The Bloch–Kato Conjecture for the Riemann Zeta Function

In 1960, Sierpiński proved that there exist infinitely many odd positive integers k such that k · 2n + 1 is composite for all integers n ≥ 0. Variations of this problem, using polynomials with integer coefficients, and considering reducibility over the rationals, have been investigated by several authors. In particular, if S is the set of all positive integers d for which there exists a polynomial f(x) ∈ ℤ[x], with f(1) ≠ -d, such that f(x)xn + d is reducible over the rationals for all integers n ≥ 0, then what are the elements of S? Interest in this problem stems partially from the fact that if S contains an odd integer, then a question of Erdös and Selfridge concerning the existence of an odd covering of the integers would be resolved. Filaseta has shown that S contains all positive integers d ≡ 0 (mod 4), and until now, nothing else was known about the elements of S. In this paper, we show that S contains infinitely many positive integers d ≡ 6 (mod 12). We also consider the corresponding problem over 𝔽p, and in that situation, we show 1 ∈ S for all primes p.

An infinite class of relations between modular forms is constructed that generalizes evaluations of the Dirichlet beta function at odd positive integers. The work is motivated by a base case appearing in Ramanujan’s Notebooks and a parallel construction for the Riemann zeta function. The identities are shown to be strong q-analogues by virtue of their reduction to the classical beta evaluations as [Formula: see text] and explicit evaluations at CM points for [Formula: see text]. We also determine asymptotic formulas for the Fourier coefficients of the associated modular forms.

This book presents the complete proof of the Bloch–Kato conjecture and several related conjectures of Beilinson and Lichtenbaum in algebraic geometry. Brought together here for the first time, these conjectures describe the structure of étale cohomology and its relation to motivic cohomology and Chow groups. Although the proof relies on the work of several people, it is credited primarily to Vladimir Voevodsky. The book draws on a multitude of published and unpublished sources to explain the large-scale structure of Voevodsky's proof and introduces the key figures behind its development. It proceeds to describe the highly innovative geometric constructions of Markus Rost, including the construction of norm varieties, which play a crucial role in the proof. It then addresses symmetric powers of motives and motivic cohomology operations. The book unites various components of the proof that until now were scattered across many sources of varying accessibility, often with differing hypotheses, definitions, and language.

We examine the remarkable connection, first discovered by Beukers, Kolk and Calabi, between ζ(2n), the value of the Riemann Zeta-function at an even positive integer, and the volume of some 2n-dimensional polytope. It can be shown that this volume is equal to the trace of a compact self-adjoint operator. We provide an explicit expression for the kernel of this operator in terms of Euler polynomials. This explicit expression makes it easy to calculate the volume of the polytope and hence ζ(2n). In the case of odd positive integers, we rediscover an integral representation for ζ(2n + 1), obtained by a different method by Cvijovic and Klinowski. Finally, we indicate that the origin of the miraculous Beukers–Kolk–Calabi change of variables in the multidimensional integral, which is at the heart of this circle of ideas, can be traced to the amoeba associated with the certain Laurent polynomial. The article is dedicated to the memory of Vladimir Arnold (1937–2010).

In this paper we show that the Beilinson-Lichtenbaum Conjecture which describes motivic cohomology of (smooth) varieties with finite coefficients is equivalent to the Bloch-Kato Conjecture, relating Milnor K-theory to Galois cohomology. The latter conjecture is known to be true in weight 2 for all primes [M-S] and in all weights for the prime 2 [V 3].

In his second notebook, Ramanujan discovered the following identity for the special values of ζ (s) at the odd positive integers where α and β are positive numbers such that αβ = π 2 and m is a positive integer. As shown by Berndt in the viewpoint of general transformation of analytic Eisenstein series, it is a natural companion of Euler’s famous formula for even zeta values. In this note, we prove an analogue of the above Ramanujan’s identity in the functions fields setting, which involves the Bernoulli-Carlitz numbers.

Let $\mathcal A$ denote a set of positive integers, and let $p(\mathcal A,n)$ denote the associated partition function. Let $\beta $ be an odd positive integer, and let $P(z)$ be a polynomial in $ \mathbb F_2[z]$ of order $\beta $ such that $P(0)=1$. J.-L

We discuss the numbers in the title, and in particular whether the minimum period of the Bell numbers modulo a prime p p can be a proper divisor of N p = ( p p − 1 ) / ( p − 1 ) N_p = (p^p-1)/(p-1) . It is known that the period always divides N p N_p . The period is shown to equal N p N_p for most primes p p below 180. The investigation leads to interesting new results about the possible prime factors of N p N_p . For example, we show that if p p is an odd positive integer and m m is a positive integer and q = 4 m 2 p + 1 q=4m^2 p+1 is prime, then q q divides p m 2 p − 1 p^{m^2p}-1 . Then we explain how this theorem influences the probability that q q divides N p N_p .

The aim of this paper is to classify all monogenic ternary semigroups, up to isomorphism. We divide them to two groups: finite and infinite. We show that every infinite monogenic ternary semigroup is isomorphic to the ternary semigroup O, the odd positive integers with ordinary addition. Then we prove that all finite monogenic ternary semigroups with the same index and the same period are isomorphic. We also investigate structure of finite monogenic ternary semigroups and we prove that any finite monogenic ternary semigroup is isomorphic to a quotient ternary semigroup.

Two-colorings of a dense subgroup of [formula omitted]n that forbid many distances

The author has previously extended the theory of regular and irregular primes to the setting of arbitrary totally real number fields. It has been conjectured that the Bernoulli numbers, or alternatively the values of the Riemann zeta function at odd negative integers, are uniformly distributed modulo p p for every p p . This is the basis of a well-known heuristic, given by Siegel, estimating the frequency of irregular primes. So far, analyses have shown that if Q ( D ) \mathbf {Q}(\sqrt {D}) is a real quadratic field, then the values of the zeta function ζ D ( 1 − 2 m ) = ζ Q ( D ) ( 1 − 2 m ) \zeta _{D}(1-2m)=\zeta _{\mathbf {Q}(\sqrt {D})}(1-2m) at negative odd integers are also distributed as expected modulo p p for any p p . We use this heuristic to predict the computational time required to find quadratic analogues of irregular primes with a given order of magnitude. We also discuss alternative ways of collecting large amounts of data to test the heuristic.

Series acceleration formulas are obtained for Dirichlet series with periodic coefficients. Special cases include Ramanujan's formula for the values of the Riemann zeta function at the odd positive integers exceeding two, and related formulas for values of Dirichlet L-series and the Lerch zeta function.

Rigidity for orientable functors

The lectures which provided the source for these notes covered several different topics which are related to each other but which do not in any reasonable sense form a coherent whole. As a result, this text is a collection of four parts which refer to each other but otherwise are independent. In the first part we introduce the motivic homotopy category and connect it with the motivic cohomology theory discussed in [7]. The exposition is a little unusual because we wanted to avoid any references to model structures and still prove the main theorem 7. We were able to do it modulo 12 where we had to refer to the next part.

The L2 geometry of spaces of harmonic maps S2→ S2 and [formula omitted