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We prove a new effective Chebotarev density theorem for Galois extensions $L/\mathbb{Q}$ that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of $L$); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of $L$, without assuming GRH. We give many different "appropriate families," including families of arbitrarily large degree. To do this, we first prove a new effective Chebotarev density theorem that requires a zero-free region of the Dedekind zeta function. Then we prove that almost all number fields in our families yield such a zero-free region. The innovation that allows us to achieve this is a delicate new method for controlling zeroes of certain families of non-cuspidal $L$-functions. This builds on, and greatly generalizes the applicability of, work of Kowalski and Michel on the average density of zeroes of a family of cuspidal $L$-functions. A surprising feature of this new method, which we expect will have independent interest, is that we control the number of zeroes in the family of $L$-functions by bounding the number of certain associated fields with fixed discriminant. As an application of the new Chebotarev density theorem, we prove the first nontrivial upper bounds for $\ell$-torsion in class groups, for all integers $\ell \geq 1$, applicable to infinite families of fields of arbitrarily large degree., Comment: 52 pages. This shorter version aligns with the published paper. Note that portions of Section 8 of the longer v1 have been developed as a separate paper with identifier arXiv:1902.02008

Generalized Zeckendorf decompositions are expansions of integers as sums of elements of solutions to recurrence relations. The simplest cases are base-$b$ expansions, and the standard Zeckendorf decomposition uses the Fibonacci sequence. The expansions are finite sequences of nonnegative integer coefficients (satisfying certain technical conditions to guarantee uniqueness of the decomposition) and which can be viewed as analogs of sequences of variable-length words made from some fixed alphabet. In this paper we present a new approach and construction for uniform measures on expansions, identifying them as the distribution of a Markov chain conditioned not to hit a set. This gives a unified approach that allows us to easily recover results on the expansions from analogous results for Markov chains, and in this paper we focus on laws of large numbers, central limit theorems for sums of digits, and statements on gaps (zeros) in expansions. We expect the approach to prove useful in other similar contexts., Comment: Version 1.0, 25 pages. Keywords: Zeckendorf decompositions, positive linear recurrence relations, distribution of gaps, longest gap, Markov processes

One of the main goals in this paper is to establish convolution sums of functions for the divisor sums $\widetilde{\sigma}_s(n)=\sum_{d|n}(-1)^{d-1}d^s$ and $\widehat{\sigma}_s(n)=\sum_{d|n}(-1)^{\frac{n}{d}-1}d^s$, for certain $s$, which were first defined by Glaisher. We first introduce three functions $\mathcal{P}(q)$, $\mathcal{E}(q)$, and $\mathcal{Q}(q)$ related to $\widetilde{\sigma}(n)$, $\widehat{\sigma}(n)$, and $\widetilde{\sigma}_3(n)$, respectively, and then we evaluate them in terms of two parameters $x$ and $z$ in Ramanujan's theory of elliptic functions. Using these formulas, we derive some identities from which we can deduce convolution sum identities. We discuss some formulae for determining $r_s(n)$ and $\delta_s(n)$, $s=4,$ $8$, in terms of $\widetilde{\sigma}(n)$, $\widehat{\sigma}(n)$, and $\widetilde{\sigma}_3(n)$, where $r_s(n)$ denotes the number of representations of $n$ as a sum of $s$ squares and $\delta_s(n)$ denotes the number of representations of $n$ as a sum of $s$ triangular numbers. Finally, we find some partition congruences by using the notion of colored partitions., Comment: This is an old paper uploaded for archival purposes

In this paper we present a p-adic algorithm to compute the zeta function of a nondegenerate curve over a finite field using Monsky-Washnitzer cohomology. The paper vastly generalizes previous work since all known cases, e.g. hyperelliptic, superelliptic and C_{ab} curves, can be transformed to fit the nondegenerate case. For curves with a fixed Newton polytope, the property of being nondegenerate is generic, so that the algorithm works for almost all curves with given Newton polytope. For a genus g curve over F_{p^n}, the expected running time is O(n^3g^6 + n^2g^{6.5}), whereas the space complexity amounts to O(n^3g^4), assuming p is fixed., 41 pages

