1648:
1456:
3944:
2674:
3661:
1779:
1154:
226:
1643:{\displaystyle K(a,b;m)={\begin{cases}2\left({\frac {\ell }{m}}\right){\sqrt {m}}{\text{ Re}}\left(\varepsilon _{m}e^{\frac {4\pi i\ell }{m}}\right)&\left({\tfrac {a}{p}}\right)=\left({\tfrac {b}{p}}\right)\\0&{\text{otherwise}}\end{cases}}}
1363:
3961:. The Kuznetsov trace formula was found by Kuznetsov while studying the growth of weight zero automorphic functions. Using estimates on Kloosterman sums he was able to derive estimates for Fourier coefficients of modular forms in cases where
685:
977:
3808:
868:
2205:
3118:
2807:
1904:
3486:
1794:
Because
Kloosterman sums occur in the Fourier expansion of modular forms, estimates for Kloosterman sums yield estimates for Fourier coefficients of modular forms as well. The most famous estimate is due to
3337:
2532:
3005:
2929:
3463:
2488:
1404:
3792:
3718:
2018:
2334:
1687:
1011:
785:
3207:
2398:
developed a new method of estimating short
Kloosterman sums. Karatsuba's method makes it possible to estimate Kloosterman's sums, the number of summands in which does not exceed
71:
2514:
3408:
2859:
2423:
1225:
3163:
4015:
1936:
2720:
3367:
2277:
2246:
2070:
as earlier reference for the idea; given Weil's rather denigratory remark on the abilities of analytic number theorists to work out this example themselves, in his
3957:
and the spectral part is a sum of
Fourier coefficients, taken over spaces of holomorphic and non-holomorphic modular forms twisted with some integral transform of
3939:{\displaystyle \sum _{c\equiv 0{\bmod {N}}}c^{-r}K(m,n,c)g\left({\frac {4\pi {\sqrt {mn}}}{c}}\right)={\text{ Integral transform }}\ +\ {\text{ Spectral terms}}.}
533:
3349:
Although the
Kloosterman sums may not be calculated in general they may be "lifted" to algebraic number fields, which often yields more convenient formulas. Let
893:
796:
4377:
2118:
3029:
2088:
This technique in fact shows much more generally that complete exponential sums 'along' algebraic varieties have good estimates, depending on the
2744:
1805:
3656:{\displaystyle K(a,b;m)=(-1)^{\Omega (m)}\sum _{\stackrel {v,w{\bmod {m}}}{v^{2}-\tau w^{2}\equiv ab{\bmod {m}}}}e^{\frac {4\pi iv}{m}}.}
33:
4515:
4476:
4442:
3277:
2338:
Up to the early 1990s, estimates for sums of this type were known mainly in the case where the number of summands was greater than
4096:
by
Kuznetsov in 1979, which contained some 'savings on average' over the square root estimate, there were further developments by
2066:
that are function fields, for which Weil gives a 1938 paper of J. Weissinger as reference (the next year he gave a 1935 paper of
2669:{\displaystyle {\sum _{n\leq x}}'\left\{{\frac {an^{*}+bn}{m}}\right\},{\sum _{p\leq x}}'\left\{{\frac {ap^{*}+bp}{m}}\right\},}
4618:
4430:
2395:
2938:
2875:
2519:
Various aspects of the method of
Karatsuba found applications in solving the following problems of analytic number theory:
4468:
2384:
3424:
2428:
280:
in short intervals, primes in arithmetic progressions, the spectral theory of automorphic functions and related topics.
3017:
2516:
is an arbitrarily small fixed number. The last paper of A.A. Karatsuba on this subject was published after his death.
2388:
2354:
1372:
3761:
3685:
4507:
37:
4093:
1956:
4067:
1774:{\displaystyle \varepsilon _{m}={\begin{cases}1&m\equiv 1{\bmod {4}}\\i&m\equiv 3{\bmod {4}}\end{cases}}}
2290:
1942:. Because of the multiplicative properties of Kloosterman sums these estimates may be reduced to the case where
1201:
4110:(1982). Subsequent applications to analytic number theory were worked out by a number of authors, particularly
4106:
1204:
suggests that none exist. The lifting formulas below, however, are often as good as an explicit evaluation. If
1149:{\displaystyle K(a,b;m)=\sum _{d\mid \gcd(a,b,m)}d\cdot K\left({\tfrac {ab}{d^{2}}},1;{\tfrac {m}{d}}\right).}
2048:
3745:
221:{\displaystyle K(a,b;m)=\sum _{\stackrel {0\leq x\leq m-1}{\gcd(x,m)=1}}e^{{\frac {2\pi i}{m}}(ax+bx^{*})}.}
2866:
the precision of approximation of an arbitrary real number in the segment by fractional parts of the form:
762:
4101:
3176:
4400:
4018:
3973:
3729:
273:
4125:, vol. I (Kendrick press, 2003). Also relevant for students and researchers interested in the field is
2493:
1358:{\displaystyle K(a,a;p)=\sum _{m=0}^{p-1}\left({\frac {m^{2}-4a^{2}}{p}}\right)e^{\frac {2\pi im}{p}},}
3372:
2834:
2401:
2074:, these ideas were presumably 'folklore' of quite long standing). The non-polar factors are of type
1709:
1492:
4495:
4082:
4051:
2350:
2028:
29:
4338:
Petersson's conjecture for forms of weight zero and Linnik's conjecture. Sums of
Kloosterman sums
4059:
3950:
3679:
3127:
254:
239:
3994:
4565:
4511:
4472:
4438:
4078:
4022:
3241:
1912:
1785:
This formula was first found by Hans Salie and there are many simple proofs in the literature.
2687:
4545:
4521:
4482:
4448:
4392:
4266:
Karatsuba, A. A. (1997). "Analogues of incomplete
Kloosterman sums and their applications".
4144:
3966:
3755:
3725:
3258:
3210:
2097:
2089:
2059:
680:{\displaystyle K(a,b;m)=K\left(n_{2}a,n_{2}b;m_{1}\right)K\left(n_{1}a,n_{1}b;m_{2}\right).}
4437:. de Gruyter Expositions in Mathematics. Vol. 39. BerlinâNew-York: Walter de Gruyter.
3352:
2255:
2225:
972:{\displaystyle \mathbb {Q} \left(\zeta _{2^{\alpha -1}}+\zeta _{2^{\alpha -1}}^{-1}\right)}
44:
in four variables, strengthening his 1924 dissertation research on five or more variables.
4549:
4525:
4486:
4452:
4118:
4111:
4030:
3981:
2358:
2040:
1165:
258:
62:
25:
4500:
4460:
4097:
4081:
on modular forms. Hans Salié introduced a form of
Kloosterman sum that is twisted by a
3962:
3262:
3214:
2093:
863:{\displaystyle \mathbb {Q} \left(\zeta _{p^{\alpha }}+\zeta _{p^{\alpha }}^{-1}\right)}
41:
4327:, Transactions of the American Mathematical Society 350(12), Pages: 5003-5015, (1998).
