We study the local zeta integrals attached to a pair of generic representations (π,τ) of GLn×GLm, n>m, over a p-adic field. Through a process of unipotent averaging we produce a pair of corresponding Whittaker functions whose zeta integral is non-zero, and we express this integral in terms of the Langlands parameters of π and τ. In many cases, these Whittaker functions also serve as a test vector for the associated Rankin–Selberg (local) L-function.

We show that a positive proportion of the derivatives of L-functions associated to holomorphic cusp forms of large weight do not vanish at the central point. Specifically, by optimizing the choice of quadratic form in the mollified moments of the L-function, we prove that this proportion is at least 39%.

Let E be a quadratic semisimple extension of a local field F of characteristic zero. We determine explicit relation between the gamma factors for the Asai representation of RE/FGL2/E defined by the Weil-Deligne representations and the local zeta integrals. When E=F×F, the results were due to Henniart and Jacquet. We completed the theory in this article based on explicit calculation.

The Jordan totient Jk(n) can be defined by Jk(n)=nk∏p|n(1−p−k). In this paper, we study the average behavior of fractions P/Q of two products P and Q of Jordan totients, which we call Jordan totient quotients. To this end, we describe two general and ready-to-use methods that allow one to deal with a larger class of totient functions. The first one is elementary and the second one uses an advanced method due to Balakrishnan and Pétermann. As an application, we determine the average behavior of the Jordan totient quotient, the kth normalized derivative of the nth cyclotomic polynomial Φn(z) at z=1, the second normalized derivative of the nth cyclotomic polynomial Φn(z) at z=−1, and the average order of the Schwarzian derivative of Φn(z) at z=1.

In this paper, we introduce modular polynomials for the congruence subgroup Γ0(M) when X0(M) has genus zero and therefore the polynomials are defined by a Hauptmodul of X0(M). We show that the intersection number of two curves defined by two modular polynomials can be expressed as the sum of the numbers of SL2(Z)-equivalence classes of positive definite binary quadratic forms over Z. We also show that the intersection numbers can be also combinatorially written by Fourier coefficients of the Siegel Eisenstein series of degree 2, weight 2 with respect to Sp2(Z).

Here we study the twists of the genus 2 curve given by the hyperelliptic equation Y2=X6+1 over any field of characteristic different from 2, 3 or 5. Since any curve of genus 2 with group of automorphisms of order 24 is isomorphic (over an algebraically closed field) to the given one, the study of this set of twists is equivalent to the classification, up to isomorphisms defined over the base field, of curves of genus 2 with that number of automorphisms. This contribution closes the series of articles on the classification of twists of curves of genus 2. The knowledge of these twists can be of interest in a wide range of arithmetical questions, such as the Sato-Tate or the Strong Lang conjectures among others.

Given a Speh representation π of GLn(F) for a non-archimedean local field F, we obtain necessary and sufficient conditions on standard parabolic subgroups for the Jacquet module of π with respect to the parabolic subgroup to be generic. Using this we describe the precise set of characters Θ of the maximal unipotent radical Un of GLn(F) such that HomUn(π,Θ) is non-zero. We then describe the behavior of this set under the base change map (with respect to a finite cyclic extension of F of prime degree).

We prove an identity between two Mahler measures. Combining it with a result of Rogers and Zudilin, this leads to a formula relating the Mahler measure of a non-tempered polynomial with L′(E,0), where E is an elliptic curve of conductor 20.

We shall give some results for an integer divisible by its unitary totient.

We give an explicit description of the inner cohomology of an adèlic locally symmetric space of a given level structure attached to the general linear group of prime rank n with coefficients in a locally constant sheaf of complex vector spaces. We show that for all primes n the inner cohomology vanishes in all degrees for nonconstant sheaves, otherwise the quotient module of the inner cohomology classes that are not cuspidal is trivial in all degrees for primes n=2,3, and for all primes n≥5 the description of the same at degrees where it is nontrivial is given in terms of algebraic Hecke characters.