In foundational papers, Gross, Zagier, and Kohnen established two formulas for arithmetic intersection numbers of certain Heegner divisors on integral models of modular curves. In [GZ1], only one imaginary quadratic discriminant plays a role. In [GZ2] and [GKZ], two quadratic discriminants play a role. In this paper we generalize the two-discriminant formula from the modular curves X0(N) to certain Shimura curves defined over Q. Our intersection formula was stated in [Ro], but the proof was only outlined there. Independently, the general formula was given, in a weaker and less explicit form, in [Ke2]; there it was proved completely. This paper is thus a synthesis of parts of [Ro] and [Ke2]. The intersection multiplicities computed here were used in [Ku] to derive a relation between height pairings and special values of the derivatives of certain Eisenstein series. We note also that Zhang [Zh] has generalized all of [GZ1] from ground field Q to general totally real ground fields F , working with general Shimura curves. So we certainly expect that all of [GKZ] should generalize similarly. Our work here can be viewed as a step in this direction. We are happy to thank B. Gross and S. Kudla for their encouragement during the long period in which this work was done.

This paper studies the multiplicative semigroup generated by all rationals of the form (3n+2)/(2n+1) for nonnegative integers n, together with 1/2. The subsemigroup of integers in this semigroup is called the wild integer semigroup, and the wild numbers are the irreducible elements in this subsemigroup. This paper presents evidence that the wild numbers consist of all the prime numbers p except 3. The subsemigroup of integers in the multiplicative semigroup generated by all rationals of the form (3n+2)/(2n+1) for nonnegative integers n is called the Wooley integer semigroup and its irreducible elements are called Wooley numbers. This semigroup is shown to be recursive, and various open problems are formulated about Wooley integers., 11 pages, latex; revised with title change and some notation changes

Let $\unicode[STIX]{x1D6FD}\in (1,2)$ be a real number. For a function $\unicode[STIX]{x1D6F9}:\mathbb{N}\rightarrow \mathbb{R}_{\geq 0}$, define $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$ to be the set of $x\in \mathbb{R}$ such that for infinitely many $n\in \mathbb{N},$ there exists a sequence $(\unicode[STIX]{x1D716}_{i})_{i=1}^{n}\in \{0,1\}^{n}$ satisfying $0\leq x-\sum _{i=1}^{n}(\unicode[STIX]{x1D716}_{i}/\unicode[STIX]{x1D6FD}^{i})\leq \unicode[STIX]{x1D6F9}(n)$. In Baker [Approximation properties of $\unicode[STIX]{x1D6FD}$-expansions. Acta Arith. 168 (2015), 269–287], the author conjectured that for Lebesgue almost every $\unicode[STIX]{x1D6FD}\in (1,2)$, the condition $\sum _{n=1}^{\infty }2^{n}\unicode[STIX]{x1D6F9}(n)=\infty$ implies that $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$ is of full Lebesgue measure within $[0,1/(\unicode[STIX]{x1D6FD}-1)]$. In this paper we make a significant step towards proving this conjecture. We prove that given a sequence of positive real numbers $(\unicode[STIX]{x1D714}_{n})_{n=1}^{\infty }$ satisfying $\lim _{n\rightarrow \infty }\unicode[STIX]{x1D714}_{n}=\infty$, for Lebesgue almost every $\unicode[STIX]{x1D6FD}\in (1.497,\ldots ,2)$, the set $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D714}_{n}\cdot 2^{-n})$ is of full Lebesgue measure within $[0,1/(\unicode[STIX]{x1D6FD}-1)]$. We also study the case where $\sum _{n=1}^{\infty }2^{n}\unicode[STIX]{x1D6F9}(n) in which the set $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$ has Lebesgue measure zero. Applying the mass transference principle developed by Beresnevich and Velani in [A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3) (2006), 971–992], we obtain some results on the Hausdorff dimension and the Hausdorff measure of $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$.

