large character sums pretentious characters and the polya vinogradov theorem in 1918 polya and vinogradov gave an upper bound for the maximal size of character sums which still remains the best known general estimate one of the main results of this paper provides a substantial improvement of the polya vinogradov bound for characters of odd bounded order in 1977 montgomery and vaughan showed how the polya vinogradov inequality may be sharpened assuming the generalized riemann hypothesis we give a simple proof of their estimate and provide an improvement for characters of odd bounded order the paper also gives characterizations of the characters for which the maximal character sum is large and it finds a hidden structure among these characters

refinement properties and extensions of filters in boolean algebras we consider the question under what conditions a given family a in a boolean algebra i13 has a disjoint refinement of course a cannot have a disjoint refinement if a is a dense subset of an atomless 03 or if 03 is complete and a generates an ultrafilter on s we show in the first two sections that these two counterexamples can be the only possible ones the third section is concerned with the question how many sets must necessarily be added to a given filter in order to obtain an ultrafilter base 0 introduction let us recall the famous disjoint refinement lemma due to bernstein kuratowski sierpifnski and others assume k to be an infinite cardinal and let a aa a k be a family of sets each of power k then there is a family d d a k such that for every a 8 k we have idal k da c a a da nd 0 the family d is called a disjoint refinement of the family a this lemma first conceived as a mere technical tool has turned out to be the birth cry of the following general disjoint refinement problem what are the conditions under which a given family has a disjoint refinement the power set of a given set a factor algebra of a set modulo some ideal a partially ordered set and most generally the boolean algebra for all these structures the question is meaningful the problem has an extensive literature e g b bf bhm bv c ch vd21 h hi ki ku2 kr p si ti the results concerning the problem have plenty of applications in boolean algebras in the theory of filters and ultrafilters in the theory of ultrapowers in the descriptive set theory and topology the aim of the present paper is to study the refinement problem in boolean algebras the notion of the disjoint refinement is the natural one 0 1 definition let si be a boolean algebra k an infinite cardinal and a aa a k a family of nonzero elements of i 3 the family a has a disjoint refinement in 513 if there is a family d da a k such that for each a 3 k we have 0 da a and da a d13 0 the paper is organized as follows 1 is devoted to the refinement property for an arbitrary family a to make the paper self contained three facts from bv1 are quoted with proofs here we shall state in terms of cardinal characteristics for received by the editors march 12 1980 1980 mathematics subject classification primary 04a20 06e05 secondary 06e10 54a25 54g05

biorthogonal systems and quotient spaces via baire category methods we show that every banach space x of density smaller that the baire category number open image in new window admits a quotient with a long schauder basis that can be taken of length 1 if x is not separable so assuming that the baire category number open image in new window does not take its minimal possible value a banach space x is separable if and only if all biorthogonal systems of x are countable

a sufficiency class for global in time solutions to the 3d navier stokes equations abstract let be an open domain of class c 2 contained in r 3 let l 2 3 be the hilbert space of square integrable functions on and let h h be the completion of the set u c 0 3 u 0 with respect to the inner product of l 2 3 a well known unsolved problem is that of the construction of a sufficient class of functions in h which will allow global in time strong solutions to the three dimensional navier stokes equations these equations describe the time evolution of the fluid velocity and pressure of an incompressible viscous homogeneous newtonian fluid in terms of a given initial velocity and given external body forces in this paper we use the analytic nature of the stokes semigroup to construct an equivalent norm for h which provides strong bounds on the nonlinear term this allows us to prove that under appropriate conditions there exists a number u depending only on the domain the viscosity the body forces and the eigenvalues of the stokes operator such that for all functions in a dense set d contained in the closed ball b b of radius 1 2 u in h the navier stokes equations have unique strong solutions in c 1 0 h

noncommutative operations on wiener functionals and feynman s operational calculus the authors recently introduced a family att 0 of banach algebras of functionals on wiener space and showed that for every f in at the functional integrals k t f exist and are given by a perturbation expansion which serves to disentangle in the sense of feynman s operational calculus the operator k t f when is purely imaginary k t f is the analytic operator valued feynman integral of f in this paper noncommutative operations and on wiener functionals are defined which enable the authors to provide a precise and rigorous interpretation of certain aspects of feynman s operational calculus for noncommuting operators in particular the authors show that if f at1and g at2then f g at1 t2 and k t1 t2 f g k t1 f k t2 g hence the product of operators which can be disentangled using their recent results can itself be disentangled the noncommutative operations the disentangling algebras at and the functional integrals possess a rich interlocking structure

