Siegel Zeros Research Articles

Analytic ranks of automorphic L$L$‐functions and Landau–Siegel zeros

AbstractWe relate the study of Landau–Siegel zeros to the ranks of Jacobians of modular curves for large primes . By a conjecture of Brumer–Murty, the rank should be equal to half of the dimension. Equivalently, almost all newforms of weight two and level have analytic rank . We show that either Landau–Siegel zeros do not exist, or that, for wide ranges of , almost all such newforms have analytic rank . In particular, in wide ranges, almost all odd newforms have analytic rank equal to one. Additionally, for a sparse set of primes in a wide range, we show that the rank of is asymptotically equal to the rank predicted by the Brumer–Murty conjecture.

Siegel Zeros, Twin Primes, Goldbach’s Conjecture, and Primes in Short Intervals

Abstract We study the distribution of prime numbers under the unlikely assumption that Siegel zeros exist. In particular, we prove for $$ \begin{align*} & \sum_{n \leq X} \Lambda(n) \Lambda(\pm n+h) \end{align*}$$an asymptotic formula that holds uniformly for $h = O(X)$. Such an asymptotic formula has been previously obtained only for fixed $h$ in which case our result quantitatively improves those of Heath-Brown (1983) and Tao and Teräväinen (2021). Since our main theorems work also for large $h$, we can derive new results concerning connections between Siegel zeros and the Goldbach conjecture and between Siegel zeros and primes in almost all very short intervals.

Non-existence of Siegel zeros for cuspidal functorial products on 𝐺𝐿(2)×𝐺𝐿(3)

Let π 1 \pi _{1} and π 2 \pi _{2} be two cuspidal automorphic forms over a number field F \mathbf {F} on G L ( 2 ) GL(2) and G L ( 3 ) GL(3) respectively. In this work, we prove the non-existence of Siegel zeros for the automorphic L L -function associated to a cuspidal functorial product π 1 × π 2 \pi _{1} \times \pi _{2} .

Smallness of Faltings heights of CM abelian varieties

We prove that assuming the Colmez conjecture and the “no Siegel zeros” conjecture, the stable Faltings height of a CM abelian variety over a number field is less than or equal to the logarithm of the root discriminant of the field of definition of the abelian variety times an effective constant depending only on the dimension of the abelian variety. In view of the fact that the Colmez conjecture for abelian CM fields, the averaged Colmez conjecture, and the “no Siegel zeros” conjecture for CM fields with no complex quadratic subfields are already proved, we prove unconditional analogues of the result above. In addition, we also prove that the logarithm of the root discriminant of the field of everywhere good reduction of CM abelian varieties can be “small”.

The Hardy–Littlewood–Chowla conjecture in the presence of a Siegel zero

Assuming that Siegel zeros exist, we prove a hybrid version of the Chowla and Hardy–Littlewood prime tuples conjectures. Thus, for an infinite sequence of natural numbers x $x$ , and any distinct integers h 1 , ⋯ , h k , h 1 ′ , ⋯ , h ℓ ′ $h_1,\dots ,h_k,h^{\prime }_1,\dots ,h^{\prime }_\ell$ , we establish an asymptotic formula for ∑ n ⩽ x Λ ( n + h 1 ) ⋯ Λ ( n + h k ) λ ( n + h 1 ′ ) ⋯ λ ( n + h ℓ ′ ) \begin{equation*} \hspace*{13pt}\sum _{n\leqslant x}\Lambda (n+h_1)\cdots \Lambda (n+h_k)\lambda (n+h_{1}^{\prime })\cdots \lambda (n+h_{\ell }^{\prime })\hspace*{-13pt} \end{equation*} for any 0 ⩽ k ⩽ 2 $0\leqslant k\leqslant 2$ and ℓ ⩾ 0 $\ell \geqslant 0$ . Specializing to either ℓ = 0 $\ell =0$ or k = 0 $k=0$ , we deduce the previously known results on the Hardy–Littlewood (or twin primes) conjecture and the Chowla conjecture under the existence of Siegel zeros, due to Heath-Brown and Chinis, respectively. The range of validity of our asymptotic formula is wider than in these previous results.

On the Conditional Bounds for Siegel Zeros

In this paper, under a weakened version of Hardy—Littlewood Conjecture on the number of representations in Goldbach problem, we shall prove bounds for the Siegel zeros of real primitive Dirichlet characters for composite moduli.

Effective Quantum Unique Ergodicity for Hecke–Maass Newforms and Landau–Siegel Zeros

Abstract We show that Landau–Siegel zeros for Dirichlet L-functions do not exist or that quantum unique ergodicity for $\mathrm{GL}_2$ Hecke–Maass newforms holds with an effective rate of convergence. This follows from a more general result: Landau–Siegel zeros of Dirichlet L-functions repel the zeros of all other automorphic L-functions from the line $\mathrm{Re}(s)=1$.