Ramanujan graphs are graphs whose spectrum is bounded optimally. Such graphs have found numerous applications in combinatorics and computer science. In recent years, a high dimensional theory has emerged. In this paper these developments are surveyed. After explaining their connection to the Ramanujan conjecture we will present some old and new results with an emphasis on random walks on these discrete objects and on the Euclidean spheres. The latter lead to "golden gates" which are of importance in quantum computation., To appear in the proceedings of the Ramanujan Centenary Celebration at the Royal Society

The structure of the group (Z/nZ). and Fermat’s little theorem are the basis for some of the best-known primality testing algorithms. Many related concepts arise: Euler’s totient function and Carmichael’s lambda function, Fermat pseudoprimes, Carmichael and cyclic numbers, Lehmer’s totient problem, Giuga’s conjecture, etc. In this paper, we present and study analogues to some of the previous concepts arising when we consider the underlying group Gn := {a + bi 2 Z[i]/nZ[i] : a2 + b2 1 (mod n)}. In particular, we characterize Gaussian Carmichael numbers via a Korselt’s criterion and present their relation with Gaussian cyclic numbers. Finally, we present the relation between Gaussian Carmichael number and 1-Williams numbers for numbers n 3 (mod 4). There are also no known composite numbers less than 1018 in this family that are both pseudoprime to base 1 + 2i and 2-pseudoprime. D. Sadornil is partially supported by the Spanish Government under projects MTM2010– 21580–C02–02 and MTM2010–16051.

In this paper we investigate the distribution of the set of values of a quadratic form Q, at integral points. In particular we are interested in the n-point correlations of the this set. The asymptotic behaviour of the counting function that counts the number of n-tuples of integral points $$\left( v_{1},\ldots ,v_{n}\right) $$ , with bounded norm, such that the $$n-1$$ differences $$Q\left( v_{1}\right) -Q\left( v_{2}\right) ,\ldots Q\left( v_{n-1}\right) -Q\left( v_{n}\right) $$ , lie in prescribed intervals is obtained. The results are valid provided that the quadratic form has rank at least 5, is not a multiple of a rational form and n is at most the rank of the quadratic form. For certain quadratic forms satisfying Diophantine conditions we obtain a rate for the limit. The proofs are based on those in the recent preprint (Distribution of values of quadratic forms at integral points. http://www.math.uni-bielefeld.de/sfb701/files/preprints/sfb13003.pdf , 2013) of Gotze and Margulis, in which they prove an ‘effective’ version of the Oppenheim conjecture. In particular, the proofs rely on Fourier analysis and estimates for certain theta series.

By two well-known results, one of Ax, one of Lubotzky and van den Dries, a profinite group is projective iff it is isomorphic to the absolute Galois group of a pseudo-algebraically closed field. This paper gives an analogous characterization of relatively projective profinite groups as absolute Galois groups of regularly closed fields.

This paper is concerned with the study of diffraction intensities of a relevant class of binary Pisot substitutions via exponential sums. Arithmetic properties of algebraic integers are used to give a new and constructive proof of the fact that there are no diffraction intensities outside the Fourier module of the underlying cut and project schemes. The results are then applied in the context of random substitutions., Comment: 9 pages

The present paper establishes qunatitative estimates on the rate of diophantine approximation in homogeneous varieties of semisimple algebraic groups. The estimates established generalize and improve previous ones, and are sharp in a number of cases. We show that the rate of diophantine approximation is controlled by the spectrum of the automorphic representation, and is thus subject to the generalised Ramanujan conjectures.

In this paper we use the generating functions and the double shuffle relations satisfied the multiple zeta values to derive some new families of identities., Comment: 21 pages, misprints corrected and reference updated

In this paper we obtain estimates for certain transcendence measures of an entire function $f$. Using these estimates, we prove Bernstein, doubling and Markov inequalities for a polynomial $P(z,w)$ in ${\Bbb C}^2$ along the graph of $f$. These inequalities provide, in turn, estimates for the number of zeros of the function $P(z,f(z))$ in the disk of radius $r$, in terms of the degree of $P$ and of $r$. Our estimates hold for arbitrary entire functions $f$ of finite order, and for a subsequence $\{n_j\}$ of degrees of polynomials. But for special classes of functions, including the Riemann $\zeta$-function, they hold for all degrees and are asymptotically best possible. From this theory we derive lower estimates for a certain algebraic measure of a set of values $f(E)$, in terms of the size of the set $E$., Comment: 40 pages