mobius disjointness for topological models of ergodic measure preserving systems with quasi discrete spectrum abstract by using a decomposition result for ergodic measure preserving system with quasi discrete spectrum we prove that a generic point of an ergodic quasi discrete spectrum measure in a topological dynamical system satisfies the required disjointness condition in sarnak s mobius disjointness conjecture as a direct application we have that sarnak s mobius disjointness conjecture holds for any topological model of an ergodic measure preserving system with quasi discrete spectrum

correlations of the von mangoldt and higher divisor functions ii divisor correlations in short ranges we study the problem of obtaining asymptotic formulas for the sums sum x n le 2x d k n d l n h and sum x n le 2x lambda n d k n h where lambda is the von mangoldt function d k is the k text th divisor function x is large and k ge l ge 2 are integers we show that for almost all h in h h with h log x 10000 k log k the expected asymptotic estimate holds in our previous paper we were able to deal also with the case of lambda n lambda n h and we obtained better estimates for the error terms at the price of having to take h x 8 33 varepsilon

lois locales de la fonction ω dans presque tous les petits intervalles for k 1 an integer and x 1 a real number let k x be the number of integers smaller than x having exactly k distinct prime divisors building on recent work of matomaki and radziwill we investigate the asymptotic behavior of k x h k x for almost all x when h is very small we obtain optimal results for k log2x and close to optimal results for 5 k log2x our method also applies to y friable integers in almost all intervals x x h when logx logy logx 1 6 e

on a problem of rolewicz about banach spaces that admit support sets we construct an example of a nonseparable banach space which does not admit a support set 2 it is a consistent and necessarily independent from the axioms of zfc example of a space c k of continuous functions on a compact hausdorff k with the supremum norm the construction depends on a construction of a boolean algebra with some combinatorial properties the space is also hereditarily lindelof in the weak topology but it doesn t have any nonseparable subspace nor any nonseparable quotient which is a c k space for k dispersed

the quantitative behaviour of polynomial orbits on nilmanifolds a theorem of leibman asserts that a polynomial orbit g n g n z on a nilmanifold g g is always equidistributed in a union of closed sub nilmanifolds of g g in this paper we give a quantitative version of leibman s result describing the uniform distribution properties of a finite polynomial orbit g n g n n in a nilmanifold more specifically we show that there is a factorisation g eg where e n is smooth n g n z is periodic and rational and g n g n p is uniformly distributed up to a specified error d inside some subnilmanifold g g of g g for all sufficiently dense arithmetic progressions p n our bounds are uniform in n and are polynomial in the error tolerance d in a companion paper we shall use this theorem to establish the mobius and nilsequences conjecture from an earlier paper of ours

propagators for relativistic systems with non abelian interactions the relativistic evolution of a system of particles in the proper time schwinger dewitt formalism is investigated for a class of interactions that can be represented as fourier transforms of bounded complex matrix valued measures a dyson series representation of the propagator is obtained this class of interactions is non abelian and includes both external electromagnetic and yang mills fields the study of the relativistic problem is facilitated by embedding the original quantum evolution into a larger class of evolution problems that result if one makes an analytic continuation of the metric tensor g this continuation is chosen so that the extended propagator shares for all signatures of g the gaussian decay properties typical of heat kernels estimates of the nth order dyson iterate kernels are found that ensure the absolute convergence of the perturbation series in this fashion a number of analytic and smoothness properties of the propagator are determined in particular it is demonstrate

time ordered operators i foundations for an alternative view of reality abstracr the purpose of this paper is to present the proper framework for the mathematical foundations of time ordered operators we introduce a new mathematical process which we call the chronological process this process generalizes the notion of a limit and allows us to recapture many properties lost in time ordering we then construct time ordered integrals and evolution operators we show that under reasonable assumptions the time ordered sum of two generators of contraction semigroups is a generator this result resolves a question that has been debated in physics for forty years

