1381:
917:
700:
1075:
2218:
207:
115:
2328:
1851:
1598:
2099:
1371:
2028:
1964:
1276:
1721:
1521:
1770:
562:
tends to infinity. The
Riemann hypothesis implies that there are no zeros at all in this region and so implies the Lindelöf hypothesis. The number of zeros with imaginary part between
1441:
1209:
1901:
3578:
Conrey, J. B.; Farmer, D. W.; Keating, Jonathan P.; Rubinstein, M. O.; Snaith, N. C. (2008), "Lower order terms in the full moment conjecture for the
Riemann zeta function",
734:
585:
1384:
The known zero-free region roughly speaking corresponds to the bottom right corner of the image, and the
Riemann hypothesis would push the entire diagram down to the x-axis
1633:
2252:
2363:
1682:
1653:
136:
47:
3356:
574:)), so the Lindelöf hypothesis seems only slightly stronger than what has already been proved, but in spite of this it has resisted all attempts to prove it.
4123:
3478:
952:
4391:
3239:
1121:
2104:
4320:
2478:
Hardy, G. H.; Littlewood, J. E. (1916). "Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes".
2257:
4067:
3899:
3706:
1300:
4305:
1778:
257:
4167:
3433:
4284:
1222:
3264:
3979:
4350:
3347:
2404:
4248:
1524:
4116:
1534:
3524:
Conrey, J. B.; Farmer, D. W.; Keating, Jonathan P.; Rubinstein, M. O.; Snaith, N. C. (2005), "Integral moments of L-functions",
4396:
312:
2041:
4289:
4274:
3690:
1973:
1912:
4386:
4310:
4095:
4214:
4109:
4090:
3720:
1687:
1446:
526:
Backlund (1918–1919) showed that the Lindelöf hypothesis is equivalent to the following statement about the
4055:
3118:
3093:
29:
1726:
1294:
3580:
3144:
2376:
1387:
1092:. The leading coefficients are conjectured to be the product of an elementary factor, a certain product over
1149:
1853:
then it is known that this bound roughly correspond to asymptotics for primes in short intervals of length
4200:
4175:
3933:
3375:
2395:. In 2010, new methods to obtain sub-convexity estimates for L-functions in the PGL(2) case were given by
4085:
3634:
Conrey, J. B.; Ghosh, A. (1998), "A conjecture for the sixth power moment of the
Riemann zeta-function",
2463:
Hardy, G. H.; Littlewood, J. E. (1923). "On Lindelöf's hypothesis concerning the
Riemann zeta-function".
4315:
4153:
2944:
Cramér, Harald (1936). "On the order of magnitude of the difference between consecutive prime numbers".
2563:
Phillips, Eric (1933). "The zeta-function of
Riemann: further developments of van der Corput's method".
1856:
912:{\displaystyle \int _{0}^{T}|\zeta (1/2+it)|^{2k}\,dt=T\sum _{j=0}^{k^{2}}c_{k,j}\log(T)^{k^{2}-j}+o(T)}
33:
3119:"New Bounds for Large Values of Dirichlet Polynomials, Part 2 - Videos | Institute for Advanced Study"
3094:"New Bounds for Large Values of Dirichlet Polynomials, Part 1 - Videos | Institute for Advanced Study"
4381:
4243:
4158:
3925:
3852:
3653:
1085:
3380:
3938:
3457:
245:
1603:
315:. It has since been lowered to slightly less than 1/6 by several authors using long and technical
4269:
4223:
3967:
3698:
3643:
3623:
3589:
3567:
3533:
3513:
3487:
3412:
3393:
3365:
3329:
3301:
3218:
3025:
2988:
2223:
1283:
527:
316:
37:
129:
can be replaced by a smaller value, the conjecture can be restated as follows: for any positive
2979:
Banks, William; Ford, Kevin; Tao, Terence (2023). "Large prime gaps and probabilistic models".
2528:
Titchmarsh, E. C. (1932). "On van der Corput's method and the zeta-function of
Riemann (III)".
2333:
1658:
4335:
4325:
4063:
4036:
4008:
3951:
3895:
3887:
3868:
3808:
3746:
3726:
3702:
3671:
3607:
3551:
3321:
3199:
3160:
3006:
2961:
2788:
2747:
2716:
2681:
2646:
2615:
2580:
2545:
2495:
4253:
4205:
3943:
3860:
3832:
3800:
3738:
3661:
3599:
3543:
3497:
3385:
3311:
3191:
3152:
2998:
2953:
2815:
2780:
2708:
2673:
2638:
2607:
2572:
2537:
2487:
2400:
2396:
2035:
1108:
300:
4077:
4048:
4020:
3963:
3909:
3894:, de Gruyter Expositions in Mathematics, vol. 5, Berlin: Walter de Gruyter & Co.,
3880:
3820:
3781:
3758:
3716:
3683:
3619:
3563:
3509:
3217:
Guth, Larry; Maynard, James (2024). "New large value estimates for
Dirichlet polynomials".
1638:
4073:
4044:
4016:
3959:
3905:
3876:
3816:
3777:
3754:
3712:
3679:
3615:
3559:
3505:
308:
265:
3996:
3929:
3856:
3827:
Ingham, A. E. (1928), "Mean-Value
Theorems in the Theory of the Riemann Zeta-Function",
3657:
695:{\displaystyle {\frac {1}{T}}\int _{0}^{T}|\zeta (1/2+it)|^{2k}\,dt=O(T^{\varepsilon })}
4027:
Motohashi, Yõichi (1995), "The
Riemann zeta-function and the non-Euclidean Laplacian",
1104:
1070:{\displaystyle {\frac {42}{9!}}\prod _{p}\left((1-p^{-1})^{4}(1+4p^{-1}+p^{-2})\right)}
543:
535:
122:
2677:
4375:
4184:
4139:
3788:
3765:
3517:
3473:
1967:
1906:
1212:
3971:
3627:
3397:
3333:
3069:
4360:
4355:
3571:
1093:
718:
304:
3916:
Keating, Jonathan P.; Snaith, N. C. (2000), "Random matrix theory and ζ(1/2+it)",
3289:
2598:
Rankin, R. A. (1955). "Van der Corput's method and the theory of exponent pairs".
3316:
2220:, therefore the latest work of Guth and Maynard gives the closest known value to
3864:
3156:
2380:
2373:
2369:
1380:
296:
217:
17:
4101:
3836:
3742:
3045:
3002:
2642:
2576:
2541:
2391:
The Riemann zeta function belongs to a more general family of functions called
2213:{\displaystyle \sigma _{I,GM}=7/10<\sigma _{H,GM}=8/10<\sigma _{I,H}=3/4}
1088:
to suggest some conjectures for the values of the coefficients for higher
4132:
3804:
3773:
3666:
3603:
3547:
3389:
2420:
2392:
2031:
25:
4040:
4012:
3955:
3872:
3812:
3750:
3675:
3611:
3555:
3325:
3240:"The Biggest Problem in Mathematics Is Finally a Step Closer to Being Solved"
3203:
3164:
3010:
2965:
2792:
2751:
2720:
2685:
2650:
2619:
2584:
2549:
2499:
724:
There is a much more precise conjecture about the asymptotic behavior of the
202:{\displaystyle \zeta \!\left({\frac {1}{2}}+it\right)\!=o(t^{\varepsilon }).}
3024:
Trudgian, Timothy S.; Yang, Andrew (2023). "Toward optimal exponent pairs".
2819:
2784:
2611:
1290:
280:(σ) is 0 for σ ≥ 1/2 and 1/2 − σ for σ ≤ 1/2.
