Introduction Arithmetic functions Elementary theory of prime numbers Characters Summation formulas Classical analytic theory of $L$-functions Elementary sieve methods Bilinear forms and the large sieve Exponential sums The Dirichlet polynomials Zero-density estimates Sums over finite fields Character sums Sums over primes Holomorphic modular forms Spectral theory of automorphic forms Sums of Kloosterman sums Primes in arithmetic progressions The least prime in an arithmetic progression The Goldbach problem The circle method Equidistribution Imaginary quadratic fields Effective bounds for the class number The critical zeros of the Riemann zeta function The spacing of zeros of the Riemann zeta-function Central values of $L$-functions Bibliography Index.

/pdf/analytic-number-theory-3xy3ackp22.pdf

Analytic Number Theory

We study the characteristic polynomials Z(U, θ) of matrices U in the Circular Unitary Ensemble (CUE) of Random Matrix Theory. Exact expressions for any matrix size N are derived for the moments of |Z| and Z/Z*, and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log Z as N→∞. In the limit, we show that these two distributions are independent and Gaussian. Costin and Lebowitz [15] previously found the Gaussian limit distribution for Im log Z using a different approach, and our result for the cumulants proves a conjecture made by them in this case. We also calculate the leading order N→∞ asymptotics of the moments of |Z| and Z/Z*. These CUE results are then compared with what is known about the Riemann zeta function ζ (s) on its critical line Re s= 1/2, assuming the Riemann hypothesis. Equating the mean density of the non-trivial zeros of the zeta function at a height T up the critical line with the mean density of the matrix eigenvalues gives a connection between N and T. Invoking this connection, our CUE results coincide with a theorem of Selberg for the value distribution of log ζ(1/2+iT) in the limit T→∞. They are also in close agreement with numerical data computed by Odlyzko [29] for large but finite T. This leads us to a conjecture for the moments of |ζ(1/2+it) |. Finally, we generalize our random matrix results to the Circular Orthogonal (COE) and Circular Symplectic (CSE) Ensembles.

/pdf/random-matrix-theory-and-z-1-2-it-3xihgkn7e2.pdf

Random Matrix Theory and ζ(1/2+it)

Consider a system ψ of nonconstant affine-linear forms ψ 1 , ... , ψ t : ℤ d → ℤ, no two of which are linearly dependent. Let N be a large integer, and let K ⊆ [-N, N] d be convex. A generalisation of a famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as N →∞, for the number of integer points n ∈ ℤ d ∩ K for which the integers ψ 1 (n), ... , ψ t (n) are simultaneously prime. This implies many other well-known conjectures, such as the twin prime conjecture and the (weak) Goldbach conjecture. It also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations, in which all unknowns are required to be prime. In this paper we (conditionally) verify this asymptotic under the assumption that no two of the affine-linear forms ψ 1 , ... , ψ t are affinely related; this excludes the important "binary" cases such as the twin prime or Goldbach conjectures, but does allow one to count "nondegenerate" configurations such as arithmetic progressions. Our result assumes two families of conjectures, which we term the inverse Gowers-norm conjecture (GI(s)) and the Mobius and nilsequences conjecture (MN(s)), where s ∈ {1,2,...} is the complexity of the system and measures the extent to which the forms ψ i depend on each other. The case s = 0 is somewhat degenerate, and follows from the prime number theorem in APs. Roughly speaking, the inverse Gowers-norm conjecture GI(s) asserts the Gowers U s+1 -norm of a function f: [N] → [-1, 1] is large if and only if f correlates with an s-step nilsequence, while the Mobius and nilsequences conjecture MN(s) asserts that the Mobius function μ is strongly asymptotically orthogonal to s-step nilsequences of a fixed complexity. These conjectures have long been known to be true for s = 1 (essentially by work of Hardy-Littlewood and Vinogradov), and were established for s = 2 in two papers of the authors. Thus our results in the case of complexity s ≤ 2 are unconditional. In particular we can obtain the expected asymptotics for the number of 4-term progressions p 1 < p 2 < p 3 < p 4 ≤ N of primes, and more generally for any (nondegenerate) problem involving two linear equations in four prime unknowns.

https://annals.math.princeton.edu/wp-content/uploads/annals-v171-n3-p08-p.pdf

Linear equations in primes

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.

https://web.williams.edu/Mathematics/sjmiller/public_html/ntandrmt/handouts/conrey/ConreyFarmerKeatingRubinsteinSnaith_IntegralMomentsLFns.pdf

Integral moments of L-functions

Etude numerique de la distribution des espacements des zeros de la fonction zeta de Riemann. Observation de certains phenomenes inatendus

http://www.dtc.umn.edu/~odlyzko/doc/arch/zeta.zero.spacing.pdf

On the distribution of spacings between zeros of the zeta function

In the study of Dirichlet series with arithmetic signi cance there has appeared (through the study of known examples) certain expectations, namely (i) if a functional equation and Euler product exists, then it is likely that a type of Riemann hypothesis will hold, (ii) that if in addition the function has a simple pole at the point s = 1, then it must be a product of the Riemann zeta-function and another Dirichlet series with similar properties, and (iii) that a type of converse theorem holds, namely that all such Dirichlet series can be obtained by considering Mellin transforms of automorphic forms associated with arithmetic groups. Guided by these ideas, consider the class S of Dirichlet series (introduced by Selberg [7]) : a Dirichlet series

On the Selberg class of Dirichlet series: small degrees

This paper deals with some questions that have received a lot of attention since they were raised by Hejhal and Rackner in their 1992 numerical computations of Maass forms. We establish sharp upper and lower bounds for the $L^2$-restrictions of these forms to certain curves on the modular surface. These results, together with the Lindelof Hypothesis and known subconvex $L^\infty$-bounds are applied to prove that locally the number of nodal domains of such a form goes to infinity with its eigenvalue.

Nodal domains of Maass forms I

LetIt is conjectured that for any k,for some constant ck. It is well known that (2) holds for k = 0, 1 and 2 with c0 = 1, C1 = 1, and c2 = (2π2)-1, but there is not even a conjectural value of ck for any other k. However, it is known that the Riemann hypothesis impliesfor all k ≥ 0 (see Ramachandra [2] and Heath-Brown [1]).

On mean values of the zeta-function

This paper deals with some questions that have received a lot of attention since they were raised by Hejhal and Rackner in their 1992 numerical computations of Maass forms. We establish sharp upper and lower bounds for the L
 2-restrictions of these forms to certain curves on the modular surface. These results, together with the Lindelof Hypothesis and known subconvex L
 ∞-bounds are applied to prove that locally the number of nodal domains of such a form goes to infinity with its eigenvalue.

Nodal Domains of Maass Forms I

Analytic Number Theory and Diophantine Problems

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