Unconditional implications of the Quasi-Riemann Hypothesis

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1. O. Klurman, A. P. Mangerel, and J. Teräväinen, Fourier uniformity in intervals longer than every fixed logarithmic power sec:walsh-fourier Walsh's published Fourier-cancellation conclusion i̧te WalshScaling assumes GRH and allows intervals longer than every fixed power of $X$. Its analytic input is a twisted prime estimate with some positive power saving. The value of that saving is not part of the conclusion. We prove that estimate, including its uniformity in the real twist, and use the published inverse argument only in the short-interval range needed here. The remaining interval lengths are covered by a previous unconditional theorem. References to intermediate numbered results are to the full author version linked from Walsh's publication page. For a complex multiplicative function $g$ with $|g(n)| 1$, put \[ D(g;T,Q)= _ |t| T\\ ḩi\ ( mod \ q),\ q Q ( _ p T 1- (g(p)p^ it ḩi(p)) p )^ 1/2 . \] Characters in this definition need not be primitive. Write $e(u)= (2 i u)$. e̱gin theorem [Walsh's Fourier-cancellation conclusion] thm:walsh-fourier If $H=H(X) X$ and $H/ X$, then, for $g=$ or $g=$, \[ _X^ 2X _ R/ Z | _ x n x+H g(n)e( n) |\,dx=o(HX). \] Here $ $ and $ $ are respectively the Möbius and Liouville functions. The frequency inside the supremum may vary with $x$. Equivalently, for every $>0$ there are $K>1$ and $X_0$ such that the integral is less than $HX$ whenever $X X_0$ and $( X)^K H X$. theorem e̱gin lemma [The precise twisted prime input] lem:walsh-prime-input There are absolute constants $c>0$ and $A>1$ such that, for every Dirichlet character $ḩi$ modulo $q$, every real $t$, and $P ( (3q(2+|t|)))^A$, \[ 1P | _ n P (n)ḩi(n)n^ it | ẹlta_ḩi 1+|t| +P^ -c , \] where $ẹlta_ḩi=1$ for a principal character and is zero otherwise. More precisely, uniformly for $P 2$, e̱gin equation eq:walsh-twisted-perron _ n P (n)ḩi(n)n^ it =ẹlta_ḩi P^ 1+it 1+it +O (P^ 15/16 ^2(3qP(2+|t|)) ). equation The same constants, after alteration, give e̱gin equation eq:walsh-prime
2. B. Green and T. Tao,
3. B. Green and T. Tao, Power decay in Möbius correlations with polynomial nilsequences sec:green-tao-power Green--Tao's unconditional nilsequence theorem has decay faster than every fixed negative power of $N$. Their published remarks immediately after Theorem 1.1 explicitly assert a stronger conclusion under GRH: decay $N^ -c$, with $c>0$ depending only on dimension and filtration degree i̧te[p. 543] GreenTaoMobius2012 . We prove that complete power-decay conclusion. The logarithmic theorem and its mere effectivity are not the assertion being counted here. We specify the terminology. A nilmanifold is $G/$, where $G$ is a connected, simply connected nilpotent Lie group and $ $ is a discrete cocompact subgroup. A degree-$d$ filtration $G_u̱llet$ is a descending sequence of closed connected subgroups $G=G_0=G_1 ḑots G_ d+1 =\ 1\ $ with $[G_i,G_j] G_ i+j$. A sequence $g: Z G$ is polynomial relative to this filtration if every iterated multiplicative difference $ _ h_i ḑots _ h_1 g$ takes values in $G_i$, where $_hg(n)=g(n+h)g(n)^ -1$. A Mal'cev basis $X=(X_1, ,X_m)$ identifies $G$ with $R^m$ by $ (t_1X_1)ḑots (t_mX_m)$, identifies $ $ with integer coordinates, and satisfies that $span (X_ j+1 , ,X_m)$ is a Lie algebra ideal for each $0 j<m$. It is adapted to $G_u̱llet$ if each filtration subgroup is spanned by an appropriate final segment of the basis, and $Q$-rational if all its Lie-bracket structure constants are rational of height at most $Q$, where the height of a reduced fraction $a/b$ is $\ |a|,|b|\$. We use the right-invariant coordinate metric and induced quotient metric $d_ X$ of i̧te[Definitions 2.1--2.4] GreenTaoOrbits2012 . For a function $F$ on this quotient, write \[ \|F\|_ Lip =\|F\|_ + _ x y |F(x)-F(y)| d_ X (x,y) . \] e̱gin theorem [Green--Tao's published GRH improvement] thm:green-tao-power For every pair of positive integers $m,d$ there are effective constants $c=c(m,d)>0$ and $C=C(m,d)>0$ such that the following holds. Suppose $G/$ has di
4. A. P. Mangerel, On a rigidity property for