4612:
4597:
4568:
4533:
2362:
2101:
1796:
1407:
1169:
304:
262:
4325:
The lifting of an exponential sum to a cyclic algebraic number field of prime degree
3222:
a lower bound for the greatest prime divisor of the product of numbers of the form:
2735:
finding the lower bound for the number of solutions of the inequalities of the form:
4132:
4074:
3985:
3795:
2067:
2063:
1161:
277:
4506:. Encyclopedia of Mathematics and Its Applications. Vol. 20 (2nd ed.).
4029:, the relative trace formula is a tool for studying the harmonic analysis on the
4396:
4241:, Transactions of the American Mathematical Society 30(1), pages: 61â62, (1971).
2085:
is a
Kloosterman sum. The estimate then follows from Weil's basic work of 1940.
737:
17:
3678:
counting multiplicity. The sum on the right can be reinterpreted as a sum over
4588:
3754:
formula connects Kloosterman sums at a deep level with the spectral theory of
3721:
269:
2200:{\displaystyle \sum _{n\in A}\exp \left(2\pi i{\frac {an^{*}+bn}{m}}\right),}
4573:
4063:
3113:{\displaystyle \pi (x;q,l)<{\frac {cx}{\varphi (q)\ln {\frac {2x}{q}}}},}
4092:
After the discovery of important formulae connecting Kloosterman sums with
2802:{\displaystyle \alpha <\left\{{\frac {an^{*}+bn}{m}}\right\}\leq \beta }
4197:
Over het splitsen van geheele positieve getallen in een some van kwadraten
4583:
4117:
The field remains somewhat inaccessible. A detailed introduction to the
1899:{\displaystyle |K(a,b;m)|\leq \tau (m){\sqrt {\gcd(a,b,m)}}{\sqrt {m}}.}
4058:, 2nd ed. (Kendrick Press, 2004). The underlying ideas here are due to
2684:
runs, one after another, through the integers satisfying the condition
4077:. In fact the sums first appeared (minus the name) in a 1912 paper of
4121:
needed to understand the Kuznetsov formulae is given in R. C. Baker,
2112:
Short Kloosterman sums are defined as trigonometric sums of the form
4296:
Karatsuba, A. A. (2010). "New estimates of short Kloosterman sums".
4135:
used Kloosterman sums in his proof of bounded gaps between primes.
691:
This reduces the evaluation of Kloosterman sums to the case where
4210:
An elementary proof of a formula of Kuznecov for Kloosterman sums
4042:. For an overview and numerous applications see the references.
2523:
finding asymptotics of the sums of fractional parts of the form:
3268:
combinatorial properties of the set of numbers (A.A.Glibichuk):
3332:{\displaystyle n^{*}{\bmod {m}},1\leq n\leq m^{\varepsilon }.}
3827:
3610:
3589:
3292:
3190:
2314:
1755:
1727:
3798:" function. Then one calls identities of the following type
3736:. Indeed, much more general exponential sums can be lifted.
4251:
Karatsuba, A. A. (1995). "Analogues of Kloostermans sums".
3247:
proving that there are infinitely many primes of the form:
2365:. The only exceptions were the special modules of the form
1767:
1636:
3758:. Originally this could have been stated as follows. Let
4073:
There are many connections between Kloosterman sums and
2092:
in dimension > 1. It has been pushed much further by
1172:. Nowadays elementary proofs of this identity are known.
4351:
The arithmetic and spectral analysis of Poincaré series
1950:. A fundamental technique of Weil reduces the estimate
261:. They occur (for example) in the Fourier expansion of
4314:, Journal of Number Theory 51, Pages: 275-287, (1995).
3000:{\displaystyle 1\leq n\leq x,(n,m)=1,x<{\sqrt {m}}}
2924:{\displaystyle \left\{{\frac {an^{*}+bn}{m}}\right\},}
2726:
runs through the primes that do not divide the module
1604:
1581:
1381:
1127:
1094:
3997:
3811:
3764:
3688:
3489:
3427:
3375:
3355:
3280:
3179:
3130:
3032:
2941:
2878:
2837:
2747:
2690:
2535:
2496:
2431:
2404:
2293:
2258:
2228:
2121:
2031:. Geometrically the sum is taken along a 'hyperbola'
1959:
1915:
1808:
1690:
1459:
1375:
1228:
1014:
896:
799:
765:
536:
74:
4281:
Karatsuba, A. A. (1999). "Kloosterman double sums".
4056:
Equations over finite fields: an elementary approach
1181:
an odd prime, there are no known simple formula for
4598:"Bombieri-Weil bound - Encyclopedia of Mathematics"
3458:{\displaystyle \left({\frac {\tau }{p}}\right)=-1.}
4499:
4212:, Resultate Math. 18(1-2), pages: 120â124, (1990).
4009:
3938:
3786:
3712:
3655:
3457:
3402:
3361:
3331:
3201:
3157:
3112:
2999:
2923:
2853:
2801:
2714:
2668:
2508:
2483:{\displaystyle \exp\{(\ln m)^{2/3+\varepsilon }\}}
2482:
2417:
2328:
2271:
2240:
2199:
2054:, and Weil showed that the local zeta-function of
2012:
1930:
1898:
1773:
1642:
1398:
1357:
1148:
971:
862:
779:
679:
220:
32:, who introduced them in 1926 when he adapted the
4435:Trigonometric sums in number theory and analysis
4340:, Mathematics of the USSR-Sbornik 39(3), (1981).
3376:
2383:increases to infinity (this case was studied by
1860:
1399:{\displaystyle \left({\tfrac {\ell }{m}}\right)}
1053:
110:
4126:
3787:{\displaystyle g:\mathbb {R} \to \mathbb {R} }
3720:This formula is due to Yangbo Ye, inspired by
3713:{\displaystyle \mathbb {Q} ({\sqrt {\tau }}).}
28:. They are named for the Dutch mathematician
8:
4165:On the representation of numbers in the form
3173:and belonging to the arithmetic progression
2477:
2438:
2235:
2229:
2013:{\displaystyle |K(a,b;p)|\leq 2{\sqrt {p}},}
4357:. Academic Press Inc., Boston, MA, (1990).
4226:Uber die Kloostermanschen Summen S(u,v; q)
2329:{\displaystyle nn^{*}\equiv 1{\bmod {m}}.}
1216:one also has the important transformation:
4467:. Colloquium Publications. Vol. 53.