110:{\displaystyle \zeta \!\left({\frac {1}{2}}+it\right)\!=O(t^{\varepsilon })}
3997:"A relation between the Riemann zeta-function and the hyperbolic Laplacian"
2771:
Kolesnik, Grigori (1982-01-01). "On the order of ζ (1/2+ it ) and Δ( R )".
2712:
943: = 2 (extending a result of Ingham who found the leading term).
4001:
Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV
3947:
3768:(2002), "Integer points, exponential sums and the Riemann zeta function",
2323:{\displaystyle N(\sigma ,T)\leq N^{{\frac {30}{13}}(1-\sigma )+\epsilon }}
3458:"Über die Beziehung zwischen Anwachsen und Nullstellen der Zeta-Funktion"
2957:
2254:
as we would expect from the Riemann hypothesis and improves the bound to
725:
3501:
3290:"Subconvexity bounds for triple L -functions and representation theory"
3195:
2491:
703:
225:
3476:(2017), "Decoupling, exponential sums and the Riemann zeta function",
3648:
3594:
3538:
3411:
Nelson, Paul D. (2021-09-30). "Bounds for standard $ L$ -functions".
3306:
1846:{\displaystyle N(\sigma ,T)\leq N^{A(\sigma )(1-\sigma )+\epsilon }}
3417:
3223:
3030:
2993:
3492:
3370:
2833:
Bombieri, E.; Iwaniec, H. (1986). "On the order of ζ (1/2+ it )".
1379:
2835:
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
1523:
of this diagram corresponds to the trivial bound coming from the
272:(1/2) = 0, which together with the above properties of
3729:(1979), "The fourth power moment of the Riemann zeta function",
4105:
3179:
2735:
2699:
Haneke, W. (1963). "Verschärfung der Abschätzung von ξ(½+it)".
3265:"'Sensational' Proof Delivers New Insights Into Prime Numbers"
4062:(2nd ed.), The Clarendon Press Oxford University Press,
3791:(2005), "Exponential sums and the Riemann zeta function. V",
295:(1/2) ≤ 1/4. The upper bound of 1/4 was lowered by
3350:; Venkatesh, Akshay (2010). "The subconvexity problem for GL
582:
The Lindelöf hypothesis is equivalent to the statement that
1593:{\displaystyle N(\sigma ,T)\leq N^{2(1-\sigma )+\epsilon }}
1116:
3980:"Quelques remarques sur la croissance de la fonction ζ(s)"
709:
and all positive real numbers ε. This has been proved for
2513:
Walfisz, Arnold (1924). "Zur Abschätzung von ζ(½ + it)".
1215:
shows that the Lindelöf hypothesis implies that, for any
2806:
Kolesnik, G. (1985). "On the method of exponent pairs".
2094:{\displaystyle A_{GM}(\sigma )={\frac {15}{5\sigma +3}}}
36:
on the critical line. This hypothesis is implied by the
3770:
Number theory for the millennium, II (Urbana, IL, 2000)
3434:"Mathematicians Clear Hurdle in Quest to Decode Primes"
2633:
Titchmarsh, E. C. (1942). "On the order of ζ(½+ it )".
1366:{\displaystyle g_{n}=O\!\left((\log p_{n})^{2}\right).}
2399:
and Andre Reznikov and in the GL(1) and GL(2) case by
2023:{\displaystyle A_{H}(\sigma )={\frac {3}{3\sigma -1}}}
256:(1 − σ) − σ + 1/2. The
3843:
Ingham, A. E. (1940), "On the estimation of N(σ,T)",
2664:
Min, Szu-Hoa (1949). "On the order of 𝜁(1/2+𝑖𝑡)".
2336:
2260:
2226:
2107:
2044:
1976:
1959:{\displaystyle A_{I}(\sigma )={\frac {3}{2-\sigma }}}
1915:
1859:
1781:
1729:
1690:
1661:
1641:
1606:
1537:
1449:
1390:
1303:
1225:
1152:
955:
737:
588:
139:
50:
1772:, and it would follow from the Lindelöf hypothesis.
4334:
4298:
4262:
4236:
4193:
4146:
3070:"Density hypothesis - Encyclopedia of Mathematics"
2383:free regions could also be expected among others.
2357:
2322:
2246:
2212:
2093:
2022:
1958:
1895:
1845:
1764:
1715:
1676:
1647:
1627:
1592:
1515:
1435:
1365:
1270:
1203:
1069:
911:
694:
244:(σ) = 0 for σ > 1, and the
201:
109:
3288:Bernstein, Joseph; Reznikov, Andre (2010-10-05).
2666:Transactions of the American Mathematical Society
1320:
1271:{\displaystyle g_{n}\ll p_{n}^{1/2+\varepsilon }}
717: = 3 seems much harder and is still an
578:Means of powers (or moments) of the zeta function
173:
143:
84:
54:
935: . This has been proved by Littlewood for
2515:Nachr. Ges. Wiss. Göttingen, math.-phys. Klasse
291:(0) = 1/2 implies that 0 ≤
3793:Proceedings of the London Mathematical Society
3731:Proceedings of the London Mathematical Society
3526:Proceedings of the London Mathematical Society
3180:"On the Difference between Consecutive Primes"
2736:"On the estimation of some trigonometric sums"
1716:{\displaystyle {\mathfrak {R}}(s)\geq \sigma }
4117:
2932:
2888:
1114:1, 1, 2, 42, 24024, 701149020, ... (sequence
8:
3479:Journal of the American Mathematical Society
1516:{\displaystyle A_{DH}(1-\sigma )=2(1-1/2)=1}
1443:. At the other extreme, the upper boundary
1293:conjecture stronger than Ingham's result is
534: > 0, the number of zeros with
2921:
2899:
4124:
4110:
4102:
3636:International Mathematics Research Notices
1765:{\displaystyle |{\mathfrak {I}}(s)|\leq T}
283:Lindelöf's convexity result together with
3937:
3665:
3647:
3593:
3537:
3491:
3416:
3379:
3369:
3315:
3305:
3222:
3029:
2992:
2345:
2341:
2335:
2287:
2286:
2259:
2236:
2225:
2202:
2184:
2169:
2148:
2133:
2112:
2106:
2101:in 2024 (preprint) and these coincide on
2070:
2049:
2043:
1999:
1981:
1975:
1938:
1920:
1914:
1874:
1864:
1858:
1807:
1780:
1751:
1736:
1735:
1730:
1728:
1692:
1691:
1689:
1660:
1640:
1605:
1563:
1536:
1496:
1454:
1448:
1416:
1395:
1389:
1349:
1339:
1308:
1302:
1252:
1248:
1243:
1230:
1224:
1189:
1170:
1157:
1151:
1050:
1034:
1012:
999:
975:
956:
954:
880:
875:
847:
835:
830:
819:
802:
793:
788:
767:
753:
747:
742:
736:
683:
663:
654:
649:
628:
614:
608:
603:
589:
587:
187:
149:
138:
98:
60:
49:
2877:
2866:
2451:
2440:
1436:{\displaystyle A_{RH}(\sigma >1/2)=0}
713: = 1 or 2, but the case
321:
4060:The theory of the Riemann zeta-function
2432:
2379:for the Legendre conjecture and better
939: = 1 and by Heath-Brown for
3918:Communications in Mathematical Physics
2910:
2855:
2851:
1204:{\displaystyle g_{n}=p_{n+1}-p_{n}.\ }
946:Conrey and Ghosh suggested the value
570: + 1 is known to be O(log(
7:
4306:Birch and Swinnerton-Dyer conjecture
3845:The Quarterly Journal of Mathematics
3357:Publications Mathématiques de l'IHÉS
3149:The Quarterly Journal of Mathematics
2635:The Quarterly Journal of Mathematics
2600:The Quarterly Journal of Mathematics
2565:The Quarterly Journal of Mathematics
2530:The Quarterly Journal of Mathematics
2330:or equivalently the asymptotics to
1737:
1693:
1527:.(Various other estimates do exist)
4392:Unsolved problems in number theory
3263:Cepelewicz, Jordana (2024-07-15).