5. A. Granville and K. Soundararajan, The spectrum Rigidity from approximate quadratic Gauss-sum symmetries sec:gauss-rigidity Mangerel's Theorem 1.3(b) in i̧te MangerelRigidity extends a rigidity statement to arbitrarily small fixed powers of the prime modulus, assuming a specified zero-free region for its Dirichlet $L$-functions. Theorem thm:input supplies that region for all sufficiently large moduli. We prove its entire conditional conclusion below. The author's unconditional Theorem 1.1 and the converse in Theorem 1.3(a) are not counted as additional transfers. Write $e(t)= (2 it)$. If $N$ is an odd prime and $f: N \ -1,1\$, define \[ S_f(a)= _ 1 n<N f(n)e(an/N), a Z/N Z. \] Here ``multiplicative'' means multiplicative on coprime arguments; complete multiplicativity of $f$ is not required. e̱gin theorem [Mangerel's conditional rigidity conclusion] thm:gauss-rigidity Fix $0<c<1$ and constants $B_1,B_2>0$. There are constants $M_0,N_0,C>0$, depending only on $c,B_1,B_2$, with the following property. Let $N N_0$ be prime, let $M_0 M N$, and let $f: N \ -1,1\$ be multiplicative. Suppose that $A$ is a set of primes less than $N$ and that signs $g(p) \ -1,1\$ are specified for the primes $p N^c$, with e̱gin align 1N _ am̱od N |S_f(ap)-g(p)S_f(a)|^2 & B_1 NM &&(p A,\ p N^c), eq:gauss-small-dilations \\ _ p<N\ A 1p & B_2 \ M, N\ . eq:gauss-missing-primes align Then a real Dirichlet character $ $ modulo $N$ satisfies \[ #\ 1 n<N:f(n) (n)\ C N M . \] The principal character and the Legendre symbol are the two possibilities for $ $. theorem e̱gin proof We first state the two published inputs and check the analytic hypothesis of the one that was conditional. Mangerel calls a prime $N$ of $A$-Littlewood type when every nonprincipal character modulo $N$ has a nonvanishing $L$-function throughout e̱gin equation eq:gauss-littlewood s>1-A N N , | s| N. equation For each fixed $A$, choose $N$ so large that $A N/ N<1/8$. The region in eq:gauss-littlewood is then contained in $ s
6. E. Sofos, Statistics of small prime quadratic
7. R. Mrazović, A random model for the Paley graph,
8. H. L. Montgomery, Topics in multiplicative Large first prime nonresidues and gaps between them sec:large-nonresidue-gaps For a prime $p$, write $n_1(p)<n_2(p)<ḑots$ for the primes that are quadratic nonresidues modulo $p$. In particular, $p$ itself is not in this list. Sofos's Theorem 4 in i̧te SofosNonresidues proves a lower bound of order $p$ unconditionally, and a lower bound of order $p p$ under GRH, for the simultaneous largeness of $n_1(p)$ and $n_2(p)-n_1(p)$. Its entire conditional conclusion follows here. e̱gin theorem [Sofos's conditional large-gap conclusion] thm:large-nonresidue-gaps There is an absolute constant $c_0>0$ such that \[ \ n_1(p),\ n_2(p)-n_1(p)\ >c_0( p)( p) \] for infinitely many primes $p$. theorem e̱gin proof We prove the prime sign-pattern estimate used in the argument, including its conductor and uniformity details. All logarithms in this proof are natural. Fix \[ ąppa= 1 32 2 , y=ąppa( x)( x), k= (y), \] where $x$ tends to infinity. Assign any signs $ǎrepsilon_ \ -1,1\ $ to the primes $y$. We will count primes $p (x,2x]$ with all $i̱gl( pi̱gr)=ǎrepsilon_ $, and, for later use, also impose $p 1 4$. Put \[ Q=8 _ 3 y\\ prime . \] The quadratic symbol $i̱gl( pi̱gr)$ on primes $p>y$ is the value at $p$ of a real Dirichlet character modulo $Q$. Indeed, one can first take the primitive character of fundamental discriminant $8$ for $=2$, and of fundamental discriminant $ $ or $4$ for an odd prime $ $, and then induce it modulo $Q$. Denote this character by $ḩi_ $. Also write $ḩi_ -4 $ for the character giving the sign of $-1$, induced modulo $Q$. For a subset $S$ of the primes at most $y$, set $d_S= _ S$, $ḩi_S= _ S ḩi_ $, and $ǎrepsilon_S= _ S ǎrepsilon_ $. The empty products are the principal character modulo $Q$ and $1$, respectively. On primes coprime to $Q$, $ḩi_S$ is the quadratic character of the positive squarefree integer $d_S$. It is principal precisely when $S$ is empty. Moreover, $ḩi_ -4 ḩi_S$ is always nonprincipal: its qua
9. M. Mourtada and V. Kumar Murty, Extreme Euler--Kronecker constants of quadratic fields sec:extreme-quadratic-euler-kronecker For a number field $K$, its Euler--Kronecker constant is \[ _K= _ s 1 ( _K' _K (s)+ 1 s-1 ). \] For a fundamental discriminant $D$, the factorization $ _ Q( D) (s)= (s)L(s,ḩi_D)$ gives \[ _ Q( D) = + L' L(1,ḩi_D), \] where $ $ is Euler's constant. Mourtada--Murty i̧te MourtadaMurtyOmega proved unconditionally that both signs attain the lower order $|D|+O(1)$ infinitely often. Their GRH result adds the term $|D|$. We recover that complete conditional lower bound, including the coefficient one on both logarithms and the published $x$ abundance. For the full statement and proof in an accessible author source, see i̧te[Chapter 4, Theorem 17] MourtadaThesis ; that chapter is explicitly identified as the work published in i̧te MourtadaMurtyOmega . e̱gin theorem [Mourtada--Murty's conditional omega conclusions] thm:extreme-quadratic-euler-kronecker