Sieving intervals and Siegel zeros

Assuming that there exist (infinitely many) Siegel zeros, we show that the (Rosser–)Jurkat–Richert bounds in the linear sieve cannot be improved, and similarly look at Iwaniec’s lower bound on Jacobsthal’s problem, as well as minor improvements to the Bru

A Bombieri–Vinogradov Theorem for Higher-Rank Groups

Abstract We establish a result of Bombieri–Vinogradov type for the Dirichlet coefficients at prime ideals of the standard $L$-function associated to a self-dual cuspidal automorphic representation $\pi $ of $\operatorname {GL}_n$ over a number field $F$ when $\pi$ is not a quadratic twist of itself. Our result does not rely on any unproven progress towards the generalized Ramanujan conjecture or the nonexistence of Landau–Siegel zeros. In particular, when $\pi $ is fixed and not equal to a quadratic twist of itself, we prove the first unconditional Siegel-type lower bound for the twisted $L$-values $|L(1,\pi \otimes \chi )|$ in the $\chi $-aspect, where $\chi $ is a primitive quadratic Hecke character over $F$. Our result improves the levels of distribution in other works that relied on these unproven hypotheses. As applications, when $n=2,3,4$, we prove a $\textrm {GL}_n$ analogue of the Titchmarsh divisor problem and a nontrivial bound for a certain $\textrm {GL}_n\times \textrm {GL}_2$ shifted convolution sum.

Zeros of Rankin–Selberg $L$-functions at the edge of the critical strip (with an appendix by Colin J. Bushnell and Guy Henniart)

Let \pi (respectively \pi_0 ) be a unitary cuspidal automorphic representation of \mathrm{GL}_m (respectively \mathrm{GL}_{m_0} ) over \mathbb{Q} . We prove log-free zero density estimates for Rankin–Selberg L -functions of the form L(s,\pi\times\pi_0) , where \pi varies in a given family and \pi_0 is fixed. These estimates are unconditional in many cases of interest; they hold in full generality assuming an average form of the generalized Ramanujan conjecture. We consider applications of these estimates related to mass equidistribution for Hecke–Maaß forms, the rarity of Landau–Siegel zeros of Rankin–Selberg L -functions, the Chebotarev density theorem, and \ell -torsion in class groups of number fields.

On a conjecture of Soundararajan

Building on recent work of A. Harper (2012), and using various results of M. C. Chang (2014) and H. Iwaniec (1974) on the zero-free regions of $L$-functions $L(s,\chi)$ for characters $\chi$ with a smooth modulus $q$, we establish a conjecture of K. Soundararajan (2008) on the distribution of smooth numbers over reduced residue classes for such moduli $q$. A crucial ingredient in our argument is that, for such $q$, there is at most one "problem character" for which $L(s,\chi)$ has a smaller zero-free region. Similarly, using the "Deuring-Heilbronn" phenomenon on the repelling nature of zeros of $L$-functions close to one, we also show that Soundararajan's conjecture holds for a family of moduli having Siegel zeros.

Inequalities between sums over prime numbers in progressions

We investigate a new type of tendency between two progressions of prime numbers which is in support of the claim that prime numbers that are congruent to 3 modulo 4 are favored over prime numbers that are congruent to 1 modulo 4. In particular, we show that the Riemann hypothesis for the corresponding L-function is equivalent to the occurrence of such a tendency. A generalization to teams of progressions of prime numbers is given, where the teams are formed by grouping according to the values of a quadratic character. In this way, it is shown that there is a tendency favoring prime numbers belonging to progressions arising from the quadratic nonresidues modulo a prime number congruent to 3 or 5 modulo 8. The scope of the tendency is extended conditionally, either by assuming the Riemann hypothesis for certain Dirichlet L-functions or by the presence of Siegel zeros. Our approach requires numerical verifications over certain ranges of the parameters, and in this respect, we freely benefit from computer software to carry out such tasks. Lastly, the divisor function is seen to be favorable over its average value along semigroups by comparing partial sums of the associated Dirichlet series.

Imaginary quadratic number fields with class groups of small exponent

Let $D<0$ be a fundamental discriminant and denote by $E(D)$ the exponent of the ideal class group $\text{Cl}(D)$ of $K={\mathbb Q}(\sqrt{D})$. Under the assumption that no Siegel zeros exist we compute all such $D$ with $E(D)$ is a divisor of $8$. We compute all $D$ with $|D|\leq 3.1\cdot 10^{20}$ such that $E(D)\leq 8$.