path integral solution of the schrodinger equation in curvilinear coordinates a straightforward procedure a new axiomatic formulation of path integrals is used to construct a path integral solution of the schrodinger equation in curvilinear coordinates an important feature of the formalism is that a coordinate transformation in the variables of the wavefunction does not imply a change of variable of integration in the path integral consequently a transformation from euclidean to curvilinear coordinates is simple to handle there is no need to introduce quantum corrections into the action functional furthermore the paths are differentiable hence issues related to stochastic paths do not arise the procedure for constructing the path integral solution of the schrodinger equation is straightforward the case of the schrodinger equation in spherical coordinates for a free particle is presented in detail

renorming banach spaces with the mazur intersection property in this paper we give new sufficient and necessary conditions for a banach space to be equivalently renormed with the mazur intersection property as a consequence several examples and applications of these results are obtained among them it is proved that every banach space embeds isometrically into a banach space with the mazur intersection property answering a question asked by giles gregory and sims we also prove that for every treet the spacec0 t admits a norm with the mazur intersection property solving a problem posed by deville godefroy and zizler finally assuming the continuum hypothesis we find an example of an asplund space admitting neither an equivalent norm with the above property nor a nicely smooth norm

linear equations in primes consider a system of nonconstant affine linear forms 1 t d no two of which are linearly dependent let n be a large integer and let k n n d be convex a generalisation of a famous and difficult open conjecture of hardy and littlewood predicts an asymptotic as n for the number of integer points n d k for which the integers 1 n t n are simultaneously prime this implies many other well known conjectures such as the twin prime conjecture and the weak goldbach conjecture it also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations in which all unknowns are required to be prime in this paper we conditionally verify this asymptotic under the assumption that no two of the affine linear forms 1 t are affinely related this excludes the important binary cases such as the twin prime or goldbach conjectures but does allow one to count nondegenerate configurations such as arithmetic progressions our result assumes two families of conjectures which we term the inverse gowers norm conjecture gi s and the mobius and nilsequences conjecture mn s where s 1 2 is the complexity of the system and measures the extent to which the forms i depend on each other the case s 0 is somewhat degenerate and follows from the prime number theorem in aps roughly speaking the inverse gowers norm conjecture gi s asserts the gowers u s 1 norm of a function f n 1 1 is large if and only if f correlates with an s step nilsequence while the mobius and nilsequences conjecture mn s asserts that the mobius function is strongly asymptotically orthogonal to s step nilsequences of a fixed complexity these conjectures have long been known to be true for s 1 essentially by work of hardy littlewood and vinogradov and were established for s 2 in two papers of the authors thus our results in the case of complexity s 2 are unconditional in particular we can obtain the expected asymptotics for the number of 4 term progressions p 1 p 2 p 3 p 4 n of primes and more generally for any nondegenerate problem involving two linear equations in four prime unknowns

foundations for relativistic quantum theory i feynman s operator calculus and the dyson conjectures in this paper we provide a representation theory for the feynman operator calculus this allows us to solve the general initial value problem and construct the dyson series we show that the series is asymptotic thus proving dyson s second conjecture for quantum electrodynamics in addition we show that the expansion may be considered exact to any finite order by producing the remainder term this implies that every nonperturbative solution has a perturbative expansion using a physical analysis of information from experiment versus that implied by our models we reformulate our theory as a sum over paths this allows us to relate our theory to feynman s path integral and to prove dyson s first conjecture that the divergences are in part due to a violation of heisenberg s uncertainly relations

fourier uniformity of bounded multiplicative functions in short intervals on average let lambda denote the liouville function we show that as x rightarrow infty begin aligned int x 2x sup alpha left sum x 0 fixed but arbitrarily small previously this was only known for theta 5 8 for smaller values of theta this is the first non trivial case of local fourier uniformity on average at this scale we also obtain the analogous statement for non pretentious 1 bounded multiplicative functions we illustrate the strength of the result by obtaining cancellations in the sum of lambda n lambda n h lambda n 2h over the ranges h x theta and n x and where lambda is the von mangoldt function