There are absolute constants $C_+,C_-$ such that, for every sufficiently large $x$, there are $x$ primes $p x$, $p 1 4$, separately for each sign, such that \[ L' L(1,ḩi_p) x+ x-C_+, \] and $x$ such primes for which \[ - L' L(1,ḩi_p) x+ x-C_-. \] Consequently the same two lower bounds hold for $_ Q( p)$ and $- _ Q( p)$, after altering the constants. theorem e̱gin proof We give the full averaging argument. Let $X$ tend to infinity, and put \[ ąppa= 1 32 2 , y=ąppa( X)( X), Z=( X)^ 32 , k= (y). \] For sufficiently large $X$ one has $2<y<Z<X$. For $\ -1,1\$ let $ P_ $ consist of the primes $p (X,2X]$ satisfying $p 1 4$ and \[ ( p )= for every prime y. \] Write $W_ = _ p P_ p$. The prime-pattern computation eq:nonresidue-pattern-projector -- eq:nonresidue-pattern-asymptotic gives e̱gin equation eq:ek-pattern-mass W_ = X 2^ k+1 +O\! (X^ 7/8 ^2(3QX) ), Q=8 _ 3 y\\ prime . equation Indeed, the characters in the projector are the quadratic characters of the squareclasses $d_S$ and $-d_S$, where $d_S= _ S
10. A. I. Badzyan, The Euler--Kronecker constant,
11. K. Ford, F. Luca and P. Moree, Cyclotomic Euler--Kronecker constants and a second asymptotic coefficient sec:cyclotomic-euler-kronecker Write $K_m= Q( _m)$, where $_m$ is a primitive $m$th root of unity, and define \[ _m= _ s 1 ( _ K_m ' _ K_m (s)+ 1 s-1 ). \] Badzyan's published conditional conclusion i̧te[Section 2] Badzyan2010 concerns every positive integer $m$, including composite conductors. We retain that scope and its full order of magnitude. The second conclusion below is the GRH clause of Ford--Luca--Moree's Theorem 2 i̧te FordLucaMoree2014 . e̱gin theorem [Badzyan's bound] thm:cyclotomic-euler-kronecker There is an effective absolute constant $C$ such that, for every integer $m 3$, \[ | _m| C m (3m). \] theorem We prove the sieve estimate needed for the prime-power terms explicitly. This also avoids applying a prime-only Brun--Titchmarsh estimate to a sum containing higher powers. e̱gin lemma lem:cyclotomic-prime-power-sieve For $m 3$, put \[ M_m=\ j N:mj+1 m^ 32 , mj+1 is a prime power \ . \] Uniformly for $1 T m^ 31$, \[ A_m(T):=#\ j M_m:j T\ m ǎrphi(m) T (2T) . \] Consequently \[ _ j M_m 1j m ǎrphi(m) (3m). \] All implied constants are effective and absolute. lemma e̱gin proof It suffices first to treat large $T$; bounded $T$ follows by enlarging the absolute constant. Put $z=T^ 1/8$ and \[ G_m(z)= _ d z\\(d,m)=1 ^2(d) ǎrphi(d) . \] Here $ $ is the Möbius function. For a squarefree $d$, the sum of $1/n$ over all positive integers with radical $d$ is $1/ǎrphi(d)$. Therefore \[ G_m(z) H_m(z):= _ n z\\(n,m)=1 1n. \] Every positive integer factors uniquely as $ab$, where every prime factor of $a$ divides $m$ and $(b,m)=1$. Writing $a m^ $ for the first condition, we obtain \[ _ n z 1n = _ a m^ 1a H_m(z/a) H_m(z) _ p m (1-p^ -1 )^ -1 . \] In particular e̱gin equation eq:cyclotomic-sieve-denominator G_m(z) ǎrphi(m) m z. equation For completeness, the required upper sieve follows by minimizing a finite quadratic form. Use real weights $_d$ supported on sq
12. J. Hoffstein and P. Lockhart,
13. D. Goldfeld, J. Hoffstein, and D. Lieman, The exponent of imaginary quadratic class groups sec:exponent For a negative fundamental discriminant $D$, let $E(D)$ be the exponent of the ideal class group of $K= Q( D)$: the least positive integer annihilating every ideal class. Boyd and Kisilevsky i̧te BK proved under ERH that $E(D)$. The following recovers that exact qualitative conclusion, and the published order of magnitude recorded in i̧te[Introduction] EKN . The sharper asymptotic coefficient $1/2$ obtained under ERH is not asserted. e̱gin lemma [A small split prime] lem:split-prime There is an effective absolute constant $C_1$ such that for every negative fundamental discriminant $D$, some rational prime $p$ satisfies \[ p D, ( Dp )=1, p C_1( (3|D|))^8. \] Here the symbol is the Kronecker symbol, including at $p=2$. lemma e̱gin proof Put $Q=|D|$ and $ḩi=ḩi_D$. This is a real primitive nonprincipal Dirichlet character modulo $Q$. Add the principal smoothed sum to eq:smoothed . The estimates proved there give \[ _ n x (n)(1+ḩi(n))(1-n/x) = x2+Oi̱gl(x^ 7/8 (3Q) +x^ 7/8 ^2(2x)+ (Qx)i̱gr). \] If there were no split prime at most $x$, all contributions with $n=p$ and $p D$ would vanish. The terms supported on prime powers with $p D$ are $O( (3Q) x)$: each such prime contributes at most $2 x$, and there are at most $Q/ 2$ such primes. The remaining prime-power terms with exponent at least two have total absolute value $O( x ^2(2x))$. Indeed, there are at most $x$ possible underlying primes, at most $x/ 2$ exponents per prime, and each von Mangoldt weight is at most $x$; the slightly sharper stated bound also follows by summing $_ a 2 (x^ 1/a )$ and the elementary bound $(t) t$. Consequently the entire left side would be bounded by these two errors. Take $x=C_1( (3Q))^8$. The ratio of the character-zero error to $x$ is $O(C_1^ -1/8 )$, while each other displayed error is $o(x)$ as $Q$. Choosing $C_1$ large gives a contradiction for all sufficiently large $Q$. The