3996:
3928:
3914:
3891:
3882:
3842:
3830:
3826:
3816:
3810:
3780:
3779:
3772:
3771:
3763:
3697:
3690:
3689:
3687:
3628:
3613:
3609:
3599:
3592:
3588:
3573:
3557:
3552:
3550:
3549:
3530:
3488:
3432:
3426:
3374:
3354:
3320:
3295:
3291:
3285:
3279:
3193:
3189:
3178:
3129:
3089:
3060:
3031:
2990:
2940:
2893:
2883:
2877:
2844:
2836:
2768:
2758:
2746:
2689:
2638:
2628:
2607:
2602:
2573:
2563:
2542:
2537:
2534:
2495:
2461:
2457:
2430:
2409:
2403:
2317:
2313:
2301:
2292:
2279:denotes the congruence class, inverse to
2263:
2257:
2227:
2168:
2158:
2126:
2120:
2000:
1989:
1960:
1958:
1914:
1886:
1858:
1838:
1809:
1807:
1758:
1754:
1730:
1726:
1704:
1695:
1689:
1628:
1603:
1580:
1547:
1537:
1523:
1516:
1502:
1487:
1458:
1380:
1374:
1330:
1310:
1294:
1287:
1271:
1260:
1227:
1126:
1108:
1093:
1046:
1013:
955:
942:
937:
916:
911:
898:
897:
895:
846:
839:
834:
819:
814:
801:
800:
798:
773:
772:
764:
663:
647:
631:
608:
592:
576:
535:
204:
167:
166:
135:
109:
107:
106:
73:
4349:Cogdell, J.W. and I. Piatetski-Shapiro,
3972:It was later translated by Jacquet to a
1164:and first proved by Kuznetsov using the
303:then the Kloosterman sum reduces to the
4220:
4218:
4156:
4228:, Math. Zeit. 34 (1931â32) pp. 91â109.
4050:Weil's estimate can now be studied in
1938:is the number of positive divisors of
2248:in which is essentially smaller than
878:ranges over all odd primes such that
787:which is the compositum of the fields
780:{\displaystyle K\subset \mathbb {R} }
328:depends only on the residue class of
7:
4536:(1948). "On some exponential sums".
4366:Lidl & Niederreiter (1997) p.253
3949:The integral transform part is some
3202:{\displaystyle p\equiv l{\bmod {q}}}
2047:elements. This curve has a ramified
2039:and we consider this as defining an
4199:, Thesis (1924) Universiteit Leiden
4114:, Fouvry, Friedlander and Iwaniec.
4429:Arkhipov, G.I.; Chubarikov, V.N.;
3674:is the number of prime factors of
3531:
2379:is a fixed prime and the exponent
284:Properties of the Kloosterman sums
14:
4253:Izv. Ross. Akad. Nauk, Ser. Math.
3410:Assume that for any prime factor
2509:{\displaystyle \varepsilon >0}
2058:has a factorization; this is the
1678:is defined as follows (note that
4123:Kloosterman Sums and Maass Forms
3403:{\displaystyle \gcd(\tau ,m)=1.}
2854:{\displaystyle x<{\sqrt {m}}}
2418:{\displaystyle m^{\varepsilon }}
4312:The lifting of Kloosterman sums
4089:have an elementary evaluation.
4017:be a subgroup. While the usual
3916: Integral transform
3872:
3854:
3776:
3704:
3694:
3540:
3534:
3527:
3517:
3511:
3493:
3391:
3379:
3152:
3134:
3080:
3074:
3054:
3036:
2972:
2960:
2703:
2691:
2454:
2441:
2396:Anatolii Alexeevitch Karatsuba
1990:
1986:
1968:
1961:
1925:
1919:
1881:
1863:
1855:
1849:
1839:
1835:
1817:
1810:
1481:
1463:
1250:
1232:
1074:
1056:
1036:
1018:
759:is an element of the subfield
558:
540:
210:
185:
125:
113:
96:
78:
36:to tackle a problem involving
34:HardyâLittlewood circle method
1:
4469:American Mathematical Society
4463:; Kowalski, Emmanuel (2004).
4378:"Bounded gaps between primes"
4127:Iwaniec & Kowalski (2004)
4094:non-holomorphic modular forms
3369:be a squarefree integer with
2349:. Such estimates were due to
4376:Zhang, Yitang (1 May 2014).
4397:10.4007/annals.2014.179.3.7
4355:Perspectives in mathematics
4268:Tatra Mountains Math. Publ.
4239:Note on the Kloosterman sum
3345:Lifting of Kloosterman sums
3158:{\displaystyle \pi (x;q,l)}
2389:Ivan Matveyevich Vinogradov
2043:over the finite field with
253:The Kloosterman sums are a
4635:
4508:Cambridge University Press
4062:and draw inspiration from
4010:{\displaystyle H\subset G}
3743:
3724:and extending the work of
2387:by means of the method of
268:There are applications to
4068:Diophantine approximation
2425:, and in some cases even
2222:, the number of elements
4107:Inventiones Mathematicae
3974:representation theoretic
3165:is the number of primes
3012:a more precise constant
1931:{\displaystyle \tau (m)}
24:is a particular kind of
3800:Kuznetsov trace formula
3746:Kuznetsov trace formula
3740:Kuznetsov trace formula
3728:and Ye on the relative
3018:BrunâTitchmarsh theorem
2715:{\displaystyle (n,m)=1}
2218:of numbers, coprime to
2062:theory for the case of
2049:ArtinâSchreier covering
736:is always an algebraic
4619:Analytic number theory
4602:encyclopediaofmath.org
4465:Analytic number theory
4104:in a seminal paper in
4011:
3940:
3788:
3714:
3657:
3468:Then for all integers
3459:
3404:
3363:
3333:
3203:
3159:
3114:
3001:
2925:
2855:
2803:
2716:
2670:
2510:
2484:
2419:
2330:
2273:
2242:
2201:
2108:Short Kloosterman sums
2027:â 0 to his results on
2014:
1932:
1900:
1775:
1644:
1400:
1359:
1282:
1150:
973:
864:
781:
681:
222:
4538:Proc. Natl. Acad. Sci
4385:Annals of Mathematics
4237:Williams, Kenneth S.
4012:
3941:
3789:
3715:
3658:
3460:
3405:
3364:
3362:{\displaystyle \tau }
3334:
3204:
3160:
3115:
3002:
2926:
2856:
2804:
2717:
2671:
2511:
2485:
2420:
2331:
2274:
2272:{\displaystyle n^{*}}
2243:
2241:{\displaystyle \|A\|}
2202:
2015:
1933:
1901:
1776:
1645:
1401:
1360:
1256:
1151:
1002:The Selberg identity:
974:
865:
782:
682:
274:Riemann zeta function
223:
4496:Niederreiter, Harald
3995:
3969:was not applicable.
3930: Spectral terms
3809:
3762:
3686:
3487:
3425:
3373:
3353:
3278:
3177:
3128:
3030:
2939:
2876:
2835:
2745:
2688:
2533:
2494:
2429:
2402:
2291:
2256:
2226:
2119:
2029:local zeta-functions
1957:
1913:
1806:
1688:
1457:
1373:
1226:
1202:SatoâTate conjecture
1012:
894:
797:
763:
534:
72:
4569:"Kloosterman's Sum"
4195:Kloosterman, H. D.
4186:(1926), pp. 407â464
4182:, Acta Mathematica
4163:Kloosterman, H. D.
4083:Dirichlet character
3794:be a sufficiently "
2214:runs through a set
963:
854:
701:for a prime number
30:Hendrik Kloosterman
4566:Weisstein, Eric W.
4300:(88:3â4): 347â359.