Let E_G be a family of hyperelliptic curves over F2^(alg cl) with general Weierstrass equation given over a very small field F. We describe in this paper an algorithm to compute the zeta function of E_g for g in a degree n extension field of F, which has as time complexity O(n^3) and memory requirements O(n^2). With a slightly different algorithm we can get time O(n^2,667) and memory O(n^2,5), and the computation of O(n) curves of the family can be done in time and space O(n^3). All these algorithms are polynomial in the genus., Comment: 23 pages

In this paper, we show that the only Pell numbers whose Euler function is also a Pell number are 1 and 2. AMS Subject Classification 2010: Primary 11B39; Secondary 11A25

Recently the authors showed that the algebraic integers of the form $-m+\zeta_k$ are bases of a canonical number system of $\mathbb{Z}[\zeta_k]$ provided $m\geq \phi(k)+1$, where $\zeta_k$ denotes a $k$-th primitive root of unity and $\phi$ is Euler's totient function. In this paper we are interested in the questions whether two bases $-m+\zeta_k$ and $-n+\zeta_k$ are multiplicatively independent. We show the multiplicative independence in case that $0 1$., Comment: 9 pages

Let $p$ be an odd prime and let $m\not\equiv 0\pmod p$ be a rational p-adic integer. In this paper we reveal the connection between quartic residues and the sum $\sum_{k=0}^{[p/4]}\binom{4k}{2k}\frac 1{m^k}$, where $[x]$ is the greatest integer not exceeding $x$. Let $q$ be a prime of the form $4k+1$ and so $q=a^2+b^2$ with $a,b\in\Bbb Z$. When $p\nmid ab(a^2-b^2)q$, we show that for $r=0,1,2,3$, $p^{\frac{q-1}4}\equiv (\frac ab)^r\pmod q$ if and only if $$\sum_{k=0}^{[p/4]}\binom{4k}{2k}\Big(\frac{a^2}{16q}\Big)^k\equiv (-1)^{\frac{p^2-1}8a+\frac{p-1}2\cdot \frac{q-1}4}\Big(\frac pq\Big) \Big(\frac ab\Big)^r\pmod p,$$ where $(\frac pq)$ is the Legendre symbol. We also establish congruences for $\sum_{k=0}^{[p/4]}\binom{4k}{2k}\frac 1{m^k}\pmod p$ in the cases $m=17,18,20,32,52,80,272$., Comment: Section 3 is new

The paper deals with establishing bounds for Eisenstein series on congruence quotients of the upper half plane, with control of both the spectral parameter and the level. The key observation in this work is that we exploit better the structure of the amplifier by just supporting on primes for the Eisenstein series, which can use both the analytic method as Young did to get a lower bound for the amplifier and the geometric method as Harcos--Templier did to obtain a more efficient treatment for the counting problem., Comment: 17 pages. Comments are welcome!

Cotangent sums are associated to the zeros of the Estermann zeta function. They have also proven to be of importance in the Nyman-Beurling criterion for the Riemann Hypothesis. The main result of the paper is the proof of the existence of a unique positive measure {\mu} on $\mathbb{R}$, with respect to which certain normalized cotangent sums are equidistributed. Improvements as well as further generalizations of asymptotic formulas regarding the relevant cotangent sums are obtained. We also prove an asymptotic formula for a more general cotangent sum as well as asymptotic results for the moments of the cotangent sums under consideration. We also give an estimate for the rate of growth of the moments of order 2k, as a function of k., Comment: 78 pages, 4 figures

