
471:
465:
428:
420:
358:
329:
289:
283:
192:
186:
100:
94:
1:
3310:Analytic class number formula
3124:Graduate Texts in Mathematics
3028:10.1215/s0012-7094-93-07225-0
2745:{\displaystyle \zeta _{F}(s)}
2462:. In fact, Murty showed that
2218:with primitive factorization
1907:is primitive. Every function
1846:if whenever it is written as
1755:whose degree is less than 1.
164:absolutely convergent for Re(
3315:Riemannāvon Mangoldt formula
2709:{\displaystyle \mathbb {Q} }
2649:are not uniquely defined by
2407:, the non-trivial zeroes of
2169:
573:
564:
378:is real and positive, Ī the
58:. The class was defined by
2621:10.4007/annals.2011.173.3.4
1946:
1941:automorphic representations
265:for any Īµ > 0;
3464:
2384:are primitive elements of
1262:. Another example, is the
792:and, for some Īø < 1/2,
3136:10.1007/978-0-387-72350-1
3006:Duke Mathematical Journal
2854:is a finite extension of
2774:{\displaystyle \zeta (s)}
1443:, and the reciprocals of
31:definition of a class of
2781:. That is, the quotient
2097: = 1 whenever
3267:Dedekind zeta functions
2956:Conrey & Ghosh 1993
2411:all lie on the line Re(
2395:Riemann hypothesis for
1538:), Selberg showed that
1435:All known examples are
452:such that the function
2928:
2868:
2848:
2828:
2775:
2746:
2710:
2688:
2534:List of zeta functions
2367:
2324:
2269:
2251:
2202:
2084:
1971:, there is an integer
1963:Conjecture 1: For all
1822:
1738:
1720:
1646:
1430:Ramanujan tau function
1418:
1350:
1321:
1240:
1181:
1109:
1014:
973:Without the condition
964:
865:Comments on definition
852:
783:
739:
667:
585:
506:
442:
365:
325:
259:
199:
155:
126:
3387:BlochāKato conjecture
3382:Beilinson conjectures
3365:Algebraic conjectures
3220:Riemann zeta function
2929:
2869:
2849:
2829:
2776:
2747:
2711:
2689:
2608:Annals of Mathematics
2484:Langlands conjectures
2478:of the rationals are
2436: = 1, then
2377:, then the functions
2368:
2304:
2270:
2231:
2203:
2085:
1947:Selberg's conjectures
1903: = 1, then
1823:
1747:It can be shown that
1739:
1700:
1647:
1454:) are polynomials in
1419:
1351:
1301:
1260:Riemann zeta function
1241:
1161:
1110:
1015:
965:
853:
784:
719:
668:
586:
507:
443:
366:
305:
260:
200:
156:
106:
3448:Zeta and L-functions
3392:Langlands conjecture
3377:Deligne's conjecture
3329:Analytic conjectures
2878:
2858:
2838:
2785:
2756:
2720:
2716:, the zeta function
2698:
2678:
2495:prime number theorem
2482:as predicted by the
2428:has a pole of order
2279:
2222:
2120:
1982:
1773:
1681:
1545:
1363:
1277:
1268:modular discriminant
1136:
1024:
977:
904:
799:
683:
616:
602:) > 1,
522:
459:
401:
277:
222:
209:Ramanujan conjecture
198:{\displaystyle F(s)}
180:
88:
3346:Lindelƶf hypothesis
2168:
1458:of bounded degree.
269:Functional equation
3336:Riemann hypothesis
3260:Algebraic examples
2924:
2864:
2844:
2824:
2771:
2742:
2706:
2684:
2476:solvable extension
2415:) = 1/2.