4007:
3951:integral transform
3936:
3837:
3784:
3710:
3680:algebraic integers
3653:
3623:
3455:
3400:
3359:
3329:
3199:
3155:
3110:
2997:
2921:
2851:
2799:
2712:
2666:
2618:
2553:
2506:
2480:
2415:
2361:, S. Uchiyama and
2326:
2269:
2238:
2197:
2137:
2010:
1946:is a prime number
1928:
1896:
1771:
1766:
1658:is chosen so that
1640:
1635:
1613:
1590:
1396:
1390:
1355:
1146:
1136:
1115:
1078:
969:
933:
860:
830:
777:
677:
235:is the inverse of
218:
161:
4584:"Kloosterman sum"
4023:harmonic analysis
3931:
3927:
3921:
3917:
3905:
3899:
3812:
3756:automorphic forms
3750:The Kuznetsov or
3702:
3647:
3620:
3545:
3440:
3242:D. R. Heath-Brown
3105:
3102:
3007:(A.A. Karatsuba);
2995:
2912:
2861:(A.A. Karatsuba);
2849:
2787:
2657:
2603:
2592:
2538:
2351:H. D. Kloosterman
2252:, and the symbol
2187:
2122:
2005:
1891:
1884:
1631:
1612:
1589:
1566:
1526:
1521:
1510:
1435:prime and assume
1389:
1349:
1320:
1135:
1114:
1042:
183:
158:
102:
38:positive definite
4626:
4605:
4593:
4579:
4578:
4553:
4529:
4505:
4490:
4456:
4416:
4415:
4413:
4411:
4405:
4399:. Archived from
4391:(3): 1121â1174.
4382:
4373:
4367:
4364:
4358:
4347:
4341:
4336:N. V. Kuznecov,
4334:
4328:
4321:
4315:
4308:
4302:
4301:
4293:
4287:
4286:
4285:(66:5): 682â687.
4278:
4272:
4271:
4263:
4257:
4256:
4248:
4242:
4235:
4229:
4222:
4213:
4206:
4200:
4193:
4187:
4161:
4041:
4016:
4014:
4013:
4008:
3979:
3967:Weil conjectures
3965:'s proof of the
3945:
3943:
3942:
3937:
3932:
3929:
3925:
3919:
3918:
3915:
3910:
3906:
3901:
3900:
3892:
3883:
3850:
3849:
3836:
3835:
3834:
3793:
3791:
3790:
3785:
3783:
3775:
3735:
3719:
3717:
3716:
3711:
3703:
3698:
3693:
3677:
3673:
3662:
3660:
3659:
3654:
3649:
3648:
3643:
3629:
3622:
3621:
3619:
3618:
3617:
3598:
3597:
3596:
3578:
3577:
3562:
3561:
3551:
3544:
3543:
3479:
3464:
3462:
3461:
3456:
3445:
3441:
3433:
3417:
3413:
3409:
3407:
3406:
3401:
3368:
3366:
3365:
3360:
3338:
3336:
3335:
3330:
3325:
3324:
3300:
3299:
3290:
3289:
3256:
3239:
3208:
3206:
3205:
3200:
3198:
3197:
3172:
3169:, not exceeding
3168:
3164:
3162:
3161:
3156:
3119:
3117:
3116:
3111:
3106:
3104:
3103:
3098:
3090:
3069:
3061:
3015:
3006:
3004:
3003:
2998:
2996:
2991:
2930:
2928:
2927:
2922:
2917:
2913:
2908:
2898:
2897:
2884:
2860:
2858:
2857:
2852:
2850:
2845:
2830:
2826:
2813:in the integers
2808:
2806:
2805:
2800:
2792:
2788:
2783:
2773:
2772:
2759:
2730:(A.A.Karatsuba);
2729:
2725:
2721:
2719:
2718:
2713:
2683:
2675:
2673:
2672:
2667:
2662:
2658:
2653:
2643:
2642:
2629:
2623:
2619:
2617:
2597:
2593:
2588:
2578:
2577:
2564:
2558:
2554:
2552:
2515:
2513:
2512:
2507:
2489:
2487:
2486:
2481:
2476:
2475:
2465:
2424:
2422:
2421:
2416:
2414:
2413:
2382:
2378:
2374:
2355:I. M. Vinogradov
2348:
2347:
2346:
2335:
2333:
2332:
2327:
2322:
2321:
2306:
2305:
2286:
2282:
2278:
2276:
2275:
2270:
2268:
2267:
2251:
2247:
2245:
2244:
2239:
2221:
2217:
2213:
2206:
2204:
2203:
2198:
2193:
2189:
2188:
2183:
2173:
2172:
2159:
2136:
2090:Weil conjectures
2084:
2080:
2072:Collected Papers
2060:Artin L-function
2057:
2053:
2046:
2019:
2017:
2016:
2011:
2006:
2001:
1993:
1964:
1949:
1945:
1941:
1937:
1935:
1934:
1929:
1905:
1903:
1902:
1897:
1892:
1887:
1885:
1859:
1842:
1813:
1780:
1778:
1777:
1772:
1770:
1769:
1763:
1762:
1735:
1734:
1700:
1699:
1681:
1677:
1668:
1657:
1649:
1647:
1646:
1641:
1639:
1638:
1632:
1629:
1618:
1614:
1605:
1595:
1591:
1582:
1573:
1569:
1568:
1567:
1562:
1548:
1542:
1541:
1527:
1524:
1522:
1517:
1515:
1511:
1503:
1446:
1434:
1424:
1405:
1403:
1402:
1397:
1395:
1391:
1382:
1364:
1362:
1361:
1356:
1351:
1350:
1345:
1331:
1325:
1321:
1316:
1315:
1314:
1299:
1298:
1288:
1281:
1270:
1215:
1199:
1180:
1155:
1153:
1152:
1147:
1142:
1138:
1137:
1128:
1116:
1113:
1112:
1103:
1095:
1077:
996:
989:
978:
976:
975:
970:
968:
964:
962:
954:
953:
952:
929:
928:
927:
926:
901:
887:
877:
869:
867:
866:
861:
859:
855:
853:
845:
844:
843:
826:
825:
824:
823:
804:
786:
784:
783:
778:
776:
758:
735:
711:
704:
700:
686:
684:
683:
678:
673:
669:
668:
667:
652:
651:
636:
635:
618:
614:
613:
612:
597:
596:
581:
580:
523:
501:
479:
470:
462:coprime. Choose
461:
452:
443:
421:
409:
374:
339:
335:
331:
327:
302:
295:
259:Bessel functions
244:
238:
227:
225:
224:
219:
214:
213:
209:
208:
184:
179:
168:
160:
159:
157:
134:
108:
60:
4634:
4633:
4629:
4628:
4627:
4625:
4624:
4623:
4609:
4608:
4596:
4582:
4564:
4563:
4560:
4532:
4518:
4493:
4479:
4461:Iwaniec, Henryk
4459:
4445:
4431:Karatsuba, A.A.