Let K be a field of characteristic 0. We consider linear equations a1*x1+...+an*xn=1 in unknowns x1,...,xn from G, where a1,...,an are non-zero elements of K, and where G is a subgroup of the multiplicative group of non-zero elements of K. Two tuples (a1,...,an) and (b1,...,bn) of non-zero elements of K are called G-equivalent if there are u1,...,un in G such that b1=a1*u1,..., bn=an*un. Denote by m(a1,...,an,G) the smallest number m such that the set of solutions of a1*x1+...+an*xn=1 in x1,...,xn from G is contained in the union of m proper linear subspaces of K^n. It is known that m(a1,...,an,G) is finite; clearly, this quantity does not change if (a1,...,an) is replaced by a G-equivalent tuple. Gyory and the author proved in 1988 that there is a constant c(n) depending only on the number of variables n, such that for all but finitely many G-equivalence classes (a1,...,an), one has m(a1,...,an,G)< c(n). It is as yet not clear what is the best possible value of c(n). Gyory and the author showed that c(n)=2^{(n+1)!} can be taken. This was improved by the author in 1993 to c(n)=(n!)^{2n+2}. In the present paper we improve this further to c(n)=2^{n+1}, and give an example showing that c(n) can not be smaller than n., 12 pages, latex file

Holomorphic almost modular forms are holomorphic functions of the complex upper half plane that can be approximated arbitrarily well (in a suitable sense) by modular forms of congruence subgroups of large index in ${\rm SL}(2,{\mathbb Z})$ . It is proved that such functions have a rotation-invariant limit distribution when the argument approaches the real axis. An example of a holomorphic almost modular form is the logarithm of $\prod_{n=1}^\infty (1-\exp(2\pi\i n^2 z))$ . The paper is motivated by the author's previous studies [ Int. Math. Res. Not. 39 (2003) 2131–2151] on the connection between almost modular functions and the distribution of the sequence $n^2x$ modulo one.

We compute equivariant Euler characteristics of locally free sheaves on curves, thereby generalizing several results of Kani and Nakajima. For instance, we extend Kani's computation of the Galois module structure of the space of global meromorphic differentials which are logarithmic along the ramification locus from the tamely ramified to the weakly ramified case., Comment: 18 pages; v2: 22 pages, further result added, paper restructured and revised

The restricted isometry property (RIP) is an important matrix condition in compressed sensing, but the best matrix constructions to date use randomness. This paper leverages pseudorandom properties of the Legendre symbol to reduce the number of random bits in an RIP matrix with Bernoulli entries. In this regard, the Legendre symbol is not special---our main result naturally generalizes to any small-bias sample space. We also conjecture that no random bits are necessary for our Legendre symbol--based construction.

In the paper, the authors discover an integral representation, some inequalities, and complete monotonicity of Bernoulli numbers of the second kind., 10 pages

Expander graphs in general, and Ramanujan graphs in particular, have been of great interest in the last three decades with many applications in computer science, combinatorics and even pure mathematics. In these notes we describe various efforts made in recent years to generalize these notions from graphs to higher dimensional simplicial complexes., The paper is based on notes prepared for the Takagi Lectures, Tokyo 2012

We develop in this paper general techniques to analyze local combinatorial structures in product sets of two subsets of a countable group which are "large" with respect to certain classes of (not necessarily invariant) means on the group. As applications of our methods, we extend and quantify a series of recent results by Jin, Bergelson-Furstenberg-Weiss, Beiglb\"ock-Bergelson-Fish, Griesmer and DiNasso-Lupini to general countable groups., Comment: 51 pages. Minor changes

We improve the error term in the van der Corput transform for exponential sums \sum_{a \le n \le b} g(n) exp(2\pi i f(n)). For many functions g and f, we can extract the next term in the asymptotic, showing that previous results, such as those of Karatsuba and Korolev, are sharp. Of particular note, the methods of this paper avoid the use of the truncated Poisson formula, and thus can be applied to much longer intervals [a,b] with far better results. We provide a detailed analysis of the error term in the case g(x)=1 and f(x)=(x/3)^{3/2}., Comment: 72 pages, 10 figures

The paper deals with lower bounds for the remainder term in asymptotics for a certain class of arithmetic functions. Typically, these are generated by a Dirichlet series which involves a product of Riemann zeta-functions of a special form., Comment: 11 pages

Schinzel's Hypothesis H is a general conjecture in number theory on prime values of polynomials that generalizes, e.g., the twin prime conjecture and Dirichlet's theorem on primes in arithmetic progression. We prove an arithmetic analog of this conjecture for polynomial rings over pseudo algebraically closed fields. This implies results over large finite fields. A main tool in the proof is an irreducibility theorems \`a la Hilbert., Comment: The paper was shorten, some errors were corrected