Bounds on short character sums and L-functions with characters to a powerful modulus

We combine a classical idea of Postnikov (1956) with the method of Korobov (1974) for estimating double Weyl sums, deriving new bounds on short character sums when the modulus q has a small core Πp|qp. Using this estimate, we improve certain bounds of Gallagher (1972) and Iwaniec (1974) for the corresponding L-functions. In turn, this allows us to improve the error term in the asymptotic formula for primes in short arithmetic progressions modulo a power of a fixed prime. As yet another application of our bounds, we substantially extend the classical zero-free region (which might include Siegel zeros). Finally, we improve the previous best value $$L=\frac{12}{5}=2.2$$ of the Linnik constant for primes in arithmetic progressions modulo powers of a fixed prime to L < 2.1115.

GOLDBACH REPRESENTATIONS IN ARITHMETIC PROGRESSIONS AND ZEROS OF DIRICHLET L ‐FUNCTIONS

Assuming a conjecture on distinct zeros of Dirichlet L-functions we get asymptotic results on the average number of representations of an integer as the sum of two primes in arithmetic progression. On the other hand the existence of good error terms gives information on the the location of zeros of L-functions and possible Siegel zeros. Similar results are obtained for an integer in a congruence class expressed as the sum of two primes.

On the least prime ideal and Siegel zeros

Let [Formula: see text] be a number field, [Formula: see text] be an integral ideal, and [Formula: see text] be the associated narrow ray class group. Suppose [Formula: see text] possesses a real exceptional character [Formula: see text], possibly principal, with a Siegel zero [Formula: see text]. For [Formula: see text] satisfying [Formula: see text] [Formula: see text], we establish an effective [Formula: see text]-uniform Linnik-type bound with explicit exponents for the least norm of a prime ideal [Formula: see text]. A special case of this result is a bound for the least rational prime represented by certain binary quadratic forms.

Quadratic forms connected with Fourier coefficients of holomorphic and Maass cusp forms

In this work we prove a prime number type theorem involving the normalised Fourier coefficients of holomorphic and Maass cusp forms, using the classical circle method. A key point is in a recent paper of Fouvry and Ganguly, based on Hoffstein–Ramakrishnan's result about the non-existence of the Siegel zeros for GL(2)L-functions, which allows us to improve preceding estimates.

Number of prime ideals in short intervals

Assuming a weaker form of the Riemann hypothesis for Dedekind zeta functions by allowing Siegel zeros, we extend a classical result of Cramér on the number of primes in short intervals to prime ideals of the ring of integers in cyclotomic extensions with norms belonging to such intervals. The extension is uniform with respect to the degree of the cyclotomic extension. Our approach is based on the arithmetic of cyclotomic fields and analytic properties of their Dedekind zeta functions together with a lower bound for the number of primes over progressions in short intervals subject to similar assumptions. Uniformity with respect to the modulus of the progression is obtained and the lower bound turns out to be best possible, apart from constants, as shown by the Brun–Titchmarsh theorem.

Common values of the arithmetic functions ϕ and σ

We show that the equation ϕ(a) = σ(b) has infinitely many solutions, where ϕ is Euler's totient function and σ is the sum-of-divisors function. This proves a fifty-year-old conjecture of Erdős. Moreover, we show that, for some c > 0, there are infinitely many integers n such that ϕ(a) = n and σ(b) = n, each having more than nc solutions. The proofs rely on the recent work of the first two authors and Konyagin on the distribution of primes p for which a given prime divides some iterate of ϕ at p, and on a result of Heath-Brown connecting the possible existence of Siegel zeros with the distribution of twin primes.

On the Ono invariants of imaginary quadratic number fields

J. Cohen, J. Sonn, F. Sairaiji and K. Shimizu proved that there are only finitely many imaginary quadratic number fields K whose Ono invariants Ono K are equal to their class numbers h K . Assuming a Restricted Riemann Hypothesis, namely that the Dedekind zeta functions of imaginary quadratic number fields K have no Siegel zeros, we determine all these K's. There are 114 such K's. We also prove that we are missing at most one such K. M. Ishibashi proved that if Ono K is large enough compared with h K , then the ideal class groups of K is cyclic. We give a short proof and a precision of Ishibashi's result. We prove that there are only finitely many imaginary quadratic number fields K satisfying Ishibashi's sufficient condition. Assuming our Restricted Riemann Hypothesis, we prove that the absolute values d K of their discriminants are less than 2.3 ⋅ 10 9 . We determine all these K's with d K ⩽ 10 6 . There are 76 such K's. We prove that there is at most one such K with d K ⩾ 1.8 ⋅ 10 11 .