the primes contain arbitrarily long polynomial progressions we establish the existence of infinitely many polynomial progressions in the primes more precisely given any integer valued polynomials p 1 p k z m in one unknown m with p 1 0 p k 0 0 and given any e 0 we show that there are infinitely many integers x and m with 1 leqslant m leqslant x varepsilon such that x p 1 m x p k m are simultaneously prime the arguments are based on those in 18 which treated the linear case p j j 1 m and e 1 the main new features are a localization of the shift parameters and the attendant gowers norm objects to both coarse and fine scales the use of pet induction to linearize the polynomial averaging and some elementary estimates for the number of points over finite fields in certain algebraic varieties

multiple recurrence and nilsequences aiming at a simultaneous extension of khintchine s and furstenberg s recurrence theorems we address the question if for a measure preserving system x mathcal x mu t and a set a in mathcal x of positive measure the set of integers n such that mu a cap t n a cap t 2n a cap ldots cap t kn a mu a k 1 epsilon is syndetic the size of this set surprisingly enough depends on the length k 1 of the arithmetic progression under consideration in an ergodic system for k 2 and k 3 this set is syndetic while for k 4 it is not the main tool is a decomposition result for the multicorrelation sequence int f x f t n x f t 2n x ldots f t kn x d mu x where k and n are positive integers and f is a bounded measurable function we also derive combinatorial consequences of these results for example showing that for a set of integers e with upper banach density d e 0 and for all e 0 the set big n in mathbb z colon d big e cap e n cap e 2n cap e 3n big d e 4 epsilon big is syndetic

differential and integral equations with henstock kurzweil integrable functions abstract in this paper we apply fixed point theorems for increasing mappings in ordered normed spaces to prove existence and comparison results for solutions of discontinuous functional differential and integral equations containing henstock kurzweil integrable functions

moments and distribution of central l values of quadratic twists of elliptic curves we show that if one can compute a little more than a particular moment for some family of l functions then one has upper bounds of the conjectured order of magnitude for all smaller positive real moments and a one sided central limit theorem holds we illustrate our method for the family of quadratic twists of an elliptic curve obtaining sharp upper bounds for all moments below the first we also establish a one sided central limit theorem supporting a conjecture of keating and snaith our work leads to a conjecture on the distribution of the order of the tate shafarevich group for rank zero quadratic twists of an elliptic curve and establishes the upper bound part of this conjecture assuming the birch swinnerton dyer conjecture

a new class of banach spaces in this paper we construct a new class of separable banach spaces for 1 p each of which contains all of the standard lp spaces as well as the space of finitely additive measures as compact dense embeddings equally important is the fact that these spaces contain all henstock kurzweil integrable functions and in particular the feynman kernel and the dirac measure as norm bounded elements as a first application we construct the elementary path integral in the manner originally intended by feynman we then suggest that is a more appropriate hilbert space for quantum theory in that it satisfies the requirements for the feynman heisenberg and schrodinger representations while the conventional choice only satisfies the requirements for the heisenberg and schrodinger representations as a second application we show that the mixed topology on the space of bounded continuous functions used to define the weak generator for a semigroup t t is stronger than the norm topology on this means that when extended to is strongly continuous so that the weak generator on becomes a strong generator on

the primes contain arbitrarily long arithmetic progressions we prove that there are arbitrarily long arithmetic progressions of primes there are three major ingredients the first is szemeredi s theorem which asserts that any subset of the integers of positive density contains progressions of arbitrary length the second which is the main new ingredient of this paper is a certain transference principle this allows us to deduce from szemeredi s theorem that any subset of a suficiently pseudorandom set or measure of positive relative density contains progressions of arbitrary length the third ingredient is a recent result of goldston and yildirim which we reproduce here using this one may place a large fraction of the primes inside a pseudorandom set of almost primes or more precisely a pseudorandom measure concentrated on almost primes with positive relative density

the mobius function is strongly orthogonal to nilsequences we show that the mobius function n is strongly asymptotically or thogonal to any polynomial nilsequence f g n n2n here g is a sim ply connected nilpotent lie group with a discrete and cocompact subgroup so g is a nilmanifold g z g is a polynomial sequence and f g r is a lipschitz function more precisely we show that j 1 n p n n 1 n f g n j f g a log a n for all a 0 in particu lar this implies the mobius and nilsequence conjecture mn s from our earlier paper for every positive integer s this is one of two major in gredients in our programme to establish a large number of cases of the generalised hardy littlewood conjecture which predicts how often a collec tion 1 t z d z of linear forms all take prime values the proof is a relatively quick application of the results in our recent companion paper we give some applications of our main theorem we show for exam ple that the mobius function is uncorrelated with any bracket polynomial such as n p 3bn p 2c we also obtain a result about the distribution of nilse quences a n x n2n as n ranges only over the primes

