35 research outputs found
On some reasons for doubting the Riemann hypothesis
Several arguments against the truth of the Riemann hypothesis are extensively
discussed. These include the Lehmer phenomenon, the Davenport-Heilbronn
zeta-function, large and mean values of on the critical line,
and zeros of a class of convolution functions. The first two topics are
classical, and the remaining ones are connected with the author's research.Comment: 30 page
Transcendental equations satisfied by the individual zeros of Riemann , Dirichlet and modular -functions
We consider the non-trivial zeros of the Riemann -function and two
classes of -functions; Dirichlet -functions and those based on level one
modular forms. We show that there are an infinite number of zeros on the
critical line in one-to-one correspondence with the zeros of the cosine
function, and thus enumerated by an integer . From this it follows that the
ordinate of the -th zero satisfies a transcendental equation that depends
only on . Under weak assumptions, we show that the number of solutions of
this equation already saturates the counting formula on the entire critical
strip. We compute numerical solutions of these transcendental equations and
also its asymptotic limit of large ordinate. The starting point is an explicit
formula, yielding an approximate solution for the ordinates of the zeros in
terms of the Lambert -function. Our approach is a novel and simple method,
that takes into account , to numerically compute non-trivial zeros of
-functions. The method is surprisingly accurate, fast and easy to implement.
Employing these numerical solutions, in particular for the -function, we
verify that the leading order asymptotic expansion is accurate enough to
numerically support Montgomery's and Odlyzko's pair correlation conjectures,
and also to reconstruct the prime number counting function. Furthermore, the
numerical solutions of the exact transcendental equation can determine the
ordinates of the zeros to any desired accuracy. We also study in detail
Dirichlet -functions and the -function for the modular form based on the
Ramanujan -function, which is closely related to the bosonic string
partition function.Comment: Matches the version to appear in Communications in Number Theory and
Physics, based on arXiv:1407.4358 [math.NT], arXiv:1309.7019 [math.NT], and
arXiv:1307.8395 [math.NT
Riemann hypothesis
This work is dedicated to the promotion of the results Hadamard, Landau E.,
Walvis A., Estarmann T and Paul R. Chernoff for pseudo zeta functions. The
properties of zeta functions are studied, these properties can lead to new
regularities of zeta functions.Comment: 10 pages. arXiv admin note: substantial text overlap with
arXiv:1605.0601
From asymptotic to closed forms for the Keiper/Li approach to the Riemann Hypothesis
The Riemann Hypothesis (RH) - that all nonreal zeros of Riemann's zeta
function shall have real part 1/2 - remains a major open problem. Its most
concrete equivalent is that an infinite sequence of real numbers, the
Keiper--Li constants, shall be everywhere positive (Li's criterion). But those
numbers are analytically elusive and strenuous to compute, hence we seek
simpler variants. The essential sensitivity to RH of that sequence lies in its
asymptotic tail; then, retaining this feature, we can modify the Keiper--Li
scheme to obtain a new sequence in elementary closed form. This makes for a
more explicit analysis, with easier and faster computations. We can moreover
show how the new sequence will signal RH-violating zeros if any, by observing
its analogs for the Davenport--Heilbronn counterexamples to RH.Comment: 16 p., 7 figs. Workshop for Prof. Yoshitsugu TAKEI's 60-th birthday,
RIMS, Kyoto Univ., Oct. 202