4428:
4425:
4420:
4419:
4409:
4407:
4403:
4380:
4375:
4374:
4370:
4365:
4361:
4353:, volume 13 of
4348:
4344:
4335:
4331:
4322:
4318:
4309:
4305:
4295:
4294:
4290:
4280:
4279:
4275:
4265:
4264:
4260:
4255:(59:5): 93â102.
4250:
4249:
4245:
4236:
4232:
4223:
4216:
4207:
4203:
4194:
4190:
4162:
4158:
4153:
4141:
4119:spectral theory
4048:
4033:
4031:symmetric space
3993:
3992:
3982:reductive group
3977:
3976:framework. Let
3884:
3878:
3838:
3807:
3806:
3760:
3759:
3748:
3742:
3733:
3684:
3683:
3675:
3667:
3630:
3624:
3569:
3553:
3526:
3485:
3484:
3477:
3428:
3423:
3422:
3415:
3411:
3371:
3370:
3351:
3350:
3347:
3316:
3281:
3276:
3275:
3248:
3223:
3175:
3174:
3170:
3166:
3126:
3125:
3091:
3070:
3062:
3028:
3027:
3013:
2937:
2936:
2889:
2885:
2879:
2874:
2873:
2833:
2832:
2828:
2814:
2764:
2760:
2754:
2743:
2742:
2727:
2723:
2686:
2685:
2681:
2634:
2630:
2624:
2601:
2569:
2565:
2559:
2536:
2531:
2530:
2492:
2491:
2453:
2427:
2426:
2405:
2400:
2399:
2380:
2376:
2366:
2342:
2340:
2339:
2297:
2289:
2288:
2284:
2280:
2259:
2254:
2253:
2249:
2224:
2223:
2219:
2215:
2211:
2164:
2160:
2148:
2144:
2117:
2116:
2110:
2082:
2075:
2055:
2051:
2044:
2041:algebraic curve
1955:
1954:
1947:
1943:
1939:
1911:
1910:
1804:
1803:
1792:
1765:
1764:
1743:
1737:
1736:
1715:
1705:
1691:
1686:
1685:
1679:
1675:
1670:
1659:
1655:
1634:
1633:
1626:
1620:
1619:
1599:
1576:
1574:
1549:
1543:
1533:
1532:
1528:
1498:
1488:
1455:
1454:
1436:
1426:
1416:
1376:
1371:
1370:
1332:
1326:
1306:
1290:
1289:
1283:
1224:
1223:
1205:
1182:
1178:
1166:spectral theory
1104:
1096:
1092:
1088:
1010:
1009:
991:
984:
938:
912:
907:
906:
902:
892:
891:
879:
875:
835:
815:
810:
809:
805:
795:
794:
761:
760:
741:
718:
706:
705:and an integer
702:
692:
659:
643:
627:
626:
622:
604:
588:
572:
571:
567:
532:
531:
522:
515:
509:
503:
500:
493:
487:
481:
478:
472:
469:
463:
460:
454:
451:
445:
442:
436:
426:
411:
376:
341:
337:
333:
329:
310:
297:
290:
286:
251:
242:
236:
200:
169:
162:
70:
69:
63:natural numbers
48:
42:quadratic forms
26:exponential sum
22:Kloosterman sum
12:
11:
5:
4632:
4630:
4622:
4621:
4611:
4610:
4607:
4606:
4594:
4580:
4559:
4558:External links
4556:
4555:
4554:
4530:
4516:
4494:Lidl, Rudolf;
4491:
4477:
4457:
4443:
4424:
4421:
4418:
4417:
4406:on 9 July 2020
4368:
4359:
4342:
4329:
4316:
4303:
4288:
4273:
4258:
4243:
4230:
4214:
4201:
4188:
4155:
4154:
4152:
4149:
4148:
4147:
4140:
4137:
4079:Henri Poincaré
4047:
4044:
4006:
4003:
4000:
3963:Pierre Deligne
3947:
3946:
3935:
3924:
3913:
3909:
3904:
3898:
3895:
3890:
3887:
3881:
3877:
3874:
3871:
3868:
3865:
3862:
3859:
3856:
3853:
3848:
3845:
3841:
3833:
3829:
3825:
3822:
3819:
3815:
3782:
3778:
3774:
3770:
3767:
3752:relative trace
3744:Main article:
3741:
3738:
3709:
3706:
3701:
3696:
3692:
3664:
3663:
3652:
3646:
3642:
3639:
3636:
3633:
3627:
3616:
3612:
3608:
3605:
3602:
3595:
3591:
3587:
3584:
3581:
3576:
3572:
3568:
3565:
3560:
3556:
3548:
3542:
3539:
3536:
3533:
3529:
3525:
3522:
3519:
3516:
3513:
3510:
3507:
3504:
3501:
3498:
3495:
3492:
3466:
3465:
3454:
3451:
3448:
3444:
3439:
3436:
3431:
3399:
3396:
3393:
3390:
3387:
3384:
3381:
3378:
3358:
3346:
3343:
3342:
3341:
3340:
3339:
3328:
3323:
3319:
3315:
3312:
3309:
3306:
3303:
3298:
3294:
3288:
3284:
3270:
3269:
3266:
3259:J. Friedlander
3245:
3219:
3218:
3211:J. Friedlander
3196:
3192:
3188:
3185:
3182:
3154:
3151:
3148:
3145:
3142:
3139:
3136:
3133:
3122:
3121:
3120:
3109:
3101:
3097:
3094:
3088:
3085:
3082:
3079:
3076:
3073:
3068:
3065:
3059:
3056:
3053:
3050:
3047:
3044:
3041:
3038:
3035:
3022:
3021:
3009:
3008:
2994:
2989:
2986:
2983:
2980:
2977:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2953:
2950:
2947:
2944:
2933:
2932:
2931:
2920:
2916:
2911:
2907:
2904:
2901:
2896:
2892:
2888:
2882:
2868:
2867:
2863:
2862:
2848:
2843:
2840:
2811:
2810:
2809:
2798:
2795:
2791:
2786:
2782:
2779:
2776:
2771:
2767:
2763:
2757:
2753:
2750:
2737:
2736:
2732:
2731:
2711:
2708:
2705:
2702:
2699:
2696:
2693:
2678:
2677:
2676:
2665:
2661:
2656:
2652:
2649:
2646:
2641:
2637:
2633:
2627:
2622:
2616:
2613:
2610:
2606:
2600:
2596:
2591:
2587:
2584:
2581:
2576:
2572:
2568:
2562:
2557:
2551:
2548:
2545:
2541:
2525:
2524:
2505:
2502:
2499:
2479:
2474:
2471:
2468:
2464:
2460:
2456:
2452:
2449:
2446:
2443:
2440:
2437:
2434:
2412:
2408:
2385:A.G. Postnikov
2325:
2320:
2316:
2312:
2309:
2304:
2300:
2296:
2266:
2262:
2237:
2234:
2231:
2208:
2207:
2196:
2192:
2186:
2182:
2179:
2176:
2171:
2167:
2163:
2157:
2154:
2151:
2147:
2143:
2140:
2135:
2132:
2129:
2125:
2109:
2106:
2094:Pierre Deligne
2021:
2020:
2009:
2004:
1999:
1996:
1992:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1963:
1927:
1924:
1921:
1918:
1907:
1906:
1895:
1890:
1883:
1880:
1877:
1874:
1871:
1868:
1865:
1862:
1857:
1854:
1851:
1848:
1845:
1841:
1837:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1812:
1791:
1788:
1787:
1786:
1783:
1782:
1781:
1768:
1761:
1757:
1753:
1750:
1747:
1744:
1742:
1739:
1738:
1733:
1729:
1725:
1722:
1719:
1716:
1714:
1711:
1710:
1708:
1703:
1698:
1694:
1673:
1652:
1651:
1650:
1637:
1627:
1625:
1622:
1621:
1617:
1611:
1608:
1602:
1598:
1594:
1588:
1585:
1579:
1575:
1572:
1565:
1561:
1558:
1555:
1552:
1546:
1540:
1536:
1531:
1520:
1514:
1509:
1506:
1501:
1497:
1494:
1493:
1491:
1486:
1483:
1480:
1477:
1474:
1471:
1468:
1465:
1462:
1449:
1448:
1412:
1411:
1394:
1388:
1385:
1379:
1367:
1366:
1365:
1354:
1348:
1344:
1341:
1338:
1335:
1329:
1324:
1319:
1313:
1309:
1305:
1302:
1297:
1293:
1286:
1280:
1277:
1274:
1269:
1266:
1263:
1259:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1218:
1217:
1174:
1173:
1160:was stated by
1158:
1157:
1156:
1145:
1141:
1134:
1131:
1125:
1122:
1119:
1111:
1107:
1102:
1099:
1091:
1087:
1084:
1081:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1049:
1045:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1004:
1003:
999:
998:
981:
980:
979:
967:
961:
958:
951:
948:
945:
941:
936:
932:
925:
922:
919:
915:
910:
905:
900:
872:
871:
870:
858:
852:
849:
842:
838:
833:
829:
822:
818:
813:
808:
803:
789:
788:
775:
771:
768:
714:
713:
689:
688:
687:
676:
672:
666:
662:
658:
655:
650:
646:
642:
639:
634:
630:
625:
621:
617:
611:
607:
603:
600:
595:
591:
587:
584:
579:
575:
570:
566:
563:
560:
557:
554:
551:
548:
545:
542:
539:
526:
525:
520:
513:
507:
498:
491:
485:
476:
467:
458:
449:
440:
434:
423:
340:. Furthermore
308:
285:
282:
272:involving the
250:
247:
229:
228:
217:
212:
207:
203:
199:
196:
193:
190:
187:
182:
178:
175:
172:
165:
156:
153:
150:
147:
144:
141:
138:
133:
130:
127:
124:
121:
118:
115:
112:
105:
101:
98:
95:
92:
89:
86:
83:
80:
77:
13:
10:
9:
6:
4:
3:
2:
4631:
4620:
4617:
4616:
4614:
4603:
4599:
4595:
4591:
4590:
4585:
4581:
4576:
4575:
4570:
4567:
4562:
4561:
4557:
4551:
4547:
4543:
4539:
4535:
4531:
4527:
4523:
4519:
4517:0-521-39231-4
4513:
4509:
4504:
4503:
4502:Finite fields
4497:
4492:
4488:
4484:
4480:
4478:0-8218-3633-1
4474:
4470:
4466:
4462:
4458:
4454:
4450:
4446:
4444:3-11-016266-0
4440:
4436:
4432:
4427:
4426:
4422:
4402:
4398:
4394:
4390:
4386:
4379:
4372:
4369:
4363:
4360:
4356:
4352:
4346:
4343:
4339:
4333:
4330:
4326:
4320:
4317:
4313:
4307:
4304:
4299:
4292:
4289:
4284:
4277:
4274:
4270:(11): 89â120.
4269:
4262:
4259:
4254:
4247:
4244:
4240:
4234:
4231:
4227:
4221:
4219:
4215:
4211:
4205:
4202:
4198:
4192:
4189:
4185:
4181:
4177:
4173:
4169:
4166:
4160:
4157:
4150:
4146:
4145:Hasse's bound
4143:
4142:
4138:
4136:
4134:
4130:
4128:
4124:
4120:
4115:
4113:
4109:
4108:
4103:
4099:
4095:
4090:
4088:
4084:
4080:
4076:
4075:modular forms
4071:
4069:
4065:
4061:
4057:
4053:
4052:W. M. Schmidt
4045:
4043:
4040:
4036:
4032:
4028:
4024:
4020:
4019:trace formula
4004:
4001:
3998:
3990:
3987:
3983:
3975:
3970:
3968:
3964:
3960:
3956:
3952:
3933:
3922:
3911:
3907:
3902:
3896:
3893:
3888:
3885:
3879:
3875:
3869:
3866:
3863:
3860:
3857:
3851:
3846:
3843:
3839:
3831:
3823:
3820:
3817:
3813:
3805:
3804:
3803:
3801:
3797:
3768:
3765:
3757:
3753:
3747:
3739:
3737:
3731:
3730:trace formula
3727:
3726:Hervé Jacquet
3723:
3707:
3699:
3682:in the field
3681:
3671:
3650:
3644:
3640:
3637:
3634:
3631:
3625:
3614:
3606:
3603:
3600:
3593:
3585:
3582:
3579:
3574:
3570:
3566:
3563:
3558:
3554:
3546:
3537:
3523:
3520:
3514:
3508:
3505:
3502:
3499:
3496:
3490:
3483:
3482:
3481:
3475:
3471:
3452:
3449:
3446:
3442:
3437:
3434:
3429:
3421:
3420:
3419:
3397:
3394:
3388:
3385:
3382:
3356:
3344:
3326:
3321:
3317:
3313:
3310:
3307:
3304:
3301:
3296:
3286:
3282:
3274:
3273:
3272:
3271:
3267:
3264:
3260:
3255:
3251:
3246:
3243:
3238:
3234:
3230:
3226:
3221:
3220:
3216:
3212:
3194:
3186:
3183:
3180:
3149:
3146:
3143:
3140:
3137:
3131:
3123:
3107:
3099:
3095:
3092:
3086:
3083:
3077:
3071:
3066:
3063:
3057:
3051:
3048:
3045:
3042:
3039:
3033:
3026:
3025:
3024:
3023:
3019:
3011:
3010:
2992:
2987:
2984:
2981:
2978:
2975:
2969:
2966:
2963:
2957:
2954:
2951:
2948:
2945:
2942:
2934:
2918:
2914:
2909:
2905:
2902:
2899:
2894:
2890:
2886:
2880:
2872:
2871:
2870:
2869:
2865:
2864:
2846:
2841:
2838:
2827:, coprime to
2825:
2821:
2817:
2812:
2796:
2793:
2789:
2784:
2780:
2777:
2774:
2769:
2765:
2761:
2755:
2751:
2748:
2741:
2740:
2739:
2738:
2734:
2733:
2709:
2706:
2700:
2697:
2694:
2679:
2663:
2659:
2654:
2650:
2647:
2644:
2639:
2635:
2631:
2625:
2620:
2614:
2611:
2608:
2604:
2598:
2594:
2589:
2585:
2582:
2579:
2574:
2570:
2566:
2560:
2555:
2549:
2546:
2543:
2539:
2529:
2528:
2527:
2526:
2522:
2521:
2520:
2517:
2503:
2500:
2497:
2472:
2469:
2466:
2462:
2458:
2450:
2447:
2444:
2435:
2432:
2410:
2406:
2397:
2394:In the 1990s
2392:
2390:
2386:
2373:
2369:
2364:
2360:
2357:, H. Salie,
2356:
2352:
2345:
2336:
2323:
2318:
2310:
2307:
2302:
2298:
2294:
2264:
2260:
2232:
2194:
2190:
2184:
2180:
2177:
2174:
2169:
2165:
2161:
2155:
2152:
2149:
2145:
2141:
2138:
2133:
2130:
2127:
2123:
2115:
2114:
2113:
2107:
2105:
2103:
2102:Nicholas Katz
2099:
2098:GĂ©rard Laumon
2095:
2091:
2086:
2079:
2073:
2069:
2065:
2064:global fields
2061:
2050:
2042:
2038:
2034:
2030:
2026:
2007:
2002:
1997:
1994:
1983:
1980:
1977:
1974:
1971:
1965:
1953:
1952:
1951:
1922:
1916:
1893:
1888:
1878:
1875:
1872:
1869:
1866:
1852:
1846:
1843:
1832:
1829:
1826:
1823:
1820:
1814:
1802:
1801:
1800:
1798:
1789:
1784:
1759:
1751:
1748:
1745:
1740:
1731:
1723:
1720:
1717:
1712:
1706:
1701:
1696:
1692:
1684:
1683:
1676:
1667:
1663:
1653:
1623:
1615:
1609:
1606:
1600:
1596:
1592:
1586:
1583:
1577:
1570:
1563:
1559:
1556:
1553:
1550:
1544:
1538:
1534:
1529:
1518:
1512:
1507:
1504:
1499:
1495:
1489:
1484:
1478:
1475:
1472:
1469:
1466:
1460:
1453:
1452:
1451:
1450:
1444:
1440:
1433:
1429:
1423:
1419:
1414:
1413:
1409:
1408:Jacobi symbol
1392:
1386:
1383:
1377:
1368:
1352:
1346:
1342:
1339:
1336:
1333:
1327:
1322:
1317:
1311:
1307:
1303:
1300:
1295:
1291:
1284:
1278:
1275:
1272:
1267:
1264:
1261:
1257:
1253:
1247:
1244:
1241:
1238:
1235:
1229:
1222:
1221:
1220:
1219:
1213:
1209:
1203:
1197:
1193:
1189:
1185:
1176:
1175:
1171:
1170:modular forms
1167:
1163:
1159:
1143:
1139:
1132:
1129:
1123:
1120:
1117:
1109:
1105:
1100:
1097:
1089:
1085:
1082:
1079:
1071:
1068:
1065:
1062:
1059:
1050:
1047:
1043:
1039:
1033:
1030:
1027:
1024:
1021:
1015:
1008:
1007:
1006:
1005:
1001:
1000:
994:
988:
982:
965:
959:
956:
949:
946:
943:
939:
934:
930:
923:
920:
917:
913:
908:
903:
890:
889:
886:
882:
873:
856:
850:
847:
840:
836:
831:
827:
820:
816:
811:
806:
793:
792:
791:
790:
769:
766:
756:
752:
748:
744:
739:
733:
729:
725:
721:
717:The value of
716:
715:
709:
699:
695:
690:
674:
670:
664:
660:
656:
653:
648:
644:
640:
637:
632:
628:
623:
619:
615:
609:
605:
601:
598:
593:
589:
585:
582:
577:
573:
568:
564:
561:
555:
552:
549:
546:
543:
537:
530:
529:
528:
527:
519:
512:
506:
497:
490:
484:
475:
466:
457:
448:
439:
433:
429:
424:
419:
415:
407:
403:
399:
395:
391:
387:
383:
379:
372:
368:
364:
360:
356:
352:
348:
344:
325:
321:
317:
313:
309:
306:
305:Ramanujan sum
300:
293:
288:
287:
283:
281:
279:
275:
271:
266:
264:
263:modular forms
260:
256:
248:
246:
241:
234:
215:
205:
201:
197:
194:
191:
188:
180:
176:
173:
170:
163:
154:
151:
148:
145:
142:
139:
136:
131:
128:
122:
119:
116:
103:
99:
93:
90:
87:
84:
81:
75:
68:
67:
66:
64:
59:
55:
51:
45:
43:
39:
35:
31:
27:
23:
19:
4601:
4587:
4572:
4541:
4537:
4501:
4464:
4434:
4408:. Retrieved
4401:the original
4388:
4384:
4371:
4362:
4354:
4350:
4345:
4337:
4332:
4324:
4319:
4311:
4306:
4298:Mat. Zametki
4297:
4291:
4283:Mat. Zametki
4282:
4276:
4267:
4261:
4252:
4246:
4238:
4233:
4225:
4224:Hans Salie,
4209:
4208:Matthes, R.
4204:
4196:
4191:
4183:
4179:
4175:
4171:
4167:
4164:
4159:
4133:Yitang Zhang
4131:
4122:
4116:
4105:
4102:Deshouillers
4091:
4086:
4072:
4055:
4049:
4038:
4034:
4026:
4021:studies the
3988:
3986:number field
3971:
3958:
3954:
3948:
3799:
3796:well behaved
3751:
3749:
3669:
3665:
3473:
3469:
3467:
3348:
3253:
3249:
3236:
3232:
3228:
3224:
2823:
2819:
2815:
2518:
2393:
2371:
2367:
2343:
2337:
2209:
2111:
2087:
2077:
2071:
2036:
2032:
2024:
2022:
1908:
1799:and states:
1793:
1671:
1665:
1661:
1442:
1438:
1431:
1427:
1421:
1417:
1406:denotes the
1211:
1207:
1195:
1191:
1187:
1183:
1162:Atle Selberg
992:
986:
884:
880:
754:
750:
746:
742:
731:
727:
723:
719:
707:
697:
693:
517:
510:
504:
495:
488:
482:
473:
464:
455:
446:
437:
431:
427:
417:
413:
405:
401:
397:
393:
389:
385:
381:
377:
370:
366:
362:
358:
354:
350:
346:
342:
323:
319:
315:
311:
298:
291:
267:
257:analogue of
252:
232:
230:
57:
53:
49:
46:
21:
15:
4544:: 204â207.