14. J.-H. Ahn and S.-H. Kwon, The complete Richaud--Degert class-number-two classification sec:rd-class-two For this section, a squarefree integer $D>1$ is of extended Richaud--Degert type if there are integers $m 1$ and $r 0$ such that \[ D=m^2+r, r 4m. \] This is the divisibility definition in i̧te[Introduction] MollinRD2009 ; no additional normalization of $r$ is imposed. We use $D$ for the squarefree radicand and $=D$ if $D 1 4$, and $=4D$ otherwise, for the fundamental discriminant. The class number is the ordinary ideal class number, without a positivity condition on generators. e̱gin theorem [Mollin's complete list] thm:rd-class-two There are exactly $107$ real quadratic fields of extended Richaud--Degert type with ordinary class number two. Their squarefree radicands are precisely the following: \[ e̱gin array rrrrrrr 10&15&26&30&34&35&39\\ 42&51&65&66&78&85&87\\ 95&102&105&110&119&122&123\\ 138&143&146&165&182&194&203\\ 205&215&221&222&230&258&285\\ 287&318&327&357&362&365&395\\ 402&429&447&482&485&527&533\\ 623&629&635&645&678&741&782\\ 843&885&902&957&965&1022&1085\\ 1157&1173&1205&1245&1298&1469&1517\\ 1533&1605&1685&1853&1965&2013&2037\\ 2045&2085&2093&2117&2301&2373&2397\\ 2613&2717&2813&3005&3237&3365&3597\\ 3605&3813&4245&4277&4773&4893&5645\\ 5757&5885&5957&6573&7157&7733&8333\\ 9005&14405&&&&& array \] The list is the one in i̧te[Theorems 3.1--3.13 and Remark 3.1] MollinRD2009 ; its completeness now requires no GRH. theorem e̱gin lemma [A regulator bound for the full divisibility family] lem:rd-regulator If $D 3$ is of extended Richaud--Degert type and $R_D$ denotes the logarithm of its fundamental unit greater than one, then \[ R_D (16D). \] lemma e̱gin proof Fix a representation $D=m^2+r$ as above and put \[ = (m+ D)^2 |r| . \] Its norm is one, and its trace is \[ 4m^2 |r| +2 sgn (r) Z, \] because $|r| 4m$. Thus $ $ is an algebraic integer unit. It is greater than one: equivalently, $(m+ D)^2>|m^2-D|$, which holds for $m,D>0$. Consequently $R_D$,
15. S. Louboutin, Prime producing quadratic
16. A. Biró and K. Lapkova, The class number
17. J. Lee, The complete determination of wide The complete class-number-one list in the same family The same finite enumeration also settles the earlier conditional class-number-one list. Louboutin i̧te[pp. 315--316] LouboutinPrime1990 records the following $43$ radicands, attributing the classification under the extended Riemann hypothesis to his work and to Mollin--Williams. That source normalizes $r$ by $-m<r m$ or permits $r= 4m/3$. Here we prove the identical list for the full divisibility definition of extended Richaud--Degert type from the beginning of this section, which contains that family. e̱gin corollary [The complete extended Richaud--Degert class-number-one list] cor:rd-class-one There are exactly $43$ real quadratic fields of extended Richaud--Degert type with ordinary class number one. Their squarefree radicands are precisely \[ e̱gin array rrrrrrr 2&3&5&6&7&11&13\\ 14&17&21&23&29&33&37\\ 38&47&53&62&69&77&83\\ 93&101&141&167&173&197&213\\ 227&237&293&398&413&437&453\\ 573&677&717&1077&1133&1253&1293\\ 1757&&&&&& array \] corollary e̱gin proof The cutoff proof for Lemma lem:rd-class-two-cutoff actually excludes every field with $h_D 2$ and $D 10^8$. Indeed, its only use of the class-number value is \[ L(1,ḩi_ )=2h_D R_D 4R_D 4 (16D), \] which holds for all such $h_D$. The same Tatuzawa lower bound contradicts this inequality for $D 10^8$. Thus every field in this corollary is among the $149024$ candidates already enumerated and unconditionally verified in the proof of Theorem thm:rd-class-two . Selecting the records with class number one gives exactly the $43$ displayed radicands. This selected list has been compared, entry by entry, with the published list on pp. 315--316 of i̧te LouboutinPrime1990 ; the two lists agree. All positive entries and all exclusions use the same complete arithmetic verification as the class-number-two classification. This proves both directions without a further hypothesis or a new enumeration. proof This corollary concerns
18. A. Bourdon and P. Pollack, Torsion subgroups of CM
