Related papers: Ordinality and Riemann Hypothesis I

Ramanujan investigated maximal order for the number of divisors function by introducing some notion such as (superior) highly composite numbers. He also studied maximal order for other arithmetic functions including the sum of powers of…

We introduce the notion of limiting theories, giving examples and providing a sufficient condition under which the first order theory of a structure is the limit of the first order theories of a collection of substructures. We also give a…

For a fixed rational number g different from -1,0,1 and integers a and d the set N_g(a,d) of primes p for which the order of g(mod p) is congruent to a(mod d) is considered. It is shown, assuming the Generalized Riemann Hypothesis (GRH),…

The well-known necessary and sufficient criteria for the Riemann hypothesis of M. Riesz and Hardy-Littlewood, based on the order of growth at infinity along the positive real axis of certain entire functions, are here imbedded in a general…

We prove some new results on existence of solutions to first--order ordinary differential equations with deviating arguments. Delay differential equations are included in our general framework, which even allows deviations to depend on the…

We study fundamental groups of non compact Riemannian manifolds. We find conditions which ensure that the fundamental group is trivial, finite or finitely generated.

In this paper we present four necessary and sufficient conditions for a natural number to be prime.

We deal with the Euler's alternating series of the Riemann zeta function to define a regularized ratio appeared in the functional equation even in the critical strip and show some evidence to indicate the hypothesis in this note.

We generalize current known distribution results on Shanks--R\'enyi prime number races to the case where arbitrarily many residue classes are involved. Our method handles both the classical case that goes back to Chebyshev and function…

Assuming the Riemann Hypothesis we establish an upper bound for the sum of the M{\" o}bius function up to $x$. Our method is based on estimating the frequency with which intervals of a given length can contain an unusual number of ordinates…

We provide two new proofs of the infinitude of prime numbers, using the additive Ramsey-theoretic result known as Folkman's theorem (alternatively, one can think of these proofs as using Hindman's theorem). This adds to the existing…

Some recent papers formulated sufficient conditions for the decomposition of matrix variances. A statement was that if we have one or two observables, then the decomposition is possible. In this paper we consider an arbitrary finite set of…

The Legendre conjecture has resisted analysis over a century, even under assumption of the Riemann Hypothesis. We present, a significant improvement on previous results by greatly reducing the assumption to a more modest statement called…

For a partial order $\preceq$ on a set X and an equivalency relation S defined on the same set X we derive a necessary and sufficient condition for the existence of such a total preorder on X whose asymmetric part contains the asymmetric…

Assuming the Riemann hypothesis, we obtain upper and lower bounds for moments of the Riemann zeta-function averaged over the extreme values between its zeros on the critical line. Our bounds are very nearly the same order of magnitude. The…

In this paper, we present a new necessary and sufficient condition for which the supremum exists with respect to the logic order. Moreover, we give out a new and much simpler representation of the supremum with respect to the order, our…

We investigate some extremal problems in Fourier analysis and their connection to a problem in prime number theory. In particular, we improve the current bounds for the largest possible gap between consecutive primes assuming the Riemann…

We show that a profinite group with the same first-order theory as the direct product over all odd primes $p$ of the dihedral group of order $2p$, is necessarily isomorphic to this direct product.

We here present a sufficient condition for general arrowing problems to be non definable in first order logic, based in well known tools of finite model theory e.g. Hanf's Theorem and known concepts in finite combinatorics, like senders and…

In this article, we prove an "equivalence" between two higher even moments of primes in short intervals under Riemann Hypothesis. We also provide numerical evidence in support of these asymptotic formulas.