Zeros of Jensen polynomials and asymptotics for the Riemann xi function
نویسندگان
چکیده
The classical criterion of Jensen for the Riemann hypothesis is that all associated polynomials have only real zeros. We find a new version this criterion, using linear combinations Hermite polynomials, and show condition holds in many cases. Detailed asymptotic expansions are given required Taylor coefficients xi function at 1/2 as well related quantities. These results build on those recent paper Griffin, Ono, Rolen Zagier.
منابع مشابه
The Riemann Zeros and Eigenvalue Asymptotics
Comparison between formulae for the counting functions of the heights tn of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the tn are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian Hcl. Many features of Hcl are provided by the analogy; for example, the “Riemann dynamics” should be chaotic and ha...
متن کاملPartition Polynomials: Asymptotics and Zeros
Let Fn(x) be the partition polynomial ∑k=1 pk(n)x where pk(n) is the number of partitions of n with k parts. We emphasize the computational experiments using degrees up to 70,000 to discover the asymptotics of these polynomials. Surprisingly, the asymptotics of Fn(x) have two scales of orders n and √ n and in three different regimes inside the unit disk. Consequently, the zeros converge to netw...
متن کاملZeros and ratio asymptotics for matrix orthogonal polynomials
Ratio asymptotics for matrix orthogonal polynomials with recurrence coefficients An and Bn having limits A and B respectively (the matrix Nevai class) were obtained by Durán. In the present paper we obtain an alternative description of the limiting ratio. We generalize it to recurrence coefficients which are asymptotically periodic with higher periodicity, or which are slowly varying in functio...
متن کاملSimple Zeros of the Riemann Zeta-function
Assuming the Riemann Hypothesis, Montgomery and Taylor showed that at least 67.25% of the zeros of the Riemann zeta-function are simple. Using Montgomery and Taylor's argument together with an elementary combinatorial argument, we prove that assuming the Riemann Hypothesis at least 67.275% of the zeros are simple.
متن کاملSome compact generalization of inequalities for polynomials with prescribed zeros
Let $p(z)=z^s h(z)$ where $h(z)$ is a polynomial of degree at most $n-s$ having all its zeros in $|z|geq k$ or in $|z|leq k$. In this paper we obtain some new results about the dependence of $|p(Rz)|$ on $|p(rz)| $ for $r^2leq rRleq k^2$, $k^2 leq rRleq R^2$ and for $Rleq r leq k$. Our results refine and generalize certain well-known polynomial inequalities.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Research in the Mathematical Sciences
سال: 2021
ISSN: ['2522-0144', '2197-9847']
DOI: https://doi.org/10.1007/s40687-020-00240-5