A post-critically finite rational map ϕ of prime degree p and a base point β yield a tower of finitely ramified iterated extensions of number fields, and sometimes provide an arboreal Galois representation with a p-adic Lie image. In this paper, we take ϕ to be the monic Chebyshev polynomial x2−2, and we examine the size of the 2-part of the ideal class group of extensions in the resulting tower. In some cases, we prove an analogue of Greenberg's conjecture from Iwasawa theory. A key tool is a general theorem on p-indivisibility of class numbers of relative cyclic extensions of degree p2.

Let K be a totally real field and GK:=Gal(K‾/K) its absolute Galois group, where K‾ is a fixed algebraic closure of K. Let ℓ be a prime and E a finite extension of Qℓ. Let S be a finite set of finite places of K not dividing ℓ. Assume that K, S, Hodge-Tate type h and a positive integer n are fixed. In this paper, we prove that if ℓ is sufficiently large, then, for any fixed E, there are only finitely many isomorphism classes of crystalline representations r:GK→GLn(E) unramified outside S∪{v:v|ℓ}, with fixed Hodge-Tate type h, such that r|GK′≃⊕ri′ for some finite totally real field extension K′ of K unramified at all places of K over ℓ, where each representation ri′ over E is an 1-dimensional representation of GK′ or a totally odd irreducible 2-dimensional representation of GK′ with distinct Hodge-Tate numbers.

The rank of partitions plays an important role in the combinatorial interpretations of several Ramanujan's famous congruence formulas. In 2005 and 2008, the D-rank and M2-rank of an overpartition were introduced by Lovejoy, respectively. Let N‾(m,n) and N2‾(m,n) denote the number of overpartitions of n with D-rank m and M2-rank m, respectively. In 2014, Chan and Mao proposed a conjecture on monotonicity properties of N‾(m,n) and N2‾(m,n). In this paper, we prove the Chan-Mao monotonicity conjecture. To be specific, we show that for any integer m and nonnegative integer n, N2‾(m,n)≤N2‾(m,n+1); and for (m,n)≠(0,4) with n≠|m|+2, we have N‾(m,n)≤N‾(m,n+1). Furthermore, when m increases, we prove that N‾(m,n)≥N‾(m+2,n) and N2‾(m,n)≥N2‾(m+2,n) for any m,n≥0, which is an analogue of Chan and Mao's result for partitions.

Let K be an imaginary quadratic field of discriminant dK, and let n be a nontrivial integral ideal of K in which N is the smallest positive integer. Let QN(dK) be the set of primitive positive definite binary quadratic forms of discriminant dK whose leading coefficients are relatively prime to N. We adopt an equivalence relation ∼n on QN(dK) so that the set of equivalence classes QN(dK)/∼n can be regarded as a group isomorphic to the ray class group of K modulo n. We further establish an explicit isomorphism of QN(dK)/∼n onto Gal(Kn/K) in terms of Fricke invariants, where Kn denotes the ray class field of K modulo n. This would be a certain extension of the classical composition theory of binary quadratic forms, originated and developed by Gauss and Dirichlet.

Let K be an imaginary quadratic field whose discriminant is congruent to one modulo 8 and O be the ring of integers of K. Let Γ denote the group SL(2,O) which acts discontinuously on the upper half space H. In this paper, we study a homomorphism φ:Γ→Z obtained from a branch of the logarithm of a theta function on H which is automorphic with respect to Γ and does not vanish on H. In particular, we determine explicitly the decomposition φ=φc+φe of φ into the cusp part φc and the Eisenstein part φe, and prove a congruence conjectured by Sczech [14] between φ and φe modulo 8 under an assumption on the 2-divisibility of a certain L-value.

A natural method how to measure sets of natural numbers is the asymptotic density which is a special case of weighted densities. These densities are based on the Riesz summation method. A completely different approach is the Abel summation method. This leads to the concept of the Abel density. In this paper we prove estimates of the values of the lower and upper Abel densities, depending on values of the lower and upper asymptotic densities.

First, we give a formula for the limiting value of the higher Mahler measures for general polynomials of one variable, which refines Biswas and Monico's result. Second, we give an analytic continuation of the zeta Mahler measures for polynomials of one variable to the whole complex plane as meromorphic functions, and we show that all of their poles are simple.