19. P. T. Joshi, The size of L(1,chi) for real
20. P. L. Clark, T. Genao, P. Pollack and F. Saia, The typical least degree of a CM point on a modular curve sec:typical-cm-degrees For a modular curve $X/ Q$, let $d_ CM (X)$ be the least residue-field degree over $Q$ of a noncuspidal closed point represented by an elliptic curve with complex multiplication. For a negative quadratic-order discriminant $ $, let $d_ , CM (X)$ denote the corresponding minimum with endomorphism order of discriminant $ $. The curves $X_0(N)$ and $X_1(N)$ classify, respectively, elliptic curves with a cyclic subgroup of order $N$ and with a point of order $N$. All orders, including nonmaximal orders, are permitted. Clark--Genao--Pollack--Saia proved the following two assertions under GRH i̧te[Theorem 1.7] CGPS2022 . The proof below preserves their exact typical exponent. Their sharper maximal-order estimates in Theorems 1.3 and 1.5 are separate assertions and are not being transferred here. e̱gin theorem [The published typical-degree conclusions] thm:typical-cm-degrees For every $>0$, all but $o(x)$ positive integers $N x$ satisfy both \[ 2^ (1/2- ) x d_ CM (X_0(N)) 2^ (1/2+ ) x \] and \[ N\,2^ (1/2- ) x d_ CM (X_1(N)) N\,2^ (1/2+ ) x . \] theorem e̱gin proof We reproduce the counting argument of i̧te[ 5.6] CGPS2022 , with the analytic estimates replaced explicitly. All logarithms in this proof are natural. It suffices by monotonicity to prove the assertions for $0< <1/4$. The algebraic inputs. Write $h_ $ for the proper class number of the order. For squarefree $B$, put $B_-= _ p B,\;( /p)=-1 p$, where the symbols are Kronecker symbols, and put $_0(B)=B _ p B (1+1/p)$. The unconditional algebraic results i̧te[Lemma 4.4 and Proposition 4.6] CGPS2022 give e̱gin align d_ , CM (X_0(B)) & h_ _0(B_-), B\ squarefree , eq:cmtyp-algebra \\ (N) 6d_ CM (X_0(N)) & d_ CM (X_1(N)) (N) 2d_ CM (X_0(N)), N 3. eq:cmtyp-tower align The constants in the first line are absolute and uniform in $ $ and $B$. Also, if $ $ is fundamental and $( /p)=1$ for every $
21. H. Pasten, Shimura curves and the abc conjecture, Wei's height bound for adjacent CM types sec:wei-height Wei's published Theorem 1.4 i̧te Wei assumes the absence of certain Siegel zeros for relative $L$-functions of CM fields, a consequence of GRH. Its hypothesis can be removed even though the imported zero-free theorem does not cover these relative $L$-functions: Stark's theorem reduces the particular exceptional zeros in question to imaginary quadratic Dirichlet $L$-functions. The argument uses the proved averaged Colmez formula, not the full Colmez conjecture. For a number field $F$, set $d_F=| disc (F)|$ and $r_F=[F: Q]^ -1 d_F$, the logarithm of its root discriminant. A CM field $E$ of degree $2g$ is a totally imaginary quadratic extension of its maximal totally real subfield $E^+$. A CM type $ $ chooses one embedding from each complex-conjugate pair of embeddings $E C$. Write $h(E, )$ for the stable Faltings height of an abelian variety with multiplication by the full ring $O_E$ and type $ $, using Wei's normalization. It depends only on $(E, )$. e̱gin theorem thm:wei-height For each positive integer $g$, there are effective constants $C_1(g)>0$ and $C_2(g)$ with the following property. Let $E$ be any CM field of degree $2g$. Let $A_i/K_i$, for $i=1,2$, be abelian varieties of dimension $g$ with embeddings \[ O_E End _ K_i (A_i), \] and let their CM types be $_i$. If $| _1a̧p _2|=g-1$, then \[ h_ Falt ^ st (A_1) +h_ Falt ^ st (A_2) 2 C_1(g) r_ K_1 +r_ K_2 2+C_2(g). \] theorem e̱gin proof We first remove the exceptional-zero hypothesis, then reconstruct the height and discriminant comparisons. Exceptional zeros. Put $F=E^+$ and $ḩi=ḩi_ E/F $, so that $L(s,ḩi)= _E(s)/ _F(s)$ is the entire quadratic Hecke $L$-function attached to $E/F$. The usual Dedekind zero-free-region theorem, in the form used by Wei, states that this function has at most one zero in \[ Re s 1- 1 4 d_E , | Im s| 1 4 d_E , \] and a zero there is real and simple. This follows also from Stark
22. X. Wei, Smallness of Faltings heights of CM abelian
23. X. Wei, arXiv:2111.00617v1 (2021),arXiv