2363:
2265:
2198:
2154:
2138:
2080:
2000:
1818:
1734:
1642:
1491:non-trivial zeroes
1414:
1346:
1236:
1105:
1010:
960:
881:Riemann hypothesis
848:
779:
663:
643:
581:
502:
438:
361:
255:
195:
151:
78:is the set of all
56:Riemann hypothesis
3435:
3434:
3213:Analytic examples
3145:978-0-387-72349-5
2867:{\displaystyle F}
2847:{\displaystyle K}
2687:{\displaystyle F}
2517:is equivalent to
2489:The functions in
2361:
2210:Conjecture 3: If
2178:
2172:
2123:
2035:
1985:
1674:. It is given by
1616:
1344:
1266:-function of the
1219:
1214:
772:
634:
576:
567:
149:
52:automorphic forms
3455:
3356:Artin conjecture
3320:Weil conjectures
3192:
3185:
3178:
3169:
3164:
3109:
3076:
3046:
3021:
3002:Conrey, J. Brian
2992:
2959:
2953:
2947:
2941:
2935:
2933:
2931:
2930:
2925:
2914:
2913:
2904:
2890:
2889:
2873:
2871:
2870:
2865:
2853:
2851:
2850:
2845:
2833:
2831:
2830:
2825:
2811:
2797:
2796:
2780:
2778:
2777:
2772:
2751:
2749:
2748:
2743:
2732:
2731:
2715:
2713:
2712:
2707:
2705:
2693:
2691:
2690:
2685:
2672:
2666:
2660:
2654:
2641:
2635:
2632:
2626:
2625:
2623:
2605:
2596:
2590:
2585:
2579:
2574:
2568:
2563:
2557:
2550:
2524: = 1.
2460:Artin conjecture
2372:
2370:
2369:
2364:
2362:
2360:
2359:
2350:
2349:
2348:
2326:
2323:
2318:
2291:
2290:
2274:
2272:
2271:
2266:
2261:
2260:
2250:
2245:
2207:
2205:
2204:
2199:
2179:
2174:
2173:
2167:
2162:
2153:
2151:
2150:
2140:
2137:
2089:
2087:
2086:
2081:
2049:
2048:
2036:
2031:
2030:
2029:
2024:
2018:
2017:
2008:
2002:
1999:
1913:
1837:
1827:
1825:
1824:
1819:
1814:
1813:
1801:
1800:
1788:
1787:
1743:
1741:
1740:
1735:
1730:
1729:
1719:
1714:
1693:
1692:
1651:
1649:
1648:
1643:
1617:
1615:
1607:
1581:
1579:
1578:
1557:
1556:
1526:
1511:
1469:Basic properties
1423:
1421:
1420:
1415:
1413:
1412:
1408:
1395:
1375:
1374:
1355:
1353:
1352:
1347:
1345:
1343:
1342:
1333:
1332:
1323:
1320:
1315:
1245:
1243:
1242:
1237:
1220:
1217:
1215:
1213:
1212:
1200:
1199:
1198:
1197:
1183:
1180:
1175:
1148:
1147:
1114:
1112:
1111:
1106:
1101:
1100:
1085:
1062:
1061:
1046:
1019:
1017:
1016:
1011:
1009:
1008:
999:
998:
989:
988:
969:
967:
966:
961:
956:
955:
928:
927:
857:
855:
854:
849:
840:
839:
818:
817:
816:
815:
788:
786:
785:
780:
778:
774:
773:
771:
770:
758:
757:
756:
755:
741:
738:
733:
695:
694:
672:
670:
669:
664:
653:
652:
642:
590:
588:
587:
582:
577:
572:
568:
560:
545:
511:
509:
508:
503:
447:
445:
444:
439:
431:
423:
414:
370:
368:
367:
362:
357:
356:
341:
340:
324:
319:
304:
303:
264:
262:
261:
256:
254:
253:
244:
243:
234:
233:
204:
202:
201:
196:
160:
158:
157:
152:
150:
148:
147:
138:
137:
128:
125:
120:
80:Dirichlet series
40:Dirichlet series
3463:
3462:
3458:
3457:
3456:
3454:
3453:
3452:
3438:
3437:
3436:
3431:
3396:
3360:
3324:
3298:
3255:
3208:
3196:
3146:
3128:Springer-Verlag
3114:
3051:
3019:math.NT/9204217
3000:
2971:
2968:
2963:
2962:
2954:
2950:
2942:
2938:
2905:
2881:
2876:
2875:
2856:
2855:
2836:
2835:
2788:
2783:
2782:
2754:
2753:
2723:
2718:
2717:
2696:
2695:
2676:
2675:
2673:
2669:
2661:
2657:
2648:
2642:
2638:
2633:
2629:
2603:
2598:
2597:
2593:
2586:
2582:
2575:
2571:
2564:
2560:
2551:
2547:
2542:
2530:
2522:
2422:
2382:
2351:
2340:
2327:
2282:
2277:
2276:
2252:
2220:
2219:
2142:
2141:
2118:
2117:
2095:
2040:
2019:
2009:
2003:
1980:
1979:
1976:
1949:
1908:
1901:
1895:
1884:
1868:
1862:
1856:
1832:
1805:
1792:
1776:
1771:
1770:
1721:
1684:
1679:
1678:
1660:
1608:
1582:
1570:
1548:
1543:
1542:
1532:
1517:
1498:
1471:
1448:
1396:
1366:
1361:
1360:
1334:
1324:
1275:
1274:
1252:
1201:
1189:
1184:
1139:
1134:
1133:
1123:
1092:
1053:
1022:
1021:
1020:there would be
1000:
990:
980:
975:
974:
941:
916:
902:
901:
888:
874:
867:
862:
828:
807:
802:
797:
796:
759:
747:
742:
718:
714:
686:
681:
680:
644:
614:
613:
546:
520:
519:
457:
456:
399:
398:
393:
387:
348:
332:
295:
275:
274:
245:
235:
225:
220:
219:
217:
178:
177:
139:
129:
86:
85:
72:
17:
12:
11:
5:
3461:
3459:
3451:
3450:
3440:
3439:
3433:
3432:
3430:
3429:
3424:
3419:
3413:
3411:
3398:
3397:
3395:
3394:
3389:
3384:
3379:
3374:
3368:
3366:
3362:
3361:
3359:
3358:
3353:
3348:
3343:
3338:
3332:
3330:
3326:
3325:
3323:
3322:
3317:
3312:
3306:
3304:
3300:
3299:
3297:
3296:
3287:
3278:
3269:
3263:
3261:
3257:
3256:
3254:
3253:
3248:
3239:
3231:
3222:
3216:
3214:
3210:
3209:
3197:
3195:
3194:
3187:
3180:
3172:
3166:
3165:
3144:
3111:
3110:
3063:, New Series,
3048:
3047:
3012:(3): 673ā693,
2998:
2967:
2964:
2961:
2960:
2948:
2936:
2923:
2920:
2917:
2912:
2908:
2903:
2899:
2896:
2893:
2888:
2884:
2863:
2843:
2823:
2820:
2817:
2814:
2810:
2806:
2803:
2800:
2795:
2791:
2770:
2767:
2764:
2761:
2741:
2738:
2735:
2730:
2726:
2704:
2683:
2667:
2655:
2644:
2636:
2627:
2591:
2580:
2569:
2558:
2544:
2543:
2541:
2538:
2537:
2536:
2529:
2526:
2520:
2421:
2418:
2417:
2416:
2393:
2380:
2358:
2354:
2347:
2343:
2339:
2336:
2333:
2330:
2322:
2317:
2314:
2311:
2307:
2303:
2300:
2297:
2294:
2289:
2285:
2264:
2259:
2255:
2249:
2244:
2241:
2238:
2234:
2230:
2227:
2208:
2197:
2194:
2191:
2188:
2185:
2182:
2177:
2171:
2166:
2161:
2157:
2149:
2145:
2136:
2133:
2130:
2126:
2112:ā² ā
2102:
2093:
2079:
2076:
2073:
2070:
2067:
2064:
2061:
2058:
2055:
2052:
2047:
2043:
2039:
2034:
2028:
2023:
2016:
2012:
2007:
1998:
1995:
1992:
1988:
1974:
1948:
1945:
1933:-functions of
1899:
1893:
1882:
1866:
1860:
1854:
1829:
1828:
1817:
1812:
1808:
1804:
1799:
1795:
1791:
1786:
1783:
1779:
1745:
1744:
1733:
1728:
1724:
1718:
1713:
1710:
1707:
1703:
1699:
1696:
1691:
1687:
1662:is called the
1658:
1653:
1652:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1620:
1614:
1611:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1577:
1573:
1569:
1566:
1563:
1560:
1555:
1551:
1530:
1483:trivial zeroes
1470:
1467:
1446:
1411:
1407:
1403:
1399:
1394:
1390:
1387:
1384:
1381:
1378:
1373:
1369:
1357:
1356:
1341:
1337:
1331:
1327:
1319:
1314:
1311:
1308:
1304:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1251:
1248:
1247:
1246:
1235:
1232:
1229:
1226:
1223:
1211:
1208:
1204:
1196:
1192:
1187:
1179:
1174:
1171:
1168:
1164:
1160:
1157:
1154:
1151:
1146:
1142:
1127:multiplicative
1121:
1104:
1099:
1095:
1091:
1088:
1084:
1080:
1077:
1074:
1071:
1068:
1065:
1060:
1056:
1052:
1049:
1045:
1041:
1038:
1035:
1032:
1029:
1007:
1003:
997:
993:
987:
983:
959:
954:
951:
948:
944:
940:
937:
934:
931:
926:
923:
919:
915:
912:
909:
884:
870:
866:
863:
861:
860:
859:
858:
846:
843:
838:
835:
831:
827:
824:
821:
814:
810:
805:
790:
789:
777:
769:
766:
762:
754:
750:
745:
737:
732:
729:
726:
722:
717:
713:
710:
707:
704:
701:
698:
693:
689:
674:
673:
662:
659:
656:
651:
647:
641:
637:
633:
630:
627:
624:
621:
593:
592:
591:
580:
575:
571:
566:
563:
558:
555:
552:
549:
542:
539:
536:
533:
530:
527:
513:
512:
500:
497:
494:
491:
488:
485:
482:
479:
476:
473:
470:
467:
464:
450:
449:
437:
434:
430:
426:
422:
417:
413:
409:
406:
389:
383:
380:gamma function
372:
371:
360:
355:
351:
347:
344:
339:
335:
331:
328:
323:
318:
315:
312:
308:
302:
298:
294:
291:
288:
285:
282:
266:
252:
248:
242:
238:
232:
228:
215:
206:
194:
191:
188:
185:
170:
162:
161:
146:
142:
136:
132:
124:
119:
116:
113:
109:
105:
102:
99:
96:
93:
71:
68:
48:zeta functions
46:-functions or
15:
13:
10:
9:
6:
4:
3:
2:
3460:
3449:
3446:
3445:
3443:
3428:
3425:
3423:
3420:
3418:
3415:
3414:
3412:
3410:
3408:
3404:
3399:
3393:
3390:
3388:
3385:
3383:
3380:
3378:
3375:
3373:
3370:
3369:
3367:
3363:
3357:
3354:
3352:
3349:
3347:
3344:
3342:
3339:
3337:
3334:
3333:
3331:
3327:
3321:
3318:
3316:
3313:
3311:
3308:
3307:
3305:
3301:
3295:
3293:
3288:
3286:
3284:
3279:
3277:
3275:
3270:
3268:
3265:
3264:
3262:
3258:
3252:
3251:Selberg class
3249:
3247:
3245:
3240:
3238:
3236:
3232:
3230:
3228:
3223:
3221:
3218:
3217:
3215:
3211:
3207:
3206:number theory
3203:
3201:
3193:
3188:
3186:
3181:
3179:
3174:
3173:
3170:
3163:
3159:
3155:
3151:
3147:
3141:
3137:
3133:
3130:, Chapter 8,
3129:
3125:
3121:
3117:
3116:Murty, M. Ram
3113:
3112:
3108:
3104:
3100:
3096:
3092:
3088:
3084:
3080:
3075:
3070:
3066:
3062:
3059:-functions",
3058:
3054:
3053:Murty, M. Ram
3050:
3049:
3045:
3041:
3037:
3033:
3029:
3025:
3020:
3015:
3011:
3007:
3003:
2999:
2996:
2991:
2987:
2983:
2979:
2975:
2970:
2969:
2965:
2957:
2952:
2949:
2946:, Theorem 4.3
2945:
2940:
2937:
2918:
2910:
2906:
2901:
2894:
2886:
2882:
2861:
2841:
2818:
2812:
2808:
2801:
2793:
2789:
2765:
2759:
2736:
2728:
2724:
2681:
2671:
2668:
2664:
2659:
2656:
2652:
2647:
2640:
2637:
2631:
2628:
2622:
2617:
2614:: 1397ā1441.