We compute generators and relations for the graded rings of paramodular forms of degree two and levels 5 and 7. The generators are expressed as quotients of Gritsenko lifts and Borcherds products. The computation is made possible by a characterization of modular forms on the Humbert surfaces of discriminant 4 that arise from paramodular forms by restriction.

In the present article, we extend previous results of the author and we show that when K is any quadratic imaginary field of class number one, Fermat's equation ap+bp+cp=0 does not have integral coprime solutions a,b,c∈K∖{0} such that 2|abc and p≥19 is prime. The results are conjectural upon the veracity of a natural generalisation of Serre's modularity conjecture.

In this paper we prove the algebraic independence of a certain power series involving the values of the Hecke-Mahler series and its derivatives of any order at any nonzero distinct algebraic numbers inside the unit circle. As corollaries, we have a refinement of our previous result and also prove that the exponential type Hecke-Mahler series and its derivatives of any order take algebraically independent values at any nonzero distinct algebraic numbers inside the unit circle.

In this paper we introduce a class of sequences connected with the m–ary partition function and investigate their congruence properties. In particular, we get facts about the sequences of m–ary partitions (bm(n))n∈N and m–ary partitions with no gaps (cm(n))n∈N. We prove, for example, that for any natural number 2

We combine A.J. Irving's sieve method with the Diamond-Halberstam-Richert (DHR) sieve to study the almost-prime values produced by products of irreducible polynomials evaluated at prime arguments. A significant improvement is made when the degrees of the irreducible components are large compared with the number of components themselves. This generalizes the previous results of Irving and Kao, who separately examined the almost-prime values of a single irreducible polynomial evaluated at prime arguments.

Extending the former work for the good reduction case, we provide a numerical criterion to verify a large portion of the “Iwasawa main conjecture without p-adic L-functions” for elliptic curves with additive reduction at an odd prime p over the cyclotomic Zp-extension. We also deduce the corresponding p-part of the Birch and Swinnerton-Dyer formula for elliptic curves of rank zero from the same numerical criterion. We give some explicit examples at the end and specify our choice of Kato's Euler system in the appendix.

The Erdős-Burgess constant of a semigroup S is the smallest positive integer k such that any sequence over S of length k contains a nonempty subsequence whose terms multiply to an idempotent element of S. In the case where S is the multiplicative semigroup of Z/nZ, we confirm a conjecture connecting the Erdős-Burgess constant of S and the Davenport constant of (Z/nZ)× for n with at most two prime factors. We also discuss the extension of our techniques to other rings.

By finding all integral points on certain elliptic and hyperelliptic curves we completely solve the Diophantine equation (nk)=(ml)+d for −3≤d≤3 and (k,l)∈{(2,3),(2,4),(2,5),(2,6),(2,8),(3,4),(3,6),(4,6),(4,8)}. Moreover, we present some other observations of computational and theoretical nature concerning the title equation.

In [KK18], the authors introduced “soft” methods to prove the effective (i.e. with power savings error) equidistribution of “shears” in cusped hyperbolic surfaces. In this paper, we study the same problem but now allow full use of the spectral theory of automorphic forms to produce explicit exponents, and uniformity in parameters. We give applications to counting square values of quadratic forms.

Let (xn)n=1∞ be a sequence on the torus T (normalized to length 1). A sequence (xn) is said to have Poissonian pair correlation if, for all s>0,limN→∞⁡1N#{1≤m≠n≤N:|xm−xn|≤sN}=2s. It is known that this implies uniform distribution of the sequence (xn). Hinrichs, Kaltenböck, Larcher, Stockinger & Ullrich extended this result to higher dimensions and showed that sequences (xn) in [0,1]d that satisfy, for all s>0,limN→∞⁡1N#{1≤m≠n≤N:‖xm−xn‖∞≤sN1/d}=(2s)d are also uniformly distributed. We prove the same result for the extension by the Euclidean norm: if a sequence (xn) in Td satisfies, for all s>0,limN→∞⁡1N#{1≤m≠n≤N:‖xm−xn‖2≤sN1/d}=ωdsd, where ωd is the volume of the unit ball, then (xn) is uniformly distributed. Our approach shows that Poissonian Pair Correlation implies an exponential sum estimate that resembles and implies the Weyl criterion.