24. H. M. Stark, Some effective cases of the Fully effective André--Oort for powers of the modular curve sec:effective-ao Identify $Y(1)$ with the affine $j$-line. A special point is a singular modulus, the $j$-invariant of an elliptic curve with complex multiplication. Its discriminant is the negative discriminant of its endomorphism order, which need not be maximal. A special subvariety of $Y(1)^n$ is an irreducible component of a locus defined by fixing some coordinates to singular moduli and imposing modular equations $_N(x_i,x_j)=0$ among the remaining coordinates. Here $_N$ is the classical modular polynomial for cyclic $N$-isogenies, and $_1(X,Y)=X-Y$. Equivalently, partition the nonconstant coordinates into blocks, choose one distinguished coordinate per block, and impose such a modular relation between it and each other coordinate of the block. For a special subvariety $V$, let $D(V)$ be the largest absolute discriminant of its constant coordinates, with $D(V)=0$ if there are none. A special subvariety of $X$ is maximal if it is not contained in a larger special subvariety of $X$. The result below is the fully effective conclusion described in i̧te[Section 1.2 and Theorem 1] BinyaminiAO under GRH for imaginary quadratic fields. It is stronger than the unconditional assertion that there are finitely many maximal special subvarieties: the bound and a terminating enumeration are computable from the input. e̱gin lemma [Effective class numbers for every imaginary quadratic order] lem:order-siegel-effective For every rational $0< <1/2$, an effectively computable positive constant $c_ $ satisfies \[ h( ) c_ | |^ 1/2- \] for every negative quadratic-order discriminant $ $. lemma e̱gin proof Write $=f^2D$, where $D$ is fundamental. Corollary cor:class gives an effective positive constant $c$ with $h(D) c |D| / (|D|)$. Since $t^ / (t)$ has an effectively computable positive lower bound for $t 3$, this implies $h(D) c'_ |D|^ 1/2-$ effectively. The order class-number
25. F. Balestrieri, A. Johnson, and R. Newton,
26. E. Gaudron and G. Remond,
27. A. Kresch and Y. Tschinkel,
28. B. Poonen, D. Testa, and R. van Luijk, Joint equidistribution of CM points sec:joint-cm-equidistribution Khayutin's published Theorems 1.3 and 3.10 i̧te Khayutin2019 assume the absence of exceptional Landau--Siegel zeros of the quadratic CM fields. We remove this analytic assumption. The two fixed splitting primes, the stated restrictions on the orders, and the genericity conditions are retained. The maximal-order case of the second statement is the published Theorem 1.4. e̱gin theorem [Khayutin's joint-packet and Galois-orbit conclusions] thm:joint-cm-equidistribution The following statements hold without an RH or GRH assumption. e̱gin enumerate Let $G$ be a $Q$-form of $PGL _2$, let $K_f G( A_f)$ be compact open, and fix a compact real torus $K_ G( R)$. Put \[ Y=G( Q)a̱ckslash G( A)/(K_ K_f). \] Let $P_i$ be $K_ $-invariant toral packets with negative fundamental discriminants $D_i$, and also write $P_i$ for their finite images in $Y$. Here a toral packet is the image of an adelic orbit of an anisotropic rational torus, as in i̧te[ 2.4.1--2.4.2] Khayutin2019 . The finite packet is a principal homogeneous space for a finite abelian group $C_i$, with a canonical surjection $C_i Pic ( O_ E_i )$, where $E_i= Q( D_i )$. Choose $_i C_i$ and put \[ N_i= \ N a: 0 a O_ E_i ,\ [ a]=[ _i]\ , _i= 1 |P_i| _ z P_i ẹlta_ (z, _i z) . \] Assume $N_i$ and that two fixed distinct primes $p_1,p_2$ split in every $E_i$. Every weak-$*$ limit point of $( _i)$ is a convex combination of the uniform probability measures on the connected components of $Y Y$. Let $G$ be a $Q$-form of $PGL _2$ split over $R$, and let $X$ be a finite product of indefinite Shimura curves relative to this same $G$, at fixed levels. Let $x_i X$ be special points such that all coordinates of $x_i$ have CM by the same imaginary quadratic order $O_i E_i$. Assume that its conductor $f_i=[ O_ E_i : O_i]$ is bounded independently of $i$. Let $_i$ be the normalized counting measure on the finite Galois orbit o
29. I. Khayutin, Effective bounds for Campagna's singular $S$ S -units sec:singular-s-units A singular modulus is the $j$-invariant of a complex elliptic curve with complex multiplication. It is an algebraic integer. For a set $S$ of rational primes, a nonzero algebraic integer $ $ is an $S$-unit if every prime divisor of $|N_ Q( )/ Q ( )|$ belongs to $S$. The set $S$ is allowed to be infinite. e̱gin theorem [Campagna's effective singular-$S$-unit conclusion] thm:singular-s-units Let $5$ be a rational prime, and put \[ S_0=\ p prime :p 1 3\ , S_ =S_0u̧p\ \ . \] There is an effectively computable $B( )$ such that every singular modulus $j$ which is an $S_ $-unit has order discriminant $ $ satisfying $| | B( )$. All these singular moduli can consequently be determined by a finite algorithm. theorem This is the precise effective conclusion of i̧te[Theorem 1.4] CampagnaBounds , published under GRH for the Dirichlet $L$-functions of imaginary quadratic