Geometric properties of the zeta function
Gram's Law and the Argument of the Riemann Zeta Function
Hermite Polynomials and the Zero-Distribution of Riemann's Zeta Function
AMS Subject Classification 2010: 11M26, 33C45, 42A38.Necessary and sufficient conditions for absence of zeros of ζ(s) in the half-plane σ ... Expansion of holomorphic functions in series of Hermite polynomials ...
Higher regularizations of zeros of cuspidal automorphic -functions of
We establish “higher depth” analogues of regularized determinants due to Milnor for zeros of cuspidal automorphic -functions of over a general number field. This is a generalization of the result of Deninger about the regularized determinant for zeros of the Riemann zeta function.
Horizontal monotonicity of the modulus of the zeta function, L-functions, and related functions
As usual, let s = σ + it. For any fixed value of t with |t| ≥ 8 and for σ < 0, we show that |ζ(s)| is strictly decreasing in σ, with the same result also holding for the related functions ξ of Riemann and η of Euler. The following inequality related to the monotonicity of all three functions is proved: ℜ (η'(s)/η(s)) < ℜ (ζ'(s)/ζ(s)) < ℜ (ξ'(s)/ξ(s)). It is also shown that extending the above monotonicity result for |ζ(s)|, |ξ(s)|, or |η(s)| from σ <...
Hybrid joint universality theorem for Dirichlet L-functions
Interpolation and the Laguerre-Pólya class.
Investigations of zeros near the central point of elliptic curve -functions (with an appendix by Eduardo Dueñez).
Irrégularités dans la distribution des nombres premiers dans les progressions arithmétiques
Large gaps between consecutive zeros of the Riemann zeta-function. II
Assuming the Riemann Hypothesis we show that there exist infinitely many consecutive zeros of the Riemann zeta-function whose gaps are greater than 2.9 times the average spacing.
Li coefficients for automorphic -functions
Xian-Jin Li gave a criterion for the Riemann hypothesis in terms of the positivity of a set of coefficients
Lower order terms in the 1-level density for families of holomorphic cuspidal newforms
Lower order terms of the second moment of S(t)
Mean value theorems for long Dirichlet polynomials and tails of Dirichlet series
We obtain formulas for computing mean values of Dirichlet polynomials that have more terms than the length of the integration range. These formulas allow one to compute the contribution of off-diagonal terms provided one knows the correlation functions for the coefficients of the Dirichlet polynomials. A smooth weight is used to control error terms, and this weight can in typical applications be removed from the final result. Similar results are obtained for the tails of Dirichlet series. Four examples...
Modular case of Levinson's theorem
We evaluate the integral mollified second moment of L-functions of primitive cusp forms and we obtain, for such L-functions, an explicit positive proportion of zeros which lie on the critical line.
More precise Pair Correlation Conjecture on the zeros of the Riemann zeta function
More than 41% of the zeros of the zeta function are on the critical line
Nearest neighbor spacing distributions for the zeros of the real or imaginary part of the Riemann xi-function on vertical lines
We show that the density functions of nearest neighbor spacing distributions for the zeros of the real or imaginary part of the Riemann xi-function on vertical lines are described by the M-function which appears in value distribution of the logarithmic derivative of the Riemann zeta-function on vertical lines.
New versions of the Nyman-Beurling criterion for the Riemann hypothesis.
Note on a Paper by h. l. Montgomery ``omega Theorems for the Riemann Zeta-function''