Integral moments of L-functions
TLDR
In this article, the authors give a new heuristic for all of the main terms in the integral moments of various families of primitive $L$-functions and show that these moments can be modelled using Random Matrix Theory.Abstract:
We give a new heuristic for all of the main terms in the integral moments of various families of primitive $L$-functions. The results agree with previous conjectures for the leading order terms. Our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary, orthogonal, or symplectic matrices, where the moments are defined by the appropriate group averages. This lends support to the idea that arithmetical $L$-functions have a spectral interpretation, and that their value distributions can be modelled using Random Matrix Theory. Numerical examples show good agreement with our conjectures.read more
Citations
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Moments of the Riemann zeta function
TL;DR: In this article, an upper bound for the moments of the Riemann zeta function on the critical line was obtained assuming that the riemann hypothesis is true, and the method extends to moments in other families of L-functions.
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An Invitation to Modern Number Theory
TL;DR: This website is an invitation to modern number theory that will be your best choice for better reading book and you can take the book as a source to make better concept.
Journal ArticleDOI
Freezing transition, characteristic polynomials of random matrices, and the Riemann zeta function.
TL;DR: It is argued that the freezing transition scenario, previously explored in the statistical mechanics of 1/f-noise random energy models, also determines the value distribution of the maximum of the modulus of the characteristic polynomials of large N×N random unitary matrices.
Journal ArticleDOI
Autocorrelation of ratios of $L$-functions
TL;DR: In this article, a new heuristic for all of the main terms in the quotient of products of L-functions averaged over a family is given. But the conjectures generalize the recent conjectures for mean values of L -functions.
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Freezing transitions and extreme values: random matrix theory, and disordered landscapes
Yan V. Fyodorov,Jon P Keating +1 more
TL;DR: In this paper, the freezing transition scenario, previously conjectured to occur in the statistical mechanics of 1/f-noise random energy models, governs the value distribution of the maximum of the modulus of the characteristic polynomials pN(θ) of large N×N random unitary (circular unitary ensemble) matrices UN; i.e.
References
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Book
Analytic Number Theory
Henryk Iwaniec,Emmanuel Kowalski +1 more
TL;DR: In this paper, the critical zeros of the Riemann zeta function are defined and the spacing of zeros is defined. But they are not considered in this paper.
Book
The Theory of the Riemann Zeta-Function
TL;DR: The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it one of the most important tools in the study of prime numbers as mentioned in this paper.
Book
A Course in Computational Algebraic Number Theory
TL;DR: The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods.
Book
Multiplicative number theory
TL;DR: In this article, the General Modulus is used to describe the distribution of the Primes in arithmetic progression. But the explicit formula for psi(x,chi) is different from the explicit Formula for xi(s) and xi (s,chi).
Book
Topics in classical automorphic forms
TL;DR: The classical modular forms Automorphic forms in general The Eisenstein and the Poincare series Kloosterman sums Bounds for the Fourier coefficients of cusp forms Hecke operators Automomorphic $L$-functions Cusp forms associated with elliptic curves Spherical functions Theta functions Representations by quadratic forms Automomorphic functions associated with number fields Convolution$L$ -functions Bibliography.