2613:
2609:
2602:
2595:
2592:
2589:
2584:
2581:
2578:
2573:
2570:
2567:
2562:
2559:
2555:
2549:
2546:
2539:
2535:
2532:
2531:
2527:
2525:
2523:
2516:
2512:
2508:
2504:
2500:
2496:
2492:
2487:
2485:
2481:
2477:
2473:
2469:
2467:
2461:
2457:
2453:
2449:
2447:
2443:
2439:
2435:
2431:
2427:
2419:
2414:
2410:
2406:
2402:
2398:
2394:
2391:
2387:
2383:
2376:
2356:
2352:
2345:
2341:
2334:
2328:
2315:
2312:
2309:
2305:
2301:
2295:
2287:
2283:
2262:
2257:
2253:
2247:
2242:
2239:
2236:
2232:
2228:
2225:
2217:
2213:
2209:
2195:
2189:
2183:
2180:
2175:
2159:
2155:
2147:
2143:
2134:
2131:
2128:
2124:
2115:
2111:
2107:
2103:
2101:is primitive.
2100:
2096:
2074:
2068:
2065:
2062:
2059:
2056:
2053:
2050:
2045:
2041:
2037:
2032:
2026:
2014:
2010:
1996:
1993:
1990:
1986:
1977:
1970:
1966:
1962:
1961:
1960:
1958:
1954:
1944:
1942:
1939:
1936:
1932:
1928:
1926:
1919:
1917:
1911:
1906:
1902:
1892:
1889: =
1888:
1881:
1878: =
1877:
1873:
1869:
1859:
1853:
1850: =
1849:
1845:
1841:
1835:
1815:
1810:
1806:
1802:
1797:
1793:
1789:
1784:
1781:
1777:
1769:
1768:
1767:
1765:
1761:
1756:
1754:
1750:
1731:
1726:
1722:
1716:
1711:
1708:
1705:
1701:
1697:
1694:
1689:
1685:
1677:
1676:
1675:
1673:
1669:
1665:
1661:
1639:
1633:
1630:
1627:
1621:
1618:
1612:
1609:
1601:
1598:
1595:
1589:
1586:
1583:
1575:
1571:
1567:
1561:
1553:
1549:
1541:
1540:
1539:
1537:
1533:
1525:
1521:
1515:
1510:
1506:
1502:
1496:
1492:
1488:
1484:
1480:
1476:
1468:
1466:
1464:
1459:
1457:
1453:
1449:
1442:
1440:
1433:
1431:
1427:
1409:
1405:
1401:
1397:
1392:
1385:
1379:
1376:
1371:
1367:
1339:
1335:
1329:
1325:
1312:
1309:
1306:
1302:
1298:
1289:
1286:
1280:
1273:
1272:
1271:
1269:
1265:
1261:
1257:
1249:
1233:
1230:
1224:
1209:
1206:
1202:
1194:
1190:
1185:
1172:
1169:
1166:
1162:
1158:
1152:
1144:
1140:
1132:
1131:
1130:
1128:
1124:
1116:
1097:
1093:
1089:
1086:
1082:
1078:
1075:
1072:
1066:
1058:
1054:
1050:
1047:
1043:
1039:
1036:
1033:
1027:
1005:
1001:
995:
991:
985:
981:
971:
952:
949:
946:
942:
938:
935:
924:
921:
917:
913:
910:
898:
896:
892:
887:
882:
878:
873:
864:
844:
836:
833:
829:
822:
819:
812:
808:
803:
795:
794:
793:
775:
767:
764:
760:
752:
748:
743:
730:
727:
724:
720:
715:
711:
708:
705:
699:
691:
687:
679:
678:
677:
657:
649:
645:
639:
635:
631:
625:
619:
612:
611:
609:
605:
601:
597:
596:Euler product
594:
578:
561:
556:
553:
540:
537:
531:
518:
517:
516:
495:
489:
483:
477:
474:
468:
455:
454:
453:
435:
432:
424:
415:
407:
404:
397:
396:
395:
392:
386:
381:
377:
353:
349:
345:
342:
337:
333:
321:
316:
313:
310:
306:
300:
296:
292:
286:
280:
273:
272:
270:
267:
250:
246:
240:
236:
230:
226:
214:
210:
207:
189:
183:
175:
172:
171:
169:
167:
144:
140:
134:
130:
117:
114:
111:
107:
103:
97:
91:
84:
83:
82:
81:
77:
69:
67:
65:
61:
57:
53:
49:
45:
41:
37:
35:
30:
26:
25:Selberg class
22:
3427:Euler system
3422:Selmer group
3406:
3402:
3291:
3282:
3273:
3250:
3243:
3242:Automorphic
3234:
3226:
3199:
3119:
3074:math/9407219
3064:
3060:
3056:
3009:
3005:
2994:
2973:
2951:
2939:
2670:
2658:
2650:
2645:
2639:
2630:
2611:
2607:
2594:
2583:
2572:
2561:
2548:
2518:
2514:
2510:
2506:
2502:
2498:
2490:
2488:
2472:Galois group
2465:
2452:M. Ram Murty
2450:
2445:
2441:
2437:
2433:
2429:
2425:
2423:
2412:
2408:
2404:
2400:
2396:
2389:
2385:
2378:
2374:
2215:
2211:
2113:
2109:
2105:
2098:
2091:
1972:
1968:
1964:
1956:
1953:Selberg 1992
1950:
1930:
1924:
1920:
1915:
1909:
1904:
1897:
1890:
1886:
1879:
1875:
1871:
1864:
1857:
1851:
1847:
1843:
1839:
1833:
1830:
1763:
1759:
1757:
1752:
1748:
1746:
1671:
1667:
1663:
1656:
1654:
1535:
1528:
1523:
1519:
1513:
1508:
1504:
1500:
1494:
1490:
1486:
1482:
1478:
1474:
1472:
1462:
1460:
1455:
1451:
1444:
1438:
1437:automorphic
1434:
1425:
1358:
1263:
1255:
1253:
1218: for Re
1119:
1117:
972:
899:
885:
876:
871:
868:
791:
675:
607:
603:
599:
514:
451:
390:
384:
382:, the ω
375:
373:
212:
165:
163:
75:
73:
64:Selberg 1992
60:Atle Selberg
43:
33:
24:
18:
3281:HasseāWeil
3067:(1): 1ā14,
2665:, Lemma 4.2
2643:While the Ļ
2480:automorphic
2454:showed in (
2373:is also in
1935:irreducible
1831:A function
891:Maass forms
883:when μ
174:Analyticity
21:mathematics
3409:-functions
3294:-functions
3285:-functions
3276:-functions
3246:-functions
3229:-functions
3225:Dirichlet
3202:-functions
3162:1190.11001
3107:0805.11062
3044:0796.11037
2990:0787.11037
2966:References
2944:Murty 1994
2663:Murty 1994
2588:Murty 1994
2577:Murty 2008
2566:Murty 2008
2554:Paul ErdÅs
2468:-functions
2456:Murty 1994
2399:: For all
1978:such that
1927:-functions
1923:Dirichlet
1842:is called
1499:1 −
1441:-functions
515:satisfies
70:Definition
36:-functions
2907:ζ
2883:ζ
2813:ζ
2790:ζ
2760:ζ
2725:ζ
2329:χ
2321:∞
2306:∑
2288:χ
2233:∏
2170:¯
2165:′
2132:≤
2125:∑
2060:
2054:
1994:≤
1987:∑
1844:primitive
1723:ω
1702:∑
1668:dimension
1631:
1613:π
1590:
1428:) is the
1380:τ
1318:∞
1303:∑
1293:Δ
1178:∞
1163:∑
1129:and that
1094:χ
1076:−
1055:χ
1006:ε
996:ε
992:≪
950:−
939:−
922:−
914:−
837:θ
736:∞
721:∑
712:
636:∏
598:: For Re(
574:¯
565:¯
557:−
548:Φ
541:α
526:Φ
478:γ
463:Φ
425:α
408:∈
405:α
350:μ
334:ω
327:Γ
307:∏
281:γ
251:ε
241:ε
237:≪
123:∞
108:∑
29:axiomatic
3442:Category
3303:Theorems
3290:Motivic
3118:(2008),
2528:See also
1938:cuspidal
1250:Examples
218:= 1 and
54:and the
3154:2376618
3091:1242382
3036:1253620
2982:1220477
2874:, then
2108:,
1874:, then
1863:, with
1518:0 ā¤ Im(
1258:is the
3405:-adic
3272:Artin
3160:
3152:
3142:
3105:
3099:265909
3097:
3089:
3042:
3034:
2988:
2980:
2464:Artin
2214:is in
1664:degree
1655:Here,
1424:and Ļ(
1359:where
374:where
27:is an
23:, the
3095:S2CID
3069:arXiv
3014:arXiv
2958:, Ā§ 4
2604:(PDF)
2540:Notes
2474:of a
1896:. If
1670:) of
1516:with
1503:ā¤ Re(
676:with
3140:ISBN
2444:)/Ī¶(
2090:and
1951:In (
1762:and
1666:(or
1522:) ā¤
1507:) ā¤
1481:has
1231:>
1125:are
62:in (
3204:in
3158:Zbl
3132:doi
3103:Zbl
3079:doi
3040:Zbl
3024:doi
2986:Zbl
2694:of
2616:doi
2612:173
2432:at
2403:in
2057:log
2051:log
1967:in
1914:of
1912:ā 1
1885:or
1870:in
1838:in
1836:ā 1
1758:If
1628:log
1587:log
1527:by
1493:of
1477:of
1270:Ī
709:exp
19:In
3444::
3156:,
3150:MR
3148:,
3138:,
3122:,
3101:,
3093:,
3087:MR
3085:,
3077:,
3065:31
3038:,
3032:MR
3030:,
3022:,
3010:72
3008:,
2984:,
2978:MR
2610:.
2606:.
2497::
2486:.
2392:).
2116:,
1959::
1432:.
1402:11
1234:0.
211::
176::
3407:L
3403:p
3292:L
3283:L
3274:L
3244:L
3235:L
3227:L
3200:L
3191:e
3184:t
3177:v
3134::
3081::
3071::
3057:L
3026::
3016::
2995:2
2922:)
2919:s
2916:(
2911:F
2902:/
2898:)
2895:s
2892:(
2887:K
2862:F
2842:K
2822:)
2819:s
2816:(
2809:/
2805:)
2802:s
2799:(
2794:F
2769:)
2766:s
2763:(
2740:)
2737:s
2734:(
2729:F
2703:Q
2682:F
2651:F
2646:i
2624:.
2618::
2521:F
2519:n
2515:F
2511:S
2507:s
2503:s
2501:(
2499:F
2491:S
2466:L
2446:s
2442:s
2440:(
2438:F
2434:s
2430:m
2426:F
2413:s
2409:F
2405:S
2401:F
2397:S
2390:F
2386:S
2381:i
2379:F
2375:S
2357:s
2353:n
2346:n
2342:a
2338:)
2335:n
2332:(
2316:1
2313:=
2310:n
2302:=
2299:)
2296:s
2293:(
2284:F
2263:,
2258:i
2254:F
2248:m
2243:1
2240:=
2237:i
2229:=
2226:F
2216:S
2212:F
2196:.
2193:)
2190:1
2187:(
2184:O
2181:=
2176:p
2160:p
2156:a
2148:p
2144:a
2135:x
2129:p
2114:S
2110:F
2106:F
2099:F
2094:F
2092:n
2078:)
2075:1
2072:(
2069:O
2066:+
2063:x
2046:F
2042:n
2038:=
2033:p
2027:2
2022:|
2015:p
2011:a
2006:|
1997:x
1991:p
1975:F
1973:n
1969:S
1965:F
1957:S
1931:L
1925:L
1916:S
1910:F
1905:F
1900:F
1898:d
1894:2
1891:F
1887:F
1883:1
1880:F
1876:F
1872:S
1867:i
1865:F
1861:2
1858:F
1855:1
1852:F
1848:F
1840:S
1834:F
1816:.
1811:G
1807:d
1803:+
1798:F
1794:d
1790:=
1785:G
1782:F
1778:d
1764:G
1760:F
1753:S
1749:F
1732:.
1727:i
1717:k
1712:1
1709:=
1706:i
1698:2
1695:=
1690:F
1686:d
1672:F
1659:F
1657:d
1640:.
1637:)
1634:T
1625:(
1622:O
1619:+
1610:2
1605:)
1602:C
1599:+
1596:T
1593:(
1584:T
1576:F
1572:d
1568:=
1565:)
1562:T
1559:(
1554:F
1550:N
1536:T
1534:(
1531:F
1529:N
1524:T
1520:s
1514:F
1509:A
1505:s
1501:A
1495:F
1487:s
1479:S
1475:F
1463:L
1456:p
1452:s
1450:(
1447:p
1445:F
1439:L
1426:n
1410:2
1406:/
1398:n
1393:/
1389:)
1386:n
1383:(
1377:=
1372:n
1368:a
1340:s
1336:n
1330:n
1326:a
1313:1
1310:=
1307:n
1299:=
1296:)
1290:,
1287:s
1284:(
1281:L
1264:L
1256:S
1228:)
1225:s
1222:(
1210:s
1207:n
1203:p
1195:n
1191:p
1186:a
1173:0
1170:=
1167:n
1159:=
1156:)
1153:s
1150:(
1145:p
1141:F
1122:n
1120:a
1103:)
1098:4
1090:,
1087:3
1083:/
1079:1
1073:s
1070:(
1067:L
1064:)
1059:4
1051:,
1048:3
1044:/
1040:1
1037:+
1034:s
1031:(
1028:L
1002:n
986:n
982:a
958:)
953:s
947:1
943:2
936:1
933:(
930:)
925:s
918:2
911:1
908:(
886:i
877:L
872:i
845:.
842:)
834:n
830:p
826:(
823:O
820:=
813:n
809:p
804:b
776:)
768:s
765:n
761:p
753:n
749:p
744:b
731:1
728:=
725:n
716:(
706:=
703:)
700:s
697:(
692:p
688:F
661:)
658:s
655:(
650:p
646:F
640:p
632:=
629:)
626:s
623:(
620:F
608:s
606:(
604:F
600:s
579:;
570:)
562:s
554:1
551:(
538:=
535:)
532:s
529:(
499:)
496:s
493:(
490:F
487:)
484:s
481:(
475:=
472:)
469:s
466:(
448:,
436:1
433:=
429:|
421:|
416:,
412:C
391:i
385:i
376:Q
359:)
354:i
346:+
343:s
338:i
330:(
322:k
317:1
314:=
311:i
301:s
297:Q
293:=
290:)
287:s
284:(
247:n
231:n
227:a
216:1
213:a
193:)
190:s
187:(
184:F
166:s
145:s
141:n
135:n
131:a
118:1
115:=
112:n
104:=
101:)
98:s
95:(
92:F
76:S
44:L
34:L
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.