fields. Mere finiteness was already unconditional. The point here is the computability of a bound valid for every discriminant. e̱gin proof We first verify the analytic hypothesis isolated in i̧te[Definition 5.1 and Theorem 5.5] CampagnaBounds . That hypothesis asks for one effective $k 0$ such that, for every negative fundamental discriminant $D$, e̱gin equation eq:campagna-P L' L (1,ḩi_D) -0.2485 |D|-k. equation Corollary cor:dirichlet-logderiv gives an effective absolute $A$ with $|L'/L(1,ḩi_D)| A (|D|)$. Because $A (t)-0.2485 t$ tends to $-$ effectively, its positive part has an effectively bounded supremum for $t 3$. Any effective upper bound for that supremum is a valid $k$. Thus the hypothesis in Campagna's stronger Theorem 5.5 is satisfied. We spell out the remaining transfer, including why its height comparison has a strict positive margin. Write $d=| |$ and $=f^2D$. In the normalization used in i̧te[Proposition 5.4] CampagnaBounds , the unconditional Chowla--Selberg and Nakkajima--Taguchi formulas give \[ h_F(E)=
30. E. Jeřábek, Integer factoring and modular square roots, Miller's equivalences between factoring and arithmetic functions sec:miller-equivalence Theorems 3--5 of Miller's published paper i̧te Miller1976 concern deterministic reductions between arithmetic functions and recognition of their graphs. These are separate from the primality-testing conclusion of that paper, which has long had an unconditional replacement. We recover the full qualitative conclusions of those three theorems. For $n 1$, let $(n)$ be the exponent of the finite group $( Z/n Z)^ $, and put \[ '(n)= lcm _ p n (p-1), '(1)=1. \] Write $Fac (n)$ for the complete prime factorization of $n$, including its exponents. All integers are supplied in binary. e̱gin theorem [Miller's functional-equivalence and recognition conclusions] thm:miller-equivalence The functions $Fac$, $ǎrphi$, $ $, and $'$ are deterministically polynomial-time equivalent in the following strong sense: given $n$ and the single value of any one of these functions at $n$, one can compute the values of all the others in time polynomial in $(2n)$. Moreover, complete integer factorization is deterministically polynomial-time Turing equivalent to computing the eventual period of the base-$b$ expansion of $1/a$, for integers $a 1$, $b 2$. Finally, the graphs of $ǎrphi$, $ $, $'$, and $Fac$ are recognizable in deterministic polynomial time. theorem e̱gin proof We first give a uniform version of Miller's splitting argument that also explains why a single supplied value suffices. Suppose that $n 3$ is odd and that a positive integer $M$ satisfying e̱gin equation eq:miller-multiple '(n) M equation is supplied. There is a deterministic algorithm factoring $n$ in time polynomial in $(2n)+ (2M)$. It does not require the prime factorization of $M$. Write $M=2^s u$ with $u$ odd, and set $B= C_0( (3n))^8$, with $C_0$ from Lemma lem:witness . For every composite factor $d n$ encountered during recursion, first test whether $d$ is a prime pow
31. M. Agrawal, N. Kayal, and N. Saxena,
32. G. L. Miller,
33. F. Morain, G. Renault, and B. Smith,
34. A. Dabrowski, J. Pomykala, and I. E. Shparlinski,
35. N. Gupta, A. Sikarov, and I. Volkovich, Deterministic extraction of roots sec:roots Adleman, Manders, and Miller i̧te AMM , Theorems I and III, proved under ERH that the least square root, and more generally the least $n$th root for every fixed positive integer $n$, can be found modulo a prime in deterministic polynomial time. Their Theorem II(a) gives deterministic polynomial-time square-root extraction modulo a supplied fully factored integer. These qualitative statements survive replacing ERH by a uniform effective polylogarithmic bound for the least power nonresidue. We give the algebraic argument, including the distinction between fixed $n$ and $n$ supplied in binary. e̱gin theorem [Adleman--Manders--Miller root-extraction conclusions] There is a deterministic algorithm that, given a prime $p$, an integer $a$, and a positive integer $n$, finds all roots of $X^n-a$ in $F_p$ in time polynomial in $n$ and $p+ (2+|a|)$. In particular, for each fixed $n$, the least nonnegative root can be found, or nonexistence decided, in deterministic polynomial time in the bit length of $(p,a)$. There is also a deterministic polynomial-time algorithm for finding a square root of $a$ modulo an integer $M$ whose complete prime factorization is supplied, or deciding that no such root exists. theorem e̱gin proof Reduce $a$ modulo $p$. If $a=0$, its only root is $0$; hence assume $a 0$. The case $p=2$ is immediate. Write $N=p-1$, and factor $n$ by trial division. This costs polynomial time in the numerical value $n$, as the theorem allows; it does not assert polynomial time in $n$. For each prime $r (n,N)$, compute $s_r=v_r(N)$ and put $R_r=r^ s_r$. Put \[ U=N / _ r (n,N) R_r. \] Then $(n,U)=1$. These calculations require no factorization of $N$ outside the primes already dividing $n$. For each such $r$, choose successively integers $z=1,2,$ until $z 0 p$ and $z^ N/r 1$ in $F_p$. A nontrivial character of order $r$ has precisely the $r$th powers as its kernel. Lemma lem:witness