In this paper we interpret the solutions to a particular Galois embedding problem over an extension K/F whose Galois group is a finite, cyclic p group in terms of certain Galois submodules within the parameterizing space of elementary p-abelian extensions of K; here p is a prime. Combined with some basic facts about the module structure of this parameterizing space, this allows us to exhibit a class of p-groups whose realization multiplicity is unbounded., 8 pages

A natural number $n$ is called {\it multiperfect} or {\it$k$-perfect} for integer $k\ge2$ if $\sigma(n)=kn$, where $\sigma(n)$ is the sum of the positive divisors of $n$. In this paper, we establish the structure theorem of odd multiperfect numbers analogous as Euler's theorem on odd perfect numbers. We prove the divisibility of the Euler part of odd multiperfect numbers and characterize the forms of odd perfect numbers $n=\pi^\alpha M^2$ such that $\pi\equiv\alpha(\text{mod}8)$. We also present some examples to show the nonexistence of odd perfect numbers as applications., Comment: 10 pages

We apply the efficient congruencing method to estimate Vino- gradov's integral for moments of order 2s, with 1 6 s 6 k 2 − 1. Thereby, we show that quasi-diagonal behaviour holds when s = o(k 2 ), we obtain near-optimal estimates for 1 6 s 6 1 k 2 + k, and optimal estimates for s > k 2 − 1. In this way we come half way to proving the main conjecture in two different directions. There are consequences for estimates of Weyl type, and in several allied applications. Thus, for example, the anticipated asymptotic formula in Waring's problem is established for sums of s kth powers of natural numbers whenever s > 2k 2 − 2k − 8 (k > 6). Estimates stemming from Vinogradov's mean value theorem deliver bounds for exponential sums of large degree, both in mean and pointwise, beyond the competence of alternate approaches. The ubiquity of such exponential sums in analytic number theory, in the analysis for example of the Riemann zeta function, in Waring's problem, and beyond, accounts for the high pro- file of Vinogradov's methods in the associated literature. In recent work, we established a version of Vinogradov's mean value theorem which achieves an essentially optimal upper bound with a number of variables only twice the number conjectured to be best possible (see (19)). For systems of degree k, previous estimates missed such a bound by a factor of order logk. Our earlier approach provides no upper bounds when the number of variables is smaller, precluding the possibility of applications involving the finer features of these mean values. Our goal in this paper is to remedy this deficiency, at the same time strengthening our previous conclusions. It transpires that we are able to come within a hair's breadth of proving the main conjecture concerning Vino- gradov's mean value theorem in half of the basic interval of relevant moments. Such developments illustrate the flexibility of the new efficient congruencing method introduced in (19). We now introduce some notation. When k ∈ N and � ∈ R k , define

In this paper, we prove a theorem on the distribution of primes in cubic progressions on average., Comment: 14 pages, to appear in Math. Z

In this paper, we prove that there is a natural correspondence between product identities for theta functions and integer matrix exact covering systems. We show that since $\mathbb{Z}^n$ can be taken as the disjoint union of a lattice generated by $n$ linearly independent vectors in $\mathbb{Z}^n$ and a finite number of its translates, certain products of theta functions can be written as linear combinations of other products of theta functions. We firstly give a general theorem to write a product of $n$ theta functions as a linear combination of other products of theta functions. Many known identities for products of theta functions are shown to be special cases of our main theorem. Several entries in Ramanujan's notebooks as well as new identities are proved as applications, including theorems for products of three and four theta functions that have not been obtained by other methods.