banach spaces of continuous functions with few operators we present two constructions of infinite separable compact hausdorff spaces k for which the banach space c k of all continuous real valued functions with the supremum norm has remarkable properties in the first construction k is zero dimensional and c k is non isomorphic to any of its proper subspaces nor any of its proper quotients in particular it is an example of a c k space where the hyperplanes one co dimensional subspaces of c k are not isomorphic to c k in the second construction k is connected and c k is indecomposable which implies that it is not isomorphic to any c k for k zero dimensional all these properties follow from the fact that there are few operators on our c k s if we assume the continuum hypothesis the spaces have few operators in the sense that every linear bounded operator t c k c k is of the form gi s where g c k and s is weakly compact or equivalently in c k spaces strictly singular

measure complexity and mobius disjointness abstract in this paper the notion of measure complexity is introduced for a topological dynamical system and it is shown that sarnak s mobius disjointness conjecture holds for any system for which every invariant borel probability measure has sub polynomial measure complexity we then apply this result to a number of situations including certain systems whose invariant measure don t all have discrete spectrum

directed sets and cofinal types we show that 1 s 1 x x xl and l are the only cofinal types of directed sets of size sl but that there exist many cofinal types of directed sets of size continuum a partially ordered set d is directed if every two elements of d have an upper bound in d in this note we consider some basic problems concerning directed sets which have their origin in the theory of moore smith convergence in topology 12 3 19 9 one such problem is to determine all essential kind of directed sets needed for defining the closure operator in a given class of spaces 3 p 47 concerning this problem the following important notion was introduced by j tukey 19 two directed sets d and e are cofinally similar if there is a partially ordered set c in which both can be embedded as cofinal subsets he showed that this is an equivalence relation and that d and e are cofinally similar iff there is a convergent map from d into e and also a convergent map from e into d the equivalence classes of this relation are called cofinal types this concept has been extensively studied since then by various authors 4 13 7 8 already from the first introduction of this concept it has been known that 1 1 x x 1 and x1 represent different cofinal types of directed sets of size btl but no more than five such types were known the main result of this paper shows that 1 1 x x 1 and x1 are the only cofinal types of convergence in spaces of character 81 which can be constructed without additional set theoretic assumptions on the other hand we shall construct many different cofinal types of directed sets of size continuum this gives a solution to problem 1 of j isbell 7 the paper also contains several results about the structure of the class of all cofinal types as well as a result about decomposing arbitrary partially ordered sets into directed sets the results of this note were proved in february march 1982 and presented to the asl in january 1983 1 a decomposition theorem in this section we show that an arbitrary partially ordered set can be decomposed into a number of its directed subsets depending on the sizes of its antichains this result is connected with an unpublished problem of f galvin concerning the dilworth decomposition theorem 6 and it generalizes a similar result of e milner and k prikry 11 the transitivity condition of a partial ordering is not used in our proof so we state our result so as to apply to an arbitrary received by the editors december 3 1984 1980 mathpmaties subjeet classification primary 03e05 03e35 secondary 06a10 18b35 54a15 c1985 american mathematical society 0002 9947 85 1 00 25 per page

time ordered operators and feynman dyson algebras an approach to time ordered operators based upon von neumann s infinite tensor product hilbert spaces is used to define feynman dyson algebras this theory is used to show that a one to one correspondence exists between path integrals and semigroups which are integral operators defined by a kernel the reproducing property of the kernel being a consequence of the semigroup property for path integrals constructed from two semigroups the results are more general than those obtained by the use of the trotter kato formula perturbation series for the feynman dyson operator calculus for time evolution and scattering operators are discussed and it is pointed out that they are asymptotic in the sense of poincare as defined in the theory of semigroups thereby giving a precise formulation to a well known conjecture of dyson stated many years ago in the context of quantum electrodynamics moreover the series converge when these operators possess suitable holomorphy properties

