Applications of theL-functions ratios conjectures

DOI10.1112/plms/pdl021zbMath1183.11050arXivmath/0509480OpenAlexW2004238784WikidataQ123124386 ScholiaQ123124386MaRDI QIDQ5292706

John Brian Conrey, Nina Claire Snaith

Publication date: 22 June 2007

Published in: Proceedings of the London Mathematical Society (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/0509480

zbMATH Keywords

Montgomery's pair correlation conjectureaverages of mollified \(L\)-functionsone-level density of family of \(L\)-functions with symplectic symmetryratios of products of \(L\)-functionstwists of Ramanujan \(\tau\)-function

Mathematics Subject Classification ID

(zeta (s)) and (L(s, chi)) (11M06) Other Dirichlet series and zeta functions (11M41) Compact groups (22C05) Nonreal zeros of (zeta (s)) and (L(s, chi)); Riemann and other hypotheses (11M26) Relations with random matrices (11M50)

Related Items

LOWER-ORDER TERMS OF THE ONE-LEVEL DENSITY OF A FAMILY OF QUADRATIC HECKE -FUNCTIONS, On the zeros of Riemann's zeta-function on the critical line, NON-VANISHING OF DIRICHLET L-FUNCTIONS AT THE CENTRAL POINT, Omega results for cubic field counts via lower-order terms in the one-level density, On mean values of mollifiers and \(L\)-functions associated to primitive cusp forms, Low-lying zeros of cubic Dirichlet \(L\)-functions and the ratios conjecture, Symplectic n-level densities with restricted support, Twin prime correlations from the pair correlation of Riemann zeros, Simple zeros of modular L -functions, Small zeros of Dirichlet \(L\)-functions of quadratic characters of prime modulus, Zeros of quadratic Dirichlet $L$-functions in the hyperelliptic ensemble, Averages of ratios of the Riemann zeta-function and correlations of divisor sums, The Ratios Conjecture and upper bounds for negative moments of 𝐿-functions over function fields, A Refined Conjecture for the Variance of Gaussian Primes across Sectors, Lower-Order Biases in the Second Moment of Dirichlet Coefficients in Families of L-Functions, The prime number theorem and hypothesis H with lower-order terms, Conjectures for the integral moments and ratios of \(L\)-functions over function fields, On the derivatives of Hardy's function Z(t), A weighted one-level density of families of \(L\)-functions, On the number variance of zeta zeros and a conjecture of Berry, Some Results in the Theory of Low-Lying Zeros of Families of L-Functions, Ratios conjecture for quadratic Hecke \(L\)-functions in the Gaussian field, Riemann zeros and random matrix theory, Determining optimal test functions for 2-level densities, Low-lying zeros of elliptic curve L-functions: Beyond the Ratios Conjecture, Negative moments of the Riemann zeta-function, Autocorrelations of characteristic polynomials for the Alternative Circular Unitary Ensemble, Type-I contributions to the one and two level densities of quadratic Dirichlet \(L\)-functions over function fields, Lower order terms for the one-level density of a symplectic family of Hecke \(L\)-functions, CENTRAL VALUES OF DERIVATIVES OF DIRICHLET L-FUNCTIONS, Variance of arithmetic sums and \(L\)-functions in \(\mathbb F_q[t\)], Correlation of shifted values of \(L\)-functions in the hyperelliptic ensemble, On a mollifier of the perturbed Riemann zeta-function, The circular unitary ensemble and the Riemann zeta function: the microscopic landscape and a new approach to ratios, The integral moments and ratios of quadratic Dirichlet \(L\)-functions over monic irreducible polynomials in \(\mathbb{F}_q [T\)], Orthogonal and symplectic 𝑛-level densities, One-level density of families of elliptic curves and the Ratios Conjecture, An elliptic curve test of the \(L\)-functions ratios conjecture, TWISTS OF AUTOMORPHIC L-FUNCTIONS AT THE CENTRAL POINT, A unitary test of the Ratios Conjecture, Discrete mean square of the Riemann zeta-function over imaginary parts of its zeros, More than five-twelfths of the zeros of \(\zeta\) are on the critical line, In support of \(n\)-correlation, THE SECOND MOMENT OF THE RIEMANN ZETA FUNCTION WITH UNBOUNDED SHIFTS, Symmetry transitions in random matrix theory \& \(L\)-functions, Book Review: Lectures on the Riemann zeta function, Triple correlation of the Riemann zeros, Dirichlet characters and low-lying zeros of \(L\)-functions, On the variance of sums of arithmetic functions over primes in short intervals and pair correlation forL-functions in the Selberg class, Unnormalized differences between zeros ofL-functions, Lower order terms for the one-level density of elliptic curve \(L\)-functions, New approach to periodic orbit theory of spectral correlations, A representation theory approach to integral moments of \(L\)-functions over function fields, Lower bounds for discrete negative moments of the Riemann zeta function, Low-lying zeros of quadratic Dirichlet -functions: lower order terms for extended support

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Mean values of the Riemann zeta-function and its derivatives
• On the averages of characteristic polynomials from classical groups
• Mean value of Dirichlet series associated with holomorphic cusp forms
• On the Berry conjecture
• Real zeros of quadratic Dirichlet \(L\)-functions
• Autocorrelation of random matrix polynomials
• Mollification of the fourth moment of automorphic \(L\)-functions and arithmetic applications.
• Nonvanishing of quadratic Dirichlet \(L\)-functions at \(s=\frac{1}{2}\)
• More than one third of zeros of Riemann's zeta-function are on \(\sigma=1/2\)
• Zeros of principal \(L\)-functions and random matrix theory
• Non-vanishing of high derivatives of Dirichlet \(L\)-functions at the central point
• Random matrix theory and the derivative of the Riemann zeta function
• Long mollifiers of the Riemann Zeta‐function
• Averages of characteristic polynomials in random matrix theory
• Semiclassical formula for the number variance of the Riemann zeros
• At least two fifths of the zeros of the Riemann zeta function are on the critical line
• More than two fifths of the zeros of the Riemann zeta function are on the critical line.
• The Fourth Power Moment of the Riemann Zeta Function
• The Riemann Zeros and Eigenvalue Asymptotics
• Random matrix theory and discrete moments of the Riemann zeta function
• Non-vanishing of high derivatives of automorphic L-functions at the center of the critical strip
• On the Distribution of Spacings Between Zeros of the Zeta Function
• Gutzwiller's Trace Formula and Spectral Statistics: Beyond the Diagonal Approximation
• Integral moments of L-functions
• The non-vanishing of central values of automorphic \(L\)-functions and Landau-Siegel zeros
• Random matrix theory and \(L\)-functions at \(s=1/2\).
• Random matrix theory and \(\zeta(1/2+it)\).
• Low lying zeros of families of \(L\)-functions