36. L. Adleman, K. Manders, and G. L. Miller, Deterministic splitting in even degree and Gao's balance test sec:even-splitting The input in this section is a prime $p$ and a monic squarefree polynomial $f F_p[X]$ of degree $n>1$ which splits completely in $F_p$. The distinct roots need not be known. These input promises can be checked in polynomial time by taking a gcd with the derivative and computing $X^pm̱od f$. A proper factor means a divisor with degree strictly between $0$ and $n$. Rónyai's published factoring theorem includes a deterministic polynomial-time algorithm under GRH to find a proper factor of every such $f$ of even degree; see i̧te RonyaiFactoring and the explicit discussion in i̧te[Section 1] GaoFactoring . Gao's i̧te[Theorems 3.1 and 3.3] GaoFactoring gives a more general decomposition based on a balance condition on the roots. We prove these conclusions with a direct construction, replacing the GRH-dependent choice of a scalar root of unity by Lemma lem:witness . Suppose first that $p$ is odd, and write $p-1=2^s w$ with $w$ odd. Fix an element $ F_p^ $ of order $2^s$. Every $a 0$ has a unique expression \[ a= ^u , 0 u<2^s, ^w=1. \] Define $_ (a)=1$ if $u<2^ s-1$ and $_ (a)=-1$ otherwise. In particular $_ (-a)=- _ (a)$. For the set $F$ of roots of $f$, put \[ t_ =#\ e̱ta F \ \ : _ ( -e̱ta)=1\ . \] Gao calls $F$ square balanced with respect to $ $ when all $t_ $ equal $(n-1)/2$. This agrees with his definition using the scalar square-root choice $_2$: $_2(a^2)=a$ exactly when $u<2^ s-1$ i̧te[Lemma 2.6(d) and Section 2.4] GaoFactoring . When $p 3 4$, $ _ $ is the quadratic character. e̱gin theorem [Gao's balance decomposition and Rónyai's even-degree case] thm:even-splitting There is a deterministic algorithm with bit complexity polynomial in $n$ and $p$ which, for odd $p$, constructs $ $ as above and computes all polynomials \[ f_j(X)= _ F\ _ =j (X- ), 0 j n-1. \] Their product is $f$, and the nonconstant members are pairwise coprime. The alg
37. The Pollack--Schorn algorithm for three squares sec:three-square-algorithm The existence of a representation by three squares is classical. The previously conditional conclusion here concerns its computational cost when the integer is supplied without a factorization. We recover the exact expected complexity in i̧te[Theorem $3'$] PollackSchorn , rather than just polynomial-time computability. As in that paper, $M(b)$ denotes a standard bit-complexity bound for multiplying two $b$-bit integers; one may use ordinary multiplication or either of the fast multiplication bounds specified there. In particular $M(b) b$ and $M(cb)=O_c(M(b))$ for every fixed $c>0$. e̱gin theorem [Pollack--Schorn's three-square algorithm] thm:three-square-algorithm There is a randomized algorithm which, given a positive integer $n$ not of the form $4^k(8m+7)$, returns integers $x,y,z$ with $n=x^2+y^2+z^2$, always correctly, and has expected bit complexity \[ O\! ( ( n)^2 n \,M( n) ) (n ). \] No factorization of $n$ is an input. theorem e̱gin proof First remove powers of $4$ and at the end rescale the three coordinates by the corresponding power of $2$. This costs $O( n)$ bit operations. It is therefore enough to treat $n 1,2,3,5,6 8$. Bounded values of $n$ can be treated by exhaustive search. Throughout the proof let $b= _2(n+1)$. We first establish the replacement for the only ERH-dependent input in i̧te[Section 3.1] PollackSchorn . There are effective absolute constants $C,c>0$ such that, for $(a,q)=1$, $q 2$, and $X Cq^ 16$, e̱gin equation eq:three-square-ap-density #\ p X:p prime ,\ p a q\ c X ǎrphi(q) X . equation Indeed, character orthogonality and Lemma lem:psi , including the principal-character term, give \[ _ m X\ a\ (q) (m) = X ǎrphi(q) +Oi̱gl(X^ 7/8 ^2(qX)i̱gr). \] The contribution from proper prime powers is $O(X^ 1/2 ^2X)$. Since $ǎrphi(q) q$, taking the effective absolute constant $C$ sufficiently large makes the two error terms at most $X/(2ǎrphi(q))$ for every $X Cq^ 16$. D
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