2421:Z function#The Lindelöf hypothesis
1896:{\displaystyle x^{1-1/A(\sigma )}}
1084:is 6, and Keating and Snaith used
522:Relation to the Riemann hypothesis
248:of the zeta function implies that
14:
4351:Main conjecture of Iwasawa theory
2678:10.1090/S0002-9947-1949-0030996-6
2368:In theory improvements to Baker,
1531:The density hypothesis says that
1080:for the leading coefficient when
268:. The Lindelöf hypothesis states
3145:"ON THE ESTIMATION OF N (σ, T )"
530:of the zeta function: for every
32:about the rate of growth of the
313:approximate functional equation
240:). It is trivial to check that
4285:Ramanujan–Petersson conjecture
4275:Generalized Riemann hypothesis
4171:-functions of Hecke characters
3462:Ofversigt Finska Vetensk. Soc.
3432:Hartnett, Kevin (2022-01-13).
2773:Pacific Journal of Mathematics
2309:
2297:
2276:
2264:
2064:
2058:
1993:
1987:
1932:
1926:
1888:
1882:
1832:
1820:
1817:
1811:
1797:
1785:
1752:
1748:
1742:
1731:
1704:
1698:
1671:
1665:
1622:
1610:
1579:
1567:
1553:
1541:
1504:
1484:
1475:
1463:
1424:
1404:
1346:
1326:
1059:
1018:
1009:
986:
906:
900:
872:
865:
789:
784:
761:
754:
689:
676:
650:
645:
622:
615:
193:
180:
104:
91:
1:
4244:Analytic class number formula
319:, as in the following table:
4249:Riemann–von Mangoldt formula
3317:10.4007/annals.2010.172.1679
1628:{\displaystyle N(\sigma ,T)}
1525:Riemann-von Mangoldt formula
482:Bombieri & Iwaniec 1986
4091:Encyclopedia of Mathematics
2933:Keating & Snaith (2000)
2247:{\displaystyle \sigma =1/2}
1635:denote the number of zeros
4413:
4056:Titchmarsh, Edward Charles
3995:Motohashi, Yõichi (1995),
3890:; Voronin, Sergei (1992),
3456:Backlund, R. (1918–1919),
3003:10.1007/s00222-023-01199-0
232:such that ζ(σ +
3892:The Riemann zeta-function
3865:10.1093/qmath/os-11.1.201
3805:10.1112/S0024611504014959
3667:10.1155/S1073792898000476
3604:10.1016/j.jnt.2007.05.013
3548:10.1112/S0024611504015175
3390:10.1007/s10240-010-0025-8
3157:10.1093/qmath/os-11.1.201
2922:Conrey & Ghosh (1998)
2358:{\displaystyle x^{17/30}}
1677:{\displaystyle \zeta (s)}
554: + 1 is o(log(
538:at least 1/2 +
258:Phragmén–Lindelöf theorem
224:(σ) is defined to be the
28:by Finnish mathematician
3978:Lindelöf, Ernst (1908),
3837:10.1112/plms/s2-27.1.273
3743:10.1112/plms/s3-38.3.385
3581:Journal of Number Theory
3184:Inventiones Mathematicae
2981:Inventiones Mathematicae
2734:Kolesnik, G. A. (1973).
2643:10.1093/qmath/os-13.1.11
2577:10.1093/qmath/os-4.1.209
2542:10.1093/qmath/os-3.1.133
307:'s method of estimating
4201:Dedekind zeta functions
4084:Voronin, S.M. (2001) ,
3829:Proc. London Math. Soc.
3695:Riemann's Zeta Function
2820:10.4064/aa-45-2-115-143
2785:10.2140/pjm.1982.98.107
2411:) case by Paul Nelson.
2407:and in 2021 for the GL(
362:Hardy & Littlewood
121:tends to infinity (see
40:. It says that for any
4397:Analytic number theory
3831:, s2-27 (1): 273–300,
3178:Huxley, M. N. (1971).
3151:. os-11 (1): 201–202.
3143:Ingham, A. E. (1940).
3074:encyclopediaofmath.org
2713:10.4064/aa-8-4-357-430
2359:
2324:
2248:
2214:
2095:
2024:
1960:
1897:
1847:
1766:
1717:
1678:
1649:
1629:
1594:
1528:
1517:
1437:
1376:The density hypothesis
1367:
1297:, which asserts that
1272:
1205:
1146:-th prime number, let
1071:
913:
842:
728:: it is believed that
696:
287:(1) = 0 and
203:
111:
30:Ernst Leonard Lindelöf
4321:Bloch–Kato conjecture
4316:Beilinson conjectures
4299:Algebraic conjectures
4154:Riemann zeta function
4086:"Lindelöf hypothesis"
3948:10.1007/s002200000261
3294:Annals of Mathematics
2612:10.1093/qmath/6.1.147
2360:
2325:
2249:
2215:
2096:
2025:
1961:
1898:
1848:
1767:
1718:
1679:
1650:
1648:{\displaystyle \rho }
1630:
1595:
1518:
1438:
1383:
1368:
1273:
1206:
1072:
914:
815:
697:
204:
112:
34:Riemann zeta function
4387:Zeta and L-functions
4326:Langlands conjecture
4311:Deligne's conjecture
4263:Analytic conjectures
3776:, pp. 275–290,
2958:10.4064/aa-2-1-23-46
2889:Backlund (1918–1919)
2637:. os-13 (1): 11–17.
2427:Notes and references
2334:
2258:
2224:
2105:
2042:
1974:
1913:
1857:
1779:
1727:
1688:
1659:
1639:
1604:
1535:
1447:
1388:
1301:
1223:
1150:
1096:, and the number of
1086:random matrix theory
953:
735:
586:
228:of all real numbers
137:
48:
4280:Lindelöf hypothesis
3930:2000CMaPh.214...57K
3857:1940QJMat..11..201I
3658:1998math......7187C
3244:Scientific American
1775:More generally let
1295:Cramér's conjecture
1267:
1219: > 0,
922:for some constants
752:
613:
303:to 1/6 by applying
246:functional equation
22:Lindelöf hypothesis
4270:Riemann hypothesis
4194:Algebraic examples
4029:Sugaku Expositions
3888:Karatsuba, Anatoly
3727:Heath-Brown, D. R.
3699:Dover Publications
3196:10.1007/BF01418933
2900:Heath-Brown (1979)
2492:10.1007/BF02422942
2355:
2320:
2244:
2210:
2091:
2020:
1956:
1893:
1843:
1762:
1713:
1674:
1645:
1625:
1590:
1529:
1513:
1433:
1363:
1284:sufficiently large
1268:
1239:
1201:
1129:Other consequences
1067:
980:
909:
738:
692:
599:
199:
107:
38:Riemann hypothesis
4369:
4368:
4147:Analytic examples
4069:978-0-19-853369-6
3901:978-3-11-013170-3
3847:, Second Series,
3708:978-0-486-41740-0
3238:Bischoff, Manon.