This paper proposes a conjecture on special values of L-functions of geometric motives over Z. This includes L-functions of mixed motives over Q and Hasse-Weil zeta-functions of schemes over Z. We conjecture the following: the order of L(M, s) at s=0 is given by the negative Euler characteristic of motivic cohomology of $M^\vee(-1)$. Up to a nonzero rational factor, the L-value at s=0 is given by the determinant of the pairing of Arakelov motivic cohomology of M with the motivic homology of M. Under standard assumptions concerning mixed motives over Q, $F_p$, and Z, this conjecture is essentially equivalent to the conjunction of Soul\'e's conjecture about pole orders of zeta-functions of schemes over Z, Beilinson's conjecture about special L-values for motives over Q and the Tate conjecture over $F_p$., Comment: Final version. To appear in Asian Journal of Mathematics. Most recent changes: Corollary 0.3. (L-values of Artin-Tate motives), Theorem 4.5 (relation of Beilinson-Soul\'e vanishing vs. perfectness of Arakelov intersection pairing)

We give new proofs of two basic results in number theory: The law of quadratic reciprocity and the sign of the Gauss sum. We show that these results are encoded in the relation between the discrete Fourier transform and the action of the Weyl element in the Weil representation modulo p,q and pq., This paper was submited for publication February 1, 2008. Key words: Quadratic reciprocity, Sign of Gauss sum, Discrete Fourier transform, Finite Weil representations, Character of the Weil representation

In a previous paper, the authors proved that in any system of three linear forms satisfying obvious necessary local conditions, there are at least two forms that infinitely often assume $E_2$-values; i.e., values that are products of exactly two primes. We use that result to prove that there are inifinitely many integers $x$ that simultaneously satisfy $$\omega(x)=\omega(x+1)=4, \Omega(x)=\Omega(x+1)=5, \text{and} d(x)=d(x+1)=24.$$ Here, $\omega(x), \Omega(x), d(x)$ represent the number of prime divisors of $x$, the number of prime power divisors of $x$, and the number of divisors of $x$, respectively. We also prove similar theorems where $x+1$ is replaced by $x+b$ for an arbitrary positive integer $b$. Our results sharpen earlier work of Heath-Brown, Pinner, and Schlage-Puchta.

This paper presents results on both the kernel and cokernel of the S-capitulation map C_{F,S}\ra C_{K,S}^{G} for arbitrary finite Galois extensions K/F (with Galois group G) and arbitrary finite sets of primes S of F (assumed to contain the archimedean primes in the number field case), Footnote added to statement of Theorem 4.1, and other minor changes. 20 pages

In the present paper, we shall show that for any prime number p, every finite p-group occurs as the Galois Group of the maximal unramified p-extension over a certain number field of finite degree. We shall also show that for any given pro-p-group G with countably many generators, there exists a number field (not necessary of finite degree) whose maximal unramified p-extension has Galois group isomorphic to G. This means that the set of the isomorphism classes of the Galois groups of the maximal unramified p-extensions over the number fields (including of infinite degree) is precisely equal to that of all the pro-p-groups with countably many generators., Comment: In this new version, a considerable improvement has been done, that is, Theorems 1 and 2 have been established for all the prime number p (without restrictive assumption C(p) on the prime number p)

Let a be an integer and q a prime number. In this paper, we find an asymptotic formula for the number of positive integers n < x with the property that no divisor d > 1 of n lies in the arithmetic progression a modulo q., 37 pages

The continued fraction expansion of an irrational number $\alpha$ is eventually periodic if and only if $\alpha$ is a quadratic irrationality. However, very little is known regarding the size of the partial quotients of algebraic real numbers of degree at least three. Because of some numerical evidence and a belief that these numbers behave like most numbers in this respect, it is often conjectured that their partial quotients form an unbounded sequence. More modestly, we may expect that if the sequence of partial quotients of an irrational number $\alpha$ is, in some sense, "simple", then $\alpha$ is either quadratic or transcendental. The term "simple" can of course lead to many interpretations. It may denote real numbers whose continued fraction expansion has some regularity, or can be produced by a simple algorithm (by a simple Turing machine, for example), or arises from a simple dynamical system... The aim of this paper is to present in a unified way several new results on these different approaches of the notion of simplicity/complexity for the continued fraction expansion of algebraic real numbers of degree at least three.