We construct an analogue of the ring of algebraic numbers, living in a quotient of the product of all finite fields of prime order. We use this ring to deduce some results about linear recurrent sequences.

We use a simple delta method to prove the Weyl bound in t-aspect for the L-function of a holomorphic or a Hecke-Maass cusp form of arbitrary level and nebentypus. In particular, this extends the results of Meurman and Jutila for the t-aspect Weyl bound, and the recent result of Booker, Milinovich and Ng to Hecke cusp forms of arbitrary level and nebentypus.

We show that, for any given c∈(1,1110), every sufficiently large integer n can be represented as n=[mc]+[pc], where m is a positive integer and p is a prime, and [t] is the integer part of the real number t. We also prove that, when c∈(1,1+52), such representation exists for almost all positive integers n. These respectively improve the results of A. Kumchev [9], and Balanzario, Garaev, and Zuazua [1].

We present a powerful method for the calculation of heat-kernel coefficients of the Laplacian on the projective complex spaces endowed with the Fubini-Study metric, and also for the Laplacian on the n-spheres equipped with the standard metric. Formulas for the regularized determinants are also given. Our formula in the case of the n-sphere recovers the result of Choi and Quine.

Bounds for max⁡{m,m˜} subject to m,m˜∈Z∩[1,p), p prime, z indivisible by p, mm˜≡zmodp and m belonging to some fixed Beatty sequence {⌊nα+β⌋:n∈N} are obtained, assuming certain conditions on α. The proof uses a method due to Banks and Shparlinski. As an intermediate step, bounds for the discrete periodic autocorrelation of the finite sequence 0,ep(y1‾),ep(y2‾),…,ep(y(p−1‾)) on average are obtained, where ep(t)=exp⁡(2πit/p) and mm‾≡1modp. The latter is accomplished by adapting a method due to Kloosterman.

Given a finite set of bases b1, b2, …, br (integers greater than 1), a multi-base representation of an integer n is a sum with summands db1α1b2α2⋯brαr, where the αj are nonnegative integers and the digits d are taken from a fixed finite set. We consider multi-base representations with at least two bases that are multiplicatively independent. Our main result states that the order of magnitude of the minimal Hamming weight of an integer n, i.e., the minimal number of nonzero summands in a representation of n, is log⁡n/(log⁡log⁡n). This is independent of the number of bases, the bases themselves, and the digit set. For the proof, the existing upper bound for prime bases is generalized to multiplicatively independent bases; for the required analysis of the natural greedy algorithm, an auxiliary result in Diophantine approximation is derived. The lower bound follows by a counting argument and alternatively by using communication complexity; thereby improving the existing bounds and closing the gap in the order of magnitude. This implies also that the greedy algorithm terminates after O(log⁡n/log⁡log⁡n) steps, and that this bound is sharp.

Let p be a prime, a≥1, and ℓa(p) be the multiplicative order of a modulo p. We prove various theorems concerning the averages of ℓa(p) over p≤x and a≤y. We prove that these theorems hold for y>exp⁡((α+ϵ)log⁡x) where α≈3.42. This is an improvement over y>exp⁡(c1log⁡x) with c1≥12e9 given in [S2]. We also provide the average of τ(ℓa(p)) over p≤x, a≤y, and y>exp⁡((α+ϵ)log⁡x), where τ(n) is the divisor function ∑d|n1.

Text For a positive integer k, let f(k) denote the largest integer t such that for every finite abelian group G and every zero-sum free subset S of G, if |S|=k then |Σ(S)|≥t. In this paper, we prove that f(k)≥16k2, which significantly improves a result of J.E. Olson. We also supply some interesting results on f(k). Video For a video summary of this paper, please visit https://youtu.be/ZEHZFRUVJKY.