2295:
2089:
2018:
1954:
1200:
971:
969:
702:for all positive
597:
519:
518:
157:
68:
4404:
4290:Artin conjecture
4254:Weil conjectures
4126:
4119:
4112:
4103:
4098:
4080:
4051:
4023:
3991:
3984:Bull. Sci. Math.
3974:
3941:
3912:
3883:
3839:
3823:
3795:, Third Series,
3784:
3761:
3733:, Third Series,
3721:2001 pbk reprint
3719:
3686:
3669:
3651:
3630:
3597:
3588:(6): 1516–1554,
3574:
3541:
3528:, Third Series,
3520:
3502:10.1090/jams/860
3495:
3469:
3448:
3447:
3445:
3444:
3429:
3423:
3422:
3420:
3408:
3402:
3401:
3383:
3373:
3348:Michel, Philippe
3344:
3338:
3337:
3319:
3309:
3300:(3): 1679–1718.
3285:
3279:
3278:
3276:
3275:
3260:
3254:
3253:
3251:
3250:
3235:
3229:
3228:
3226:
3214:
3208:
3207:
3175:
3169:
3168:
3140:
3134:
3133:
3131:
3130:
3115:
3109:
3108:
3106:
3105:
3090:
3084:
3083:
3081:
3080:
3066:
3060:
3059:
3057:
3056:
3042:
3036:
3035:
3033:
3021:
3015:
3014:
2996:
2987:(3): 1471–1518.
2976:
2970:
2969:
2946:Acta Arithmetica
2941:
2935:
2930:
2924:
2919:
2913:
2908:
2902:
2897:
2891:
2886:
2880:
2875:
2869:
2864:
2858:
2849:
2843:
2842:
2830:
2824:
2823:
2808:Acta Arithmetica
2803:
2797:
2796:
2768:
2762:
2761:
2759:
2758:
2740:Acta Arithmetica
2731:
2725:
2724:
2701:Acta Arithmetica
2696:
2690:
2689:
2661:
2655:
2654:
2630:
2624:
2623:
2595:
2589:
2588:
2560:
2554:
2553:
2525:
2519:
2518:
2510:
2504:
2503:
2480:Acta Mathematica
2475:
2469:
2468:
2460:
2454:
2449:
2443:
2437:
2401:Akshay Venkatesh
2397:Joseph Bernstein
2364:
2362:
2361:
2356:
2354:
2353:
2349:
2329:
2327:
2326:
2321:
2319:
2318:
2296:
2288:
2253:
2251:
2250:
2245:
2240:
2219:
2217:
2216:
2211:
2206:
2195:
2194:
2173:
2162:
2161:
2137:
2126:
2125:
2100:
2098:
2097:
2092:
2090:
2088:
2071:
2057:
2056:
2029:
2027:
2026:
2021:
2019:
2017:
2000:
1986:
1985:
1965:
1963:
1962:
1957:
1955:
1953:
1939:
1925:
1924:
1902:
1900:
1899:
1894:
1892:
1891:
1878:
1852:
1850:
1849:
1844:
1842:
1841:
1771:
1769:
1768:
1763:
1755:
1741:
1740:
1734:
1722:
1720:
1719:
1714:
1697:
1696:
1683:
1681:
1680:
1675:
1654:
1652:
1651:
1646:
1634:
1632:
1631:
1626:
1599:
1597:
1596:
1591:
1589:
1588:
1522:
1520:
1519:
1514:
1500:
1462:
1461:
1442:
1440:
1439:
1434:
1420:
1403:
1402:
1372:
1370:
1369:
1364:
1359:
1355:
1354:
1353:
1344:
1343:
1313:
1312:
1277:
1275:
1274:
1269:
1266:
1256:
1247:
1235:
1234:
1210:
1208:
1207:
1202:
1198:
1194:
1193:
1181:
1180:
1162:
1161:
1119:
1076:
1074:
1073:
1068:
1066:
1062:
1058:
1057:
1042:
1041:
1017:
1016:
1007:
1006:
979:
970:
968:
957:
918:
916:
915:
910:
893:
892:
885:
884:
858:
857:
841:
840:
839:
829:
801:
800:
792:
771:
757:
751:
746:
701:
699:
698:
693:
688:
687:
662:
661:
653:
632:
618:
612:
607:
598:
590:
416:Titchmarsh 1942
384:Titchmarsh 1932
351:Convexity bound
322:
309:exponential sums
208:
206:
205:
200:
192:
191:
172:
168:
158:
150:
116:
114:
113:
108:
103:
102:
83:
79:
69:
61:
4412:
4411:
4407:
4406:
4405:
4403:
4402:
4401:
4372:
4371:
4370:
4365:
4330:
4294:
4258:
4232:
4189:
4142:
4130:
4083:
4070:
4054:
4026:
3994:
3977:
3915:
3902:
3886:
3842:
3826:
3787:
3764:
3725:
3709:
3689:
3642:(15): 775–780,
3633:
3577:
3523:
3472:
3455:
3452:
3451:
3442:
3440:
3438:Quanta Magazine
3431:
3430:
3426:
3410:
3409:
3405:
3381:10.1.1.750.8950
3353:
3346:
3345:
3341:
3287:
3286:
3282:
3273:
3271:
3269:Quanta Magazine
3262:
3261:
3257:
3248:
3246:
3237:
3236:
3232:
3216:
3215:
3211:
3177:
3176:
3172:
3142:
3141:
3137:
3128:
3126:
3117:
3116:
3112:
3103:
3101:
3092:
3091:
3087:
3078:
3076:
3068:
3067:
3063:
3054:
3052:
3044:
3043:
3039:
3023:
3022:
3018:
2978:
2977:
2973:
2943:
2942:
2938:
2931:
2927:
2920:
2916:
2909:
2905:
2898:
2894:
2887:
2883:
2878:Bourgain (2017)
2876:
2872:
2867:Bourgain (2017)
2865:
2861:
2850:
2846:
2832:
2831:
2827:
2805:
2804:
2800:
2770:
2769:
2765:
2756:
2754:
2733:
2732:
2728:
2698:
2697:
2693:
2663:
2662:
2658:
2632:
2631:
2627:
2597:
2596:
2592:
2562:
2561:
2557:
2527:
2526:
2522:
2512:
2511:
2507:
2477:
2476:
2472:
2465:Proc. R. Soc. A
2462:
2461:
2457:
2452:Lindelöf (1908)
2450:
2446:
2441:Lindelöf (1908)
2438:
2434:
2429:
2417:
2405:Philippe Michel
2389:
2337:
2332:
2331:
2282:
2256:
2255:
2222:
2221:
2180:
2144:
2108:
2103:
2102:
2075:
2045:
2040:
2039:
2004:
1977:
1972:
1971:
1943:
1916:
1911:
1910:
1860:
1855:
1854:
1803:
1777:
1776:
1725:
1724:
1686:
1685:
1657:
1656:
1637:
1636:
1602:
1601:
1559:
1533:
1532:
1450:
1445:
1444:
1391:
1386:
1385:
1378:
1345:
1335:
1325:
1321:
1304:
1299:
1298:
1226:
1221:
1220:
1185:
1166:
1153:
1148:
1147:
1141:
1131:
1115:
1100: ×
1046:
1030:
1008:
995:
985:
981:
961:
951:
950:
934:
876:
871:
843:
831:
787:
733:
732:
679:
648:
584:
583:
580:
524:
266:convex function
214:
183:
148:
144:
135:
134:
94:
59:
55:
46:
45:
12:
11:
5:
4410:
4408:
4400:
4399:
4394:
4389:
4384:
4374:
4373:
4367:
4366:
4364:
4363:
4358:
4353:
4347:
4345:
4332:
4331:
4329:
4328:
4323:
4318:
4313:
4308:
4302:
4300:
4296:
4295:
4293:
4292:
4287:
4282:
4277:
4272:
4266:
4264:
4260:
4259:
4257:
4256:
4251:
4246:
4240:
4238:
4234:
4233:
4231:
4230:
4221:
4212:
4203:
4197:
4195:
4191:
4190:
4188:
4187:
4182:
4173:
4165:
4156:
4150:
4148:
4144:
4143:
4131:
4129:
4128:
4121:
4114:
4106:
4100:
4099:
4081:
4068:
4052:
4024:
4007:(2): 299–313,
3992:
3975:
3939:10.1.1.15.8362
3913:
3900:
3884:
3851:(1): 291–292,
3840:
3824:
3785:
3762:
3737:(3): 385–422,
3723:
3707:
3691:Edwards, H. M.