4534:Weil, André
4410:17 November
4066:'s work in
4060:S. Stepanov
3476:coprime to
738:real number
270:mean values
255:finite ring
18:mathematics
4589:PlanetMath
4550:0032.26102
4526:0866.11069
4487:1059.11001
4453:1074.11043
4423:References
4087:Salié sums
3722:Don Zagier
3263:H. Iwaniec
3215:H. Iwaniec
2359:L. Carlitz
1797:André Weil
1200:, and the
740:. In fact
480:such that
4574:MathWorld
4064:Axel Thue
4002:⊂
3889:π
3844:−
3821:≡
3814:∑
3777:→
3700:τ
3635:π
3580:≡
3567:τ
3564:−
3547:∑
3532:Ω
3521:−
3450:−
3435:τ
3383:τ
3357:τ
3322:ε
3314:≤
3308:≤
3287:∗
3184:≡
3132:π
3087:
3072:φ
3034:π
2952:≤
2946:≤
2895:∗
2797:β
2794:≤
2770:∗
2749:α
2640:∗
2612:≤
2605:∑
2575:∗
2547:≤
2540:∑
2498:ε
2473:ε
2448:
2436:
2411:ε
2308:≡
2303:∗
2265:∗
2236:‖
2230:‖
2170:∗
2153:π
2142:
2131:∈
2124:∑
1995:≤
1917:τ
1847:τ
1844:≤
1790:Estimates
1749:≡
1721:≡
1693:ε
1682:is odd):
1630:otherwise
1560:ℓ
1554:π
1535:ε
1505:ℓ
1384:ℓ
1337:π
1301:−
1276:−
1258:∑
1083:⋅
1051:∣
1044:∑
957:−
947:−
944:α
935:ζ
921:−
918:α
909:ζ
848:−
841:α
832:ζ
821:α
812:ζ
770:⊂
206:∗
174:π
152:−
146:≤
140:≤
104:∑
40:diagonal
4613:Category
4498:(1997).
4433:(2004).
4139:See also
4112:Bombieri
3480:we have
3418:we have
2621:′
2556:′
2490:, where
2375:, where
2081:, where
1525: Re
1430:> 1,
516:⥠1 mod
494:⥠1 mod
4323:Ye, Y.
4310:Ye, Y.
4098:Iwaniec
4085:: Such
4046:History
3984:over a
3020: :
3016:in the
2363:A. Weil
2341:√
2283:modulo
1447:. Then:
336:modulo
249:Context
65:. Then
4548:
4524:
4514:
4485:
4475:
4451:
4441:
3926:
3920:
3124:where
2935:where
2818:, 1 â€
2722:, and
2680:where
2210:where
2100:, and
1654:where
1369:where
995:> 3
874:where
524:. Then
278:primes
240:modulo
4404:(PDF)
4381:(PDF)
4151:Notes
3980:be a
3734:GL(2)
3666:Here
3231:<
3227:+ 2,
2068:Hasse
2023:when
1909:Here
1445:) = 1
1425:with
1214:) = 1
990:with
985:2 ||
444:with
420:) = 1
231:Here
4512:ISBN
4473:ISBN
4439:ISBN
4412:2022
4100:and
3991:and
3732:for
3058:<
2988:<
2842:<
2752:<
2501:>
2076:1 â
1669:and
1664:mod
1660:â âĄ
1437:gcd(
1415:Let
1206:gcd(
1177:For
983:for
888:and
502:and
471:and
453:and
425:Let
412:gcd(
392:) =
375:and
357:) =
332:and
47:Let
20:, a
4546:Zbl
4522:Zbl
4483:Zbl
4449:Zbl
4393:doi
4389:179
4025:on
3953:of
3828:mod
3611:mod
3590:mod
3414:of
3377:gcd
3293:mod
3235:†2
3191:mod
2433:exp
2391:).
2315:mod
2139:exp
1861:gcd
1756:mod
1728:mod
1441:, 2
1168:of
1054:gcd
883:||
710:â„ 1
410:if
301:= 0
296:or
294:= 0
289:If
111:gcd
61:be
16:In
4615::
4600:.
4586:.
4571:.
4542:34
4540:.
4520:.
4510:.
4481:.
4471:.
4447:.
4387:.
4383:.
4217:^
4184:49
4180:dt
4178:+
4176:cz
4174:+
4172:by
4170:+
4168:ax
4129:.
4070:.
4054:,
3802::
3668:Ω(
3472:,
3453:1.
3398:1.
3265:);
3261:,
3257:.(
3252:+
3244:);
3240:.(
3217:);
3213:,
3084:ln
2831:,
2822:â€
2445:ln
2370:=
2353:,
2287::
2104:.
2096:,
2078:Kt
2037:ab
2035:=
2033:XY
2025:ab
1662:ab
1443:ab
1420:=
1210:,
1194:;
1190:,
753:;
749:,
730:;
726:,
696:=
430:=
416:,
404:;
402:bc
400:,
388:;
384:,
382:ac
369:;
365:,
353:;
349:,
322:;
318:,
276:,
265:.
245:.
233:x*
56:,
52:,
4604:.
4592:.
4577:.
4552:.
4528:.
4489:.
4455:.
4414:.
4395::
4039:H
4037:/
4035:G
4027:G
4005:G
3999:H
3989:F
3978:G
3959:g
3955:g
3934:.
3923:+
3912:=
3908:)
3903:c
3897:n
3894:m
3886:4
3880:(
3876:g
3873:)
3870:c
3867:,
3864:n
3861:,
3858:m
3855:(
3852:K
3847:r
3840:c
3832:N
3824:0
3818:c
3781:R
3773:R
3769::
3766:g
3708:.
3705:)
3695:(
3691:Q
3676:m
3672:)
3670:m
3651:.
3645:m
3641:v
3638:i
3632:4
3626:e
3615:m
3607:w
3604:,
3601:v
3594:m
3586:b
3583:a
3575:2
3571:w
3559:2
3555:v
3541:)
3538:m
3535:(
3528:)
3524:1
3518:(
3515:=
3512:)
3509:m
3506:;
3503:b
3500:,
3497:a
3494:(
3491:K
3478:m
3474:b
3470:a
3447:=
3443:)
3438:p
3430:(
3416:m
3412:p
3395:=
3392:)
3389:m
3386:,
3380:(
3327:.
3318:m
3311:n
3305:1
3302:,
3297:m
3283:n
3254:b
3250:a
3237:N
3233:n
3229:N
3225:n
3209:(
3195:q
3187:l
3181:p
3171:x
3167:p
3153:)
3150:l
3147:,
3144:q
3141:;
3138:x
3135:(
3108:,
3100:q
3096:x
3093:2
3081:)
3078:q
3075:(
3067:x
3064:c
3055:)
3052:l
3049:,
3046:q
3043:;
3040:x
3037:(
3014:c
2993:m
2985:x
2982:,
2979:1
2976:=
2973:)
2970:m
2967:,
2964:n
2961:(
2958:,
2955:x
2949:n
2943:1
2919:,
2915:}
2910:m
2906:n
2903:b
2900:+
2891:n
2887:a
2881:{
2847:m
2839:x
2829:m
2824:x
2820:n
2816:n
2790:}
2785:m
2781:n
2778:b
2775:+
2766:n
2762:a
2756:{
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