Muto, Narita and Pitale construct counterexamples to the Generalized Ramanujan Conjecture for GL2(B) over the division quaternion algebra B with discriminant two via a lift from SL2. In this paper, we try to exactly characterize the image of this lift. The previous methods of Maass, Kohnen or Kojima do not apply here, hence we approach this problem via a combination of classical and representation theory techniques to identify the image. Crucially, we use the Jacquet Langlands correspondence described by Badulescu and Renard to characterize the representations.

We classify the dual strongly perfect lattices in dimension 16. There are four pairs of such lattices, the famous Barnes-Wall lattice Λ16, the extremal 5-modular lattice N16, the odd Barnes-Wall lattice O16 and its dual, and one pair of new lattices Γ16 and its dual. The latter pair belongs to a new infinite series of dual strongly perfect lattices, the sandwiched Barnes-Wall lattices, described by the authors in a previous paper. An updated table of all known strongly perfect lattices up to dimension 26 is available in the catalogue of lattices [15].

We investigate the algebraic degree of circulant graphs, i.e. the dimension of the splitting field of the characteristic polynomial of the associated adjacency matrix over the rationals. Studying the algebraic degree of graphs seems more natural than characterizing graphs with integral spectra only. We prove that the algebraic degree of circulant graphs on n vertices is bounded above by φ(n)/2, where φ denotes Euler's totient function, and that the family of cycle graphs provides a family of maximum algebraic degree within the family of all circulant graphs. Moreover, we precisely determine the algebraic degree of circulant graphs on a prime number of vertices.

For big Galois representations attached to Hida families of elliptic cuspforms, Howard constructed the systems of big Heegner points. On the other hand, for Galois representations attached to cuspforms, there are Euler systems of generalized Heegner cycles, which are constructed by Bertolini-Darmon-Prasanna. Castella recently proved that the systems of big Heegner points interpolate the systems of generalized Heegner cycles. In this paper, by a different approach we slightly improve his work.

Let K be a field complete with respect to a discrete valuation whose residue field is algebraically closed of an odd positive characteristic. We study the ramification in the cohomology of a smooth proper surface X defined over K, under the assumption that X admits an integral model X whose special fibre has at worst ordinary double points. We will introduce a numerical invariant of X, in terms of which the ramification in the cohomology of X is determined.

We present a set of diagonal matrices which index enough Fourier coefficients for a complete characterization of all Siegel cusp forms of degree 2, weight k, level N and character χ, where k is an even integer ≥4, N is an odd, square-free positive integer, and χ has conductor equal to N. As an application, we show that the Koecher-Maass series of any F∈Sk2 twisted by the set of Maass waveforms whose eigenvalues are in the continuum spectrum of the hyperbolic Laplacian determines F. We also generalize a result due to Skogman about the non-vanishing of all theta components of a Jacobi cusp form of even weight and prime index, which may have some independent interest.

We give a reformulation of the Birch and Swinnerton-Dyer conjecture over global function fields in terms of Weil-étale cohomology of the curve with coefficients in the Néron model, and show that it holds under the assumption of finiteness of the Tate-Shafarevich group.

Text We report the results of our empirical investigations on the Bateman-Horn conjecture. This conjecture, in its commonly known form, produces rather large deviations when the polynomials involved are not monic. We propose a modified version of the conjecture which empirically demonstrates remarkable accuracy even for modest values of primes. Video For a video summary of this paper, please visit https://youtu.be/mINg-0n7nOY.

Garcia, Kahoro, and Luca showed that the Bateman–Horn conjecture implies φ(p−1)⩾φ(p+1) for a majority of twin-primes pairs p,p+2 and that the reverse inequality holds for a small positive proportion of the twin primes. That is, p tends to have more primitive roots than does p+2. We prove that Dickson's conjecture, which is much weaker than Bateman–Horn, implies that the quotients φ(p+1)φ(p−1), as p,p+2 range over the twin primes, are dense in the positive reals. We also establish several Schinzel-type theorems, some of them unconditional, about the behavior of φ(p+1)φ(p) and σ(p+1)σ(p), in which σ denotes the sum-of-divisors function.