3687:
3631:
3575:
3521:
3486:(1): 205–224,
3474:Bourgain, Jean
3470:
3450:
3449:
3424:
3403:
3364:(1): 171–271.
3351:
3339:
3280:
3255:
3230:
3209:
3190:(2): 164–170.
3170:
3135:
3110:
3085:
3061:
3037:
3016:
2971:
2936:
2925:
2914:
2903:
2892:
2881:
2870:
2859:
2844:
2825:
2814:(2): 115–143.
2798:
2779:(1): 107–122.
2763:
2742:(in Russian).
2726:
2707:(4): 357–430.
2691:
2672:(3): 448–472.
2656:
2625:
2606:(1): 147–153.
2590:
2571:(1): 209–225.
2555:
2536:(1): 133–141.
2520:
2505:
2470:
2455:
2444:
2431:
2430:
2428:
2425:
2424:
2423:
2416:
2413:
2388:
2385:
2352:
2348:
2344:
2340:
2317:
2314:
2311:
2308:
2305:
2302:
2299:
2294:
2291:
2285:
2281:
2278:
2275:
2272:
2269:
2266:
2263:
2243:
2239:
2235:
2232:
2229:
2209:
2205:
2201:
2198:
2193:
2190:
2187:
2183:
2179:
2176:
2172:
2168:
2165:
2160:
2157:
2154:
2151:
2147:
2143:
2140:
2136:
2132:
2129:
2124:
2121:
2118:
2115:
2111:
2087:
2084:
2081:
2078:
2074:
2069:
2066:
2063:
2060:
2055:
2052:
2048:
2030:in 1971, and
2016:
2013:
2010:
2007:
2003:
1998:
1995:
1992:
1989:
1984:
1980:
1952:
1949:
1946:
1942:
1937:
1934:
1931:
1928:
1923:
1919:
1890:
1887:
1884:
1881:
1877:
1873:
1870:
1867:
1863:
1840:
1837:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1806:
1802:
1799:
1796:
1793:
1790:
1787:
1784:
1761:
1758:
1754:
1750:
1747:
1744:
1739:
1733:
1712:
1709:
1706:
1703:
1700:
1695:
1673:
1670:
1667:
1664:
1644:
1624:
1621:
1618:
1615:
1612:
1609:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1562:
1558:
1555:
1552:
1549:
1546:
1543:
1540:
1512:
1509:
1506:
1503:
1499:
1495:
1492:
1489:
1486:
1483:
1480:
1477:
1474:
1471:
1468:
1465:
1460:
1457:
1453:
1432:
1429:
1426:
1423:
1419:
1415:
1412:
1409:
1406:
1401:
1398:
1394:
1377:
1374:
1362:
1358:
1352:
1348:
1342:
1338:
1334:
1331:
1328:
1324:
1319:
1316:
1311:
1307:
1265:
1262:
1259:
1255:
1251:
1246:
1242:
1238:
1233:
1229:
1197:
1192:
1188:
1184:
1179:
1176:
1173:
1169:
1165:
1160:
1156:
1137:
1130:
1127:
1126:
1125:
1105:Young tableaux
1078:
1077:
1065:
1061:
1056:
1053:
1049:
1045:
1040:
1037:
1033:
1029:
1026:
1023:
1020:
1015:
1011:
1005:
1002:
998:
994:
991:
988:
984:
978:
974:
967:
964:
960:
926:
920:
919:
908:
905:
902:
899:
896:
891:
888:
883:
879:
874:
870:
867:
864:
861:
856:
853:
850:
846:
838:
834:
828:
825:
822:
818:
814:
811:
808:
805:
799:
796:
791:
786:
783:
780:
777:
774:
770:
766:
763:
760:
756:
750:
745:
741:
691:
686:
682:
678:
675:
672:
669:
666:
660:
657:
652:
647:
644:
641:
638:
635:
631:
627:
624:
621:
617:
611:
606:
602:
596:
593:
579:
576:
544:imaginary part
523:
520:
517:
516:
513:
510:
506:
505:
502:
499:
495:
494:
491:
488:
484:
483:
480:
477:
473:
472:
471:Kolesnik 1985
469:
466:
462:
461:
460:Kolesnik 1982
458:
455:
451:
450:
449:Kolesnik 1973
447:
444:
440:
439:
436:
433:
429:
428:
425:
422:
418:
417:
414:
411:
407:
406:
403:
400:
397:
396:
395:Phillips 1933
393:
390:
386:
385:
382:
379:
375:
374:
371:
368:
364:
363:
360:
357:
353:
352:
349:
346:
343:
339:
338:
335:
329:
213:
212:The μ function
210:
198:
195:
190:
186:
182:
179:
176:
171:
167:
164:
161:
156:
153:
147:
142:
123:big O notation
106:
101:
97:
93:
90:
87:
82:
78:
75:
72:
67:
64:
58:
53:
13:
10:
9:
6:
4:
3:
2:
4409:
4398:
4395:
4393:
4390:
4388:
4385:
4383:
4380:
4379:
4377:
4362:
4359:
4357:
4354:
4352:
4349:
4348:
4346:
4344:
4342:
4338:
4333:
4327:
4324:
4322:
4319:
4317:
4314:
4312:
4309:
4307:
4304:
4303:
4301:
4297:
4291:
4288:
4286:
4283:
4281:
4278:
4276:
4273:
4271:
4268:
4267:
4265:
4261:
4255:
4252:
4250:
4247:
4245:
4242:
4241:
4239:
4235:
4229:
4227:
4222:
4220:
4218:
4213:
4211:
4209:
4204:
4202:
4199:
4198:
4196:
4192:
4186:
4185:Selberg class
4183:
4181:
4179:
4174:
4172:
4170:
4166:
4164:
4162:
4157:
4155:
4152:
4151:
4149:
4145:
4141:
4140:number theory
4137:
4135:
4127:
4122:
4120:
4115:
4113:
4108:
4107:
4104:
4097:
4093:
4092:
4087:
4082:
4079:
4075:
4071:
4065:
4061:
4057:
4053:
4050:
4046:
4042:
4038:
4034:
4030:
4025:
4022:
4018:
4014:
4010:
4006:
4002:
3998:
3993:
3989:
3985:
3981:
3976:
3973:
3969:
3965:
3961:
3957:
3953:
3949:
3945:
3940:
3935:
3931:
3927:
3923:
3919:
3914:
3911:
3907:
3903:
3897:
3893:
3889:
3885:
3882:
3878:
3874:
3870:
3866:
3862:
3858:
3854:
3850:
3846:
3841:
3838:
3834:
3830:
3825:
3822:
3818:
3814:
3810:
3806:
3802:
3798:
3794:
3790:
3789:Huxley, M. N.