well posedness results for differential and integral equations containing henstock kurzweil integrable vector valued functions abstract in this paper we prove existence uniqueness and comparison results for solutions of differential and integral equations in banach spaces containing henstock kurzweil integrable functions from a compact real interval to an ordered banach space

on sharp transitions in making squares in the fastest performing integer factoring algorithms one creates a sequence of integers in a pseudo random way and wishes to rapidly determine a subsequence whose product is a square in 1994 pomerance stated the following problem which encapsulates all of the key issues select integers a1 a2 at random from the interval 1 x until some nonempty subsequence has product equal to a square find a good estimate for the expected stopping time of this process a good solution should allow one to determine the optimal choice of parameters in many factoring algorithms pomerance 1994 using an idea of schroeppel 1985 showed that with probability 1 o 1 the rst subsequence whose product equals a square

a density theorem on automorphic l functions and some applications we establish a density theorem of automorphic l functions and give some applications on the extreme values of these l functions at s 1 and the distribution of the hecke eigenvalue of holomorphic cusp forms contents

monotone convergence theorems for henstock kurzweil integrable functions and applications abstract in this paper we prove monotone convergence theorems for henstock kurzweil integrable functions from a compact real interval to an ordered banach space these theorems are then applied to prove existence results for solutions of a discontinuous functional integral equation containing henstock kurzweil integrable functions

rigidity theorems for multiplicative functions we establish several results concerning the expected general phenomenon that given a multiplicative function f mathbb n rightarrow mathbb c the values of f n and f n a are generally independent unless f is of a special form first we classify all bounded completely multiplicative functions having uniformly large gaps between its consecutive values this implies the solution of the following folklore conjecture for any completely multiplicative function f mathbb n rightarrow mathbb t we have begin aligned liminf n rightarrow infty f n 1 f n 0 end aligned second we settle an old conjecture due to chudakov on the generalized characters in actes du congres international des mathematiciens nice 1970 tome 1 p 487 gauthier villars paris that states that any completely multiplicative function f mathbb n rightarrow mathbb c that a takes only finitely many values b vanishes at only finitely many primes and c has bounded discrepancy is a dirichlet character this generalizes previous work of tao on the erd s discrepancy problem finally we show that if many of the binary correlations of a 1 bounded multiplicative function are asymptotically equal to those of a dirichlet character chi mod q then f n chi n n it for all n where chi is a dirichlet character modulo q and t in mathbb r this establishes a variant of a conjecture of h cohn for multiplicative arithmetic functions the main ingredients include the work of tao on logarithmic elliott conjecture correlation formulas for pretentious multiplicative functions developed earlier by the first author and szemeredi s theorem for long arithmetic progressions

correlations of the von mangoldt and higher divisor functions i long shift ranges we show that the expected asymptotic for the sums x n 2x n n h x n 2x d k n d l n h and x n 2x n d k n h hold for almost all h h h provided that x8 33 e h x 1 e with an error term saving on average an arbitrary power of the logarithm over the trivial bound previous work of mikawa and baier browning marasingha zhao covered the range h x 1 3 e we also obtain an analogous result for n n n n our proof uses the circle method and some oscillatory integral estimates following a paper of zhan to reduce matters to establishing some mean value estimates for certain dirichlet polynomials associated to type d 3 and type d 4 sums as well as some other sums that are easier to treat after applying holder s inequality to the type d 3 sum one is left with two expressions one of which we can control using a short interval mean value theorem of jutila and the other we can control using exponential sum estimates of robert and sargos the type d 4 sum is treated similarly using the classical l2 mean value theorem and the classical van der corput exponential sum estimates

thin very tall compact scattered spaces which are hereditarily separable we strengthen the property of a function f 2 2 2 considered by baumgartner and shelah this allows us to consider new types of amalgamations in the forcing used by rabus juhasz and soukup to construct thin very tall compact scattered spaces we consistently obtain spaces k as above where k n is hereditarily separable for each n n this serves as a counterexample concerning cardinal functions on compact spaces as well as having some applications in banach spaces the banach space c k is an asplund space of density 2 which has no frechet smooth renorming nor an uncountable biorthogonal system

the feynman measure as a limit of complex measures abstract the fundamental solution of the schrodinger equation for a free particle is a distribution such distribution is approximated by a sequence of smooth functions for each one of these functions it is defined a complex measure on the space of paths for certain test functions there exists the limit of integrals of a test function with respect to the complex measures we define the feynman measure of one such function by this limit