Let Fq[T] be the polynomial ring over a finite field Fq. We study the endomorphism rings of Drinfeld Fq[T]-modules of arbitrary rank over finite fields. We compare the endomorphism rings to their subrings generated by the Frobenius endomorphism and deduce from this a refinement of a reciprocity law for division fields of Drinfeld modules proved in our earlier paper. We then use these results to give an efficient algorithm for computing the endomorphism rings and discuss some interesting examples produced by our algorithm.

In this article we develop the multilinear theory of Drinfeld displays and use it to construct tensor products, symmetric and exterior powers of z-divisible local Anderson modules, which are the function fields analogs of p-divisible groups.

Let X(D,1)=Γ(D,1)\H denote the Shimura curve of level N=1 arising from an indefinite quaternion algebra of fixed discriminant D. We study the discrete average of the error term in the hyperbolic circle problem over Heegner points of discriminant d<0 on X(D,1) as d→−∞. We prove that if |d| is sufficiently large compared to the radius r≈log⁡X of the circle, we can improve on the classical O(X2/3)-bound of Selberg. Our result extends the result of Petridis and Risager for the modular surface to arithmetic compact Riemann surfaces.

We investigate Lagrange's equation with almost-prime variables. We establish the result that every sufficiently large integer of the form 24k+4 can be represented as the sum of four squares of almost-primes, three of them being P3-numbers and the other a P4-number. A major component of the idea of the proof is the use of Selberg's Λ2-sieve method. This sieve method has been successfully used in some linear problems and we demonstrate its use on a non-linear equation.

Let k be a perfect field of positive characteristic and Z an effective Cartier divisor in the projective line over k with complement U. In this note, we establish some results about the formal deformation theory of overconvergent isocrystals on U with fixed “local monodromy” along Z. En route, we show that a Hochschild cochain complex governs deformations of a module over an arbitrary associative algebra. We also relate this Hochschild cochain complex to a de Rham complex in order to understand the deformation theory of a differential module over a differential ring.

For 1≤i≤r, let χi be primitive Dirichlet characters modulo qi and Z(t,χi) be the Z-function corresponding to the Dirichlet L-series L(s,χi). Let Ω(t) be a real linear combination of Z(t,χi). Since Z(t,χi) is real for real t, Ω(t) is real for real t. In this paper, we show that the Lebesgue measure of the set, where the functional values of Ω(t) is positive or negative in the interval [T,2T] is at least Tr2. We also study the Lebesgue measure of the set that the certain complex linear combinations of Z(t,χi) takes positive or negative values respectively. In particular, we study the distribution of signs of the Z-function correspond to the Davenport-Heilbronn function. Moreover, we prove that for sufficiently large T, the generalized Davenport-Heilbronn function has at least H(logT)2φ(q)−ϵ odd order zeros along the critical line on the interval [T,T+H].

Let E be an elliptic curve defined over Q and let N be a positive integer. Then, ME(N) counts the number of primes p such that the group Ep(Fp) is of order N. For a given positive integer ℓ, we study the probability of the event {ME(N)=ℓ}. In an earlier joint work with Balasubramanian, we showed that ME(N) follows the Poisson distribution when an average is taken over a family of elliptic curves with parameters A and B where A,B≥Nℓ+12(log⁡N)1+γ and AB>N3(ℓ+1)2(log⁡N)2+γ for a positive integer ℓ and any γ>0. In this paper, we show that for sufficiently large N, the same result holds even if we take A and B in the range exp⁡(Nϵ220(ℓ+1))≥A,B>Nϵ and AB>N3(ℓ+1)2(log⁡N)3+2γ(log⁡log⁡N)ℓ+8 for any ϵ>0.

In this paper we discuss metric theory associated with the affine (inhomogeneous) linear forms in the so called doubly metric settings within the classical and the mixed setups. We consider the system of affine forms given by q↦qX+α, where q∈Zm (viewed as a row vector), X is an m×n real matrix and α∈Rn. The classical setting refers to the dist(qX+α,Zm) to measure the closeness of the integer values of the system (X,α) to integers. The absolute value setting is obtained by replacing dist(qX+α,Zm) with dist(qX+α,0); and the more general mixed settings are obtained by replacing dist(qX+α,Zm) with dist(qX+α,Λ), where Λ is a subgroup