We show that the Gaussian primes $P[i] \subseteq \Z[i]$ contain infinitely constellations of any prescribed shape and orientation. More precisely, given any distinct Gaussian integers $v_0,...,v_{k-1}$, we show that there are infinitely many sets $\{a+rv_0,...,a+rv_{k-1}\}$, with $a \in \Z[i]$ and $r \in \Z \backslash \{0\}$, all of whose elements are Gaussian primes. The proof is modeled on a recent paper by Green and Tao and requires three ingredients. The first is a hypergraph removal lemma of Gowers and R\"odl-Skokan; this hypergraph removal lemma can be thought of as a generalization of the Szemer\'edi-Furstenberg-Katznelson theorem concerning multidimensional arithmetic progressions. The second ingredient is the transference argument of Green and Tao, which allows one to extend this hypergraph removal lemma to a relative version, weighted by a pseudorandom measure. The third ingredient is a Goldston-Yildirim type analysis for the Gaussian integers, which yields a pseudorandom measure which is concentrated on Gaussian "almost primes"., Comment: 58 pages, no figures. An issue (pointed out by Lilian Matthiesen) regarding the need to ensure the linear forms are not commensurate to their conjugate has been addressed

We study the absolute Galois group by looking for invariants and orbits of its faithful action on Grothendieck's dessins d'enfants. We define a class of functions called Belyi-extending maps, which we use to construct new Galois invariants of dessins from previously known invariants. Belyi-extending maps are the source of the ``new-type'' relations on the injection of the absolute Galois group into the Grothendieck-Teichmuller group. We make explicit how to get from a general Belyi-extending map to formula for its associated invariant which can be implemented in a computer algebra package. We give an example of a new invariant differing on two dessins which have the same values for the other readily computable invariants., Comment: 13 pages, 7 figures; submitted for publication; revisions are that the paper now deals only with Galois invariants of dessins, and that material is slightly expanded

In this paper we extend a conjecture of A. Ash and W. Sinnott relating niveau 1 Galois representations to the $\mod p$ cohomology of congruence subgroups of ${\rm SL}\sb n(\mathbb {Z})$ to include Galois representations of higher niveau. We then present computational evidence for our conjecture in the case $n=3$ in the form of three-dimensional Galois representations which appear to correspond to cohomology eigenclasses as predicted by the conjecture. Our examples include Galois representations with nontrivial weight and level, as well as irreducible three-dimensional representations that are in no obvious way related to lower-dimensional representations. In addition, we prove that certain symmetric square representations are actually attached to cohomology eigenclasses predicted by the conjecture.

This paper considers a higher-dimensional generalization of the notion of Ramanujan graphs, defined by Lubotzky, Phillips, and Sarnak. Specifically the Ramanujan property is studied for cubical complexes which are uniformized by an ordered product of regular trees. We construct explicit arithmetic examples using quaternion algebras over totally real fields. Here we reduce the Ramanujan property to special cases of the Ramanujan-Petersson conjecture for holomorphic Hilbert modular forms, many of which are known.

This paper has a two-fold purpose. First, by considering a reformulation of a deep theorem of G\"ollnitz, we obtain a new weighted partition identity involving the Rogers-Ramanujan partitions, namely, partitions into parts differing by at least two. Consequences of this include Jacobi's celebrated triple product identity for theta functions, Sylvester's famous refinement of Euler's theorem, as well as certain weighted partition identities. Next, by studying partitions with prescribed bounds on successive ranks and replacing these with weighted Rogers-Ramanujan partitions, we obtain two new sets of theorems - a set of three theorems involving partitions into parts $\not\equiv 0, \pm i$ ($mod$ 6), and a set of three theorems involving partitions into parts $\not\equiv 0, \pm i$ ($mod$ 7), $i=1,2,3$., Comment: 31 pages, 4 figures

In a previous paper \cite{BorGunn}, we defined the space of toric forms $\TTT(l)$, and showed that it is a finitely generated subring of the holomorphic modular forms of integral weight on the congruence group $\Gamma_1(l)$. In this article we prove the following theorem: modulo Eisenstein series, the weight two toric forms coincide exactly with the vector space generated by all cusp eigenforms f such that $L(f,1) \not = 0$. The proof uses work of Merel, and involves an explicit computation of the intersection pairing on Manin symbols., Comment: 14 pp., 1 figure, AMS-LaTeX