3786:
3783:
3779:
3775:
3771:
3767:
3766:Huxley, M. N.
3763:
3760:
3756:
3752:
3748:
3744:
3740:
3736:
3732:
3728:
3724:
3722:
3718:
3714:
3710:
3704:
3700:
3696:
3692:
3688:
3685:
3681:
3677:
3673:
3668:
3663:
3659:
3655:
3650:
3645:
3641:
3637:
3632:
3629:
3625:
3621:
3617:
3613:
3609:
3605:
3601:
3596:
3591:
3587:
3583:
3582:
3576:
3573:
3569:
3565:
3561:
3557:
3553:
3549:
3545:
3540:
3535:
3532:(1): 33–104,
3531:
3527:
3522:
3519:
3515:
3511:
3507:
3503:
3499:
3494:
3489:
3485:
3481:
3480:
3475:
3471:
3467:
3463:
3459:
3454:
3453:
3439:
3435:
3428:
3425:
3419:
3414:
3407:
3404:
3399:
3395:
3391:
3387:
3382:
3377:
3372:
3367:
3363:
3359:
3358:
3349:
3343:
3340:
3335:
3331:
3327:
3323:
3318:
3313:
3308:
3303:
3299:
3295:
3291:
3284:
3281:
3270:
3266:
3259:
3256:
3245:
3241:
3234:
3231:
3225:
3220:
3213:
3210:
3205:
3201:
3197:
3193:
3189:
3185:
3181:
3174:
3171:
3166:
3162:
3158:
3154:
3150:
3146:
3139:
3136:
3124:
3120:
3114:
3111:
3099:
3095:
3089:
3086:
3075:
3071:
3065:
3062:
3051:
3047:
3041:
3038:
3032:
3027:
3020:
3017:
3012:
3008:
3004:
3000:
2995:
2990:
2986:
2982:
2975:
2972:
2967:
2963:
2959:
2955:
2951:
2947:
2940:
2937:
2934:
2929:
2926:
2923:
2918:
2915:
2912:
2911:Ingham (1928)
2907:
2904:
2901:
2896:
2893:
2890:
2885:
2882:
2879:
2874:
2871:
2868:
2863:
2860:
2857:
2856:Huxley (2005)
2853:
2852:Huxley (2002)
2848:
2845:
2841:(3): 449–472.
2840:
2836:
2829:
2826:
2821:
2817:
2813:
2809:
2802:
2799:
2794:
2790:
2786:
2782:
2778:
2774:
2767:
2764:
2753:
2749:
2745:
2741:
2737:
2730:
2727:
2722:
2718:
2714:
2710:
2706:
2703:(in German).
2702:
2695:
2692:
2687:
2683:
2679:
2675:
2671:
2667:
2660:
2657:
2652:
2648:
2644:
2640:
2636:
2629:
2626:
2621:
2617:
2613:
2609:
2605:
2601:
2594:
2591:
2586:
2582:
2578:
2574:
2570:
2566:
2559:
2556:
2551:
2547:
2543:
2539:
2535:
2531:
2524:
2521:
2516:
2509:
2506:
2501:
2497:
2493:
2489:
2485:
2481:
2474:
2471:
2466:
2459:
2456:
2453:
2448:
2445:
2442:
2436:
2433:
2426:
2422:
2419:
2418:
2414:
2412:
2410:
2406:
2402:
2398:
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2386:
2384:
2382:
2378:
2375:
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2366:
2350:
2346:
2342:
2338:
2315:
2312:
2306:
2303:
2300:
2292:
2289:
2283:
2279:
2273:
2270:
2267:
2261:
2241:
2237:
2233:
2230:
2227:
2207:
2203:
2199:
2196:
2191:
2188:
2185:
2181:
2177:
2174:
2170:
2166:
2163:
2158:
2155:
2152:
2149:
2145:
2141:
2138:
2134:
2130:
2127:
2122:
2119:
2116:
2113:
2109:
2085:
2082:
2079:
2076:
2072:
2067:
2061:
2053:
2050:
2046:
2037:
2033:
2014:
2011:
2008:
2005:
2001:
1996:
1990:
1982:
1978:
1969:
1950:
1947:
1944:
1940:
1935:
1929:
1921:
1917:
1908:
1904:
1885:
1879:
1875:
1871:
1868:
1865:
1861:
1838:
1835:
1829:
1826:
1823:
1814:
1808:
1804:
1800:
1794:
1791:
1788:
1782:
1773:
1759:
1756:
1745:
1710:
1707:
1701:
1668:
1662:
1642:
1619:
1616:
1613:
1607:
1585:
1582:
1576:
1573:
1570:
1564:
1560:
1556:
1550:
1547:
1544:
1538:
1526:
1510:
1507:
1501:
1497:
1493:
1490:
1487:
1481:
1478:
1472:
1469:
1466:
1458:
1455:
1451:
1430:
1427:
1421:
1417:
1413:
1410:
1407:
1399:
1396:
1392:
1382:
1375:
1373:
1360:
1356:
1350:
1340:
1336:
1332:
1329:
1322:
1317:
1314:
1309:
1305:
1296:
1292:
1287:
1285:
1281:
1263:
1260:
1257:
1253:
1249:
1244:
1240:
1236:
1231:
1227:
1218:
1214:
1213:Albert Ingham
1195:
1190:
1186:
1182:
1177:
1174:
1171:
1167:
1163:
1158:
1154:
1145:
1140:
1136:
1128:
1123:
1118:
1113:
1112:
1111:
1110:
1107:given by the
1106:
1103:
1099:
1095:
1091:
1087:
1083:
1063:
1054:
1051:
1047:
1043:
1038:
1035:
1031:
1027:
1024:
1021:
1013:
1003:
1000:
996:
992:
989:
982:
976:
972:
965:
962:
958:
949:
948:
947:
944:
942:
938:
933:
929:
925:
903:
897:
894:
889:
886:
881:
877:
868:
862:
859:
854:
851:
848:
844:
836:
832:
826:
823:
820:
816:
812:
809:
806:
803:
797:
794:
781:
778:
775:
772:
768:
764:
758:
748:
743:
739:
731:
730:
729:
727:
722:
720:
716:
712:
708:
705:
684:
680:
673:
670:
667:
664:
658:
655:
642:
639:
636:
633:
629:
625:
619:
609:
604:
600:
594:
591:
577:
575:
573:
569:
565:
561:
557:
553:
549:
545:
541:
537:
533:
529:
521:
514:
511:
508:
507:
503:
500:
497:
496:
492:
489:
486:
485:
481:
478:
475:
474:
470:
467:
464:
463:
459:
456:
453:
452:
448:
445:
442:
441:
437:
434:
431:
430:
426:
423:
420:
419:
415:
412:
409:
408:
404:
401:
399:
398:
394:
391:
388:
387:
383:
380:
377:
376:
373:Walfisz 1924
372:
369:
366:
365:
361:
358:
355:
354:
350:
347:
344:
341:
340:
336:
334:(1/2) ≤
333:
330:
328:(1/2) ≤
327:
324:
323:
320:
318:
314:
310:
306:
302:
298:
294:
290:
286:
281:
279:
276:implies that
275:
271:
267:
263:
260:implies that
259:
255:
251:
247:
243:
239:
236: ) = O(
235:
231:
227:
223:
219:
211:
209:
196:
188:
184:
177:
174:
169:
165:
162:
159:
154:
151:
145:
140:
132:
128:
124:
120:
99:
95:
88:
85:
80:
76:
73:
70:
65:
62:
56:
51:
43:
39:
35:
31:
27:
23:
19:
4361:Euler system
4356:Selmer group
4340:
4336:
4279:
4225:
4216:
4207:
4177:
4176:Automorphic
4168:
4160:
4133:
4089:
4059:
4035:(1): 59–87,
4032:
4028:
4004:
4000:
3987:
3983:
3924:(1): 57–89,
3921:
3917:
3891:
3848:
3844:
3828:
3796:
3792:
3769:
3734:
3730:
3697:, New York:
3694:
3649:math/9807187
3639:
3635:
3595:math/0612843
3585:
3579:
3539:math/0206018
3529:
3525:
3483:
3477:
3465:
3461:
3441:. Retrieved
3437:
3427:
3406:
3361:
3355:
3342:
3307:math/0608555
3297:
3293:
3283:
3272:. Retrieved
3268:
3258:
3247:. Retrieved
3243:
3233:
3212:
3187:
3183:
3173:
3148:
3138:
3127:. Retrieved
3125:. 2024-06-04
3122:
3113:
3102:. Retrieved
3100:. 2024-06-04
3097:
3088:
3077:. Retrieved
3073:
3064:
3053:. Retrieved
3049:
3040:
3019:
2984:
2980:
2974:
2952:(1): 23–46.
2949:
2945:
2939:
2928:
2917:
2906:
2895:
2884:
2873:
2862:
2847:
2838:
2834:
2828:
2811:
2807:
2801:
2776:
2772:
2766:
2755:. Retrieved
2743:
2739:
2729:
2704:
2700:
2694:
2669:
2665:
2659:
2634:
2628:
2603:
2599:
2593:
2568:
2564:
2558:
2533:
2529:
2523:
2514:
2508:
2483:
2479:
2473:
2464:
2458:
2447:
2435:
2408:
2390:
2381:Siegel zeros
2367:
1909:showed that
1905:
1774:
1530:
1288:
1279:
1216:
1211:A result by
1143:
1138:
1134:
1133:Denoting by
1132:
1101:
1097:
1089:
1081:
1079:
945:
940:
936:
931:
927:
923:
921:
723:
719:open problem
714:
710:
706:
581:
571:
567:
563:
559:
555:
551:
547:
539:
531:
525:
438:Haneke 1962
405:Rankin 1955
331:
325:
292:
288:
284:
282:
277:
273:
269:
261:
253:
249:
241:
237:
233:
229:
221:
215:
130:
126:
118:
41:
21:
15:
4382:Conjectures
4215:Hasse–Weil
3799:(1): 1–41,
3123:www.ias.edu
3098:www.ias.edu
2746:(1): 7–30.
2486:: 119–196.
2393:L-functions
2387:L-functions
18:mathematics
4376:Categories
4343:-functions
4228:-functions
4219:-functions
4210:-functions
4180:-functions
4163:-functions
4159:Dirichlet
4136:-functions
3774:A K Peters
3443:2022-02-17
3418:2109.15230
3274:2024-07-16
3249:2024-07-16
3224:2405.20552
3129:2024-07-16
3104:2024-07-16
3079:2024-07-16
3055:2024-07-16
3050:aimath.org
3031:2306.05599
2994:1908.08613
2757:2024-02-05
2517:: 155–158.
2467:: 403–412.
301:Littlewood
26:conjecture
4096:EMS Press
4041:0898-9583
4013:0391-173X
3990:: 341–356
3956:0010-3616
3934:CiteSeerX
3873:0033-5606
3813:0024-6115
3751:0024-6115
3676:1073-7928
3612:0022-314X
3556:0024-6115
3518:118064221
3493:1408.5794
3376:CiteSeerX
3371:0903.3591
3326:0003-486X
3204:0020-9910
3165:0033-5606
3011:0020-9910
2966:0065-1036
2793:0030-8730
2752:0065-1036
2721:0065-1036
2686:0002-9947
2651:0033-5606
2620:0033-5606
2585:0033-5606
2550:0033-5606
2500:0001-5962
2377:estimates
2316:ϵ
2307:σ
2304:−
2280:≤
2268:σ
2228:σ
2182:σ
2146:σ
2110:σ
2080:σ
2062:σ
2012:−
2009:σ
1991:σ
1966:in 1940,
1951:σ
1948:−
1930:σ
1886:σ
1869:−
1839:ϵ
1830:σ
1827:−
1815:σ
1801:≤
1789:σ
1757:≤
1711:σ
1708:≥
1663:ζ
1643:ρ
1614:σ
1586:ϵ
1577:σ
1574:−
1557:≤
1545:σ
1491:−
1473:σ
1470:−
1408:σ
1333:
1291:prime gap
1264:ε
1237:≪
1183:−
1052:−
1036:−
1001:−
993:−
973:∏
887:−
863:
817:∑
759:ζ
740:∫
685:ε
620:ζ
601:∫
536:real part
515:Bourgain
504:Bourgain
443:173/1067
427:Min 1949
402:0.164511
392:0.164512
389:229/1392
348:Lindelöf
189:ε
141:ζ
125:). Since
100:ε
52:ζ
4237:Theorems
4224:Motivic
4058:(1986),
3972:11095649
3693:(1974),
3628:15922788
3398:14155294
3334:14745024
2415:See also
1600:, where
1109:sequence
726:integral
704:integers
546:between
468:0.16201
465:139/858
457:0.16204
446:0.16214
435:0.16217
367:163/988
216:If σ is
44:> 0,
4078:0882550
4049:1335956
4021:1354909
3964:1794265
3926:Bibcode
3910:1183467
3881:0003649
3853:Bibcode
3821:2107036
3782:1956254
3759:0532980
3717:0466039
3684:1639551
3654:Bibcode
3620:2419176
3572:1435033
3564:2149530
3510:3556291
2036:Maynard
1120:in the
1117:A039622
512:0.1548
501:0.1550
498:53/342
493:Huxley
490:0.1561
487:32/205
479:0.1608
454:35/216
424:0.1631
413:0.1638
410:19/116
381:0.1647
378:27/164
370:0.1650
359:0.1667
337:Author
311:to the
226:infimum
220:, then
4339:-adic
4206:Artin
4076:
4066:
4047:
4039:
4019:
4011:
3970:
3962:
3954:
3936:
3908:
3898:
3879:
3871:
3819:
3811:
3780:
3757:
3749:
3715:
3705:
3682:
3674:
3626:
3618:
3610:
3570:
3562:
3554:
3516:
3508:
3396:
3378:
3332:
3324:
3202:
3163:
3009:
2964:
2791:
2750:
2719:
2684:
2649:
2618:
2583:
2548:
2498:
2372:, and
2370:Harman
1968:Huxley
1907:Ingham
1199:
1094:primes
558:)) as
509:13/84
421:15/92
317:proofs
252:(σ) =
20:, the
3968:S2CID
3644:arXiv
3624:S2CID
3590:arXiv
3568:S2CID
3534:arXiv
3514:S2CID
3488:arXiv
3413:arXiv
3394:S2CID
3366:arXiv
3330:S2CID
3302:arXiv
3219:arXiv
3046:"25a"
3026:arXiv
2989:arXiv
2374:Pintz
2038:that
1970:that
1684:with
528:zeros
476:9/56
432:6/37
345:0.25
297:Hardy
264:is a
24:is a
4064:ISBN
4037:ISSN
4009:ISSN
3952:ISSN
3896:ISBN
3869:ISSN
3809:ISSN
3747:ISSN
3703:ISBN
3672:ISSN
3640:1998
3608:ISSN
3552:ISSN
3322:ISSN
3200:ISSN
3161:ISSN
3007:ISSN
2962:ISSN
2789:ISSN
2748:ISSN
2717:ISSN
2682:ISSN
2647:ISSN
2616:ISSN
2581:ISSN
2569:os-4
2546:ISSN
2534:os-3
2496:ISSN
2439:see
2403:and
2178:<
2142:<
2034:and
2032:Guth
1723:and
1411:>
1142:the
1122:OEIS
566:and
550:and
542:and
356:1/6
342:1/4
305:Weyl
299:and
218:real
4138:in
3944:doi
3922:214
3861:doi
3833:doi
3801:doi
3739:doi
3662:doi
3600:doi
3586:128
3544:doi
3498:doi
3468:(9)
3386:doi
3362:111
3354:".
3312:doi
3298:172
3192:doi
3153:doi
2999:doi
2985:233
2954:doi
2816:doi
2781:doi
2709:doi
2674:doi
2639:doi
2608:doi
2573:doi
2538:doi
2488:doi
1655:of
1330:log
1282:is
1278:if
860:log
117:as
16:In
4378::
4094:,
4088:,
4074:MR
4072:,
4045:MR
4043:,
4031:,
4017:MR
4015:,
4005:22
4003:,
3999:,
3988:32
3986:,
3982:,
3966:,
3960:MR
3958:,
3950:,
3942:,
3932:,
3920:,
3906:MR
3904:,
3877:MR
3875:,
3867:,
3859:,
3849:11
3817:MR
3815:,
3807:,
3797:90
3778:MR
3772:,
3755:MR
3753:,
3745:,
3735:38
3713:MR
3711:,
3701:,
3680:MR
3678:,
3670:,
3660:,
3652:,
3638:,
3622:,
3616:MR
3614:,
3606:,
3598:,
3584:,
3566:,
3560:MR
3558:,
3550:,
3542:,
3530:91
3512:,
3506:MR
3504:,
3496:,
3484:30
3482:,
3466:61
3464:,
3460:,
3436:.
3392:.
3384:.
3374:.
3360:.
3328:.
3320:.
3310:.
3296:.
3292:.
3267:.
3242:.
3198:.
3188:15
3186:.
3182:.
3159:.
3147:.
3121:.
3096:.
3072:.
3048:.
3005:.
2997:.
2983:.
2960:.
2948:.
2854:,
2839:13
2837:.
2812:45
2810:.
2787:.
2777:98
2775:.
2744:25
2738:.
2715:.
2680:.
2670:65
2668:.
2645:.
2614:.
2602:.
2579:.
2567:.
2544:.
2532:.
2494:.
2484:41
2482:.
2365:.
2351:30
2343:17
2293:13
2290:30
2175:10
2139:10
2073:15
1903:.
1289:A
1286:.
1124:).
959:42
721:.
234:iT
133:,
4341:L
4337:p
4226:L
4217:L
4208:L
4178:L
4169:L
4161:L
4134:L
4125:e
4118:t
4111:v
4033:8
3946::
3928::
3863::
3855::
3835::
3803::
3741::
3664::
3656::
3646::
3602::
3592::
3546::
3536::
3500::
3490::
3446:.
3421:.
3415::
3400:.
3388::
3368::
3352:2
3336:.
3314::
3304::
3277:.
3252:.
3227:.
3221::
3206:.
3194::
3167:.
3155::
3132:.
3107:.
3082:.
3058:.
3034:.
3028::
3013:.
3001::
2991::
2968:.
2956::
2950:2
2822:.
2818::
2795:.
2783::
2760:.
2723:.
2711::
2705:8
2688:.
2676::
2653:.
2641::
2622:.
2610::
2604:6
2587:.
2575::
2552:.
2540::
2502:.
2490::
2409:n
2347:/
2339:x
2313:+
2310:)
2301:1
2298:(
2284:N
2277:)
2274:T
2271:,
2265:(
2262:N
2242:2
2238:/
2234:1
2231:=
2208:4
2204:/
2200:3
2197:=
2192:H
2189:,
2186:I
2171:/
2167:8
2164:=
2159:M
2156:G
2153:,
2150:H
2135:/
2131:7
2128:=
2123:M
2120:G
2117:,
2114:I
2086:3
2083:+
2077:5
2068:=
2065:)
2059:(
2054:M
2051:G
2047:A
2015:1
2006:3
2002:3
1997:=
1994:)
1988:(
1983:H
1979:A
1945:2
1941:3
1936:=
1933:)
1927:(
1922:I
1918:A
1889:)
1883:(
1880:A
1876:/
1872:1
1866:1
1862:x
1836:+
1833:)
1824:1
1821:(
1818:)
1812:(
1809:A
1805:N
1798:)
1795:T
1792:,
1786:(
1783:N
1760:T
1753:|
1749:)
1746:s
1743:(
1738:I
1732:|
1705:)
1702:s
1699:(
1694:R
1672:)
1669:s
1666:(
1623:)
1620:T
1617:,
1611:(
1608:N
1583:+
1580:)
1571:1
1568:(
1565:2
1561:N
1554:)
1551:T
1548:,
1542:(
1539:N
1511:1
1508:=
1505:)
1502:2
1498:/
1494:1
1488:1
1485:(
1482:2
1479:=
1476:)
1467:1
1464:(
1459:H
1456:D
1452:A
1431:0
1428:=
1425:)
1422:2
1418:/
1414:1
1405:(
1400:H
1397:R
1393:A
1361:.
1357:)
1351:2
1347:)
1341:n
1337:p
1327:(
1323:(
1318:O
1315:=
1310:n
1306:g
1280:n
1261:+
1258:2
1254:/
1250:1
1245:n
1241:p
1232:n
1228:g
1217:ε
1196:.
1191:n
1187:p
1178:1
1175:+
1172:n
1168:p
1164:=
1159:n
1155:g
1144:n
1139:n
1135:p
1102:n
1098:n
1090:k
1082:k
1064:)
1060:)
1055:2
1048:p
1044:+
1039:1
1032:p
1028:4
1025:+
1022:1
1019:(
1014:4
1010:)
1004:1
997:p
990:1
987:(
983:(
977:p
966:!
963:9
941:k
937:k
932:j
930:,
928:k
924:c
907:)
904:T
901:(
898:o
895:+
890:j
882:2
878:k
873:)
869:T
866:(
855:j
852:,
849:k
845:c
837:2
833:k
827:0
824:=
821:j
813:T
810:=
807:t
804:d
798:k
795:2
790:|
785:)
782:t
779:i
776:+
773:2
769:/
765:1
762:(
755:|
749:T
744:0
715:k
711:k
707:k
690:)
681:T
677:(
674:O
671:=
668:t
665:d
659:k
656:2
651:|
646:)
643:t
640:i
637:+
634:2
630:/
626:1
623:(
616:|
610:T
605:0
595:T
592:1
572:T
568:T
564:T
560:T
556:T
552:T
548:T
540:ε
532:ε
332:μ
326:μ
293:μ
289:μ
285:μ
278:μ
274:μ
270:μ
262:μ
254:μ
250:μ
242:μ
238:T
230:a
222:μ
197:.
194:)
185:t
181:(
178:o
175:=
170:)
166:t
163:i
160:+
155:2
152:1
146:(
131:ε
127:ε
119:t
105:)
96:t
92:(
89:O
86:=
81:)
77:t
74:i
71:+
66:2
63:1
57:(
42:ε
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