multiple recurrence and convergence for sequences related to the prime numbers for any measure preserving system x mathcal x mu t and a in mathcal x with mu a 0 we show that there exist infinitely many primes p such that mu bigl a cap t p 1 a cap t 2 p 1 a bigr 0 furthermore we show the existence of the limit in l 2 mu of the associated double average over the primes a key ingredient is a recent result of green and tao on the von mangoldt function a combinatorial consequence is that every subset of the integers with positive upper density contains an arithmetic progression of length three and common difference of the form p 1 for some prime p

on fourier coefficients of modular forms of half integral weight at squarefree integers we show that the dirichlet series associated to the fourier coefficients of a half integral weight hecke eigenform at squarefree integers extends analytically to a holomorphic function in the half plane re s tfrac 1 2 this exhibits a high fluctuation of the coefficients at squarefree integers

time ordered operators ii apres avoir construit les fondements mathematiques generaux pour des equations d evolution lineaires en temps ordonne on applique les resultats pour montrer qu a la fois le developpement en perturbation et la methode des diagrammes de feymann sont mathematiquement sains

extreme values of zeta and l functions we introduce a resonance method to produce large values of the riemann zeta function on the critical line and large and small central values of l functions

an inverse theorem for the gowers u s 1 n norm we prove the inverse conjecture for the gowers u s 1 n norm for all s 1 this is new for s 4 more precisely we establish that if f n 1 1 is a function with f u s 1 n d then there is a bounded complexity s step nilsequence f g n g that correlates with f where the bounds on the complexity and correlation depend only on s and d from previous results this conjecture implies the hardy littlewood prime tuples conjecture for any linear system of finite complexity

bombieri vinogradov for multiplicative functions and beyond the x1 2 barrier abstract part and parcel of the study of multiplicative number theory is the study of the distribution of multiplicative functions in arithmetic progressions although appropriate analogies to the bombieri vingradov theorem have been proved for particular examples of multiplicative functions there has not previously been headway on a general theory seemingly none of the different proofs of the bombieri vingradov theorem for primes adapt well to this situation in this article we find out why such a result has been so elusive and discover what can be proved along these lines and develop some limitations for a fixed residue class a we extend such averages out to moduli x 20 39

first countable countably compact spaces and the continuum hypothesis we build a model of zfc ch in which every first countable countably compact space is either compact or contains a homeomorphic copy of 1 with the order topology the majority of the paper consists of developing forcing technology that allows us to conclude that our iteration adds no reals our results generalize saharon shelah s iteration theorems appearing in chapters v and viii of proper and improper forcing 1998 as well as eisworth and roitman s 1999 iteration theorem we close the paper with a zfc example constructed using shelah s club guessing sequences that shows similar results do not hold for closed pre images of 2

on constructing primitive roots in finite fields with advice finding primitive roots in a finite field of p n elements of characteristic p remains to be a hard computational problem with the bottlenecks coming from both locating a small set of possible candidates and also factoring p n 1 in order to test these candidates kopparty et al 2016 have introduced a question of designing a fast algorithm to find primitive roots with a short advice from an oracle trivially for an m bit prime p such a primitive root can be fully described by about mn bits of information received from an all powerful oracle here we have shown that one can achieve this in polynomial time and with about 1 2 o 1 m o log n bits of advice

on disjointness properties of some parabolic flows the ratner property a quantitative form of divergence of nearby trajectories is a central feature in the study of parabolic homogeneous flows discovered by marina ratner and used in her 1980th seminal works on horocycle flows it pushed forward the disjointness theory of such systems in this paper exploiting a recent variation of the ratner property we prove new disjointness phenomena for smooth parabolic flows beyond the homogeneous world in particular we establish a general disjointness criterion based on the switchable ratner property we then apply this new criterion to study disjointness properties of smooth time changes of horocycle flows and smooth arnol d flows on mathbb t 2 focusing in particular on disjointness of distinct flow rescalings as a consequence we answer a question by marina ratner on the mobius orthogonality of time changes of horocycle flows in fact we prove mobius orthogonality for all smooth time changes of horocycle flows and uniquely ergodic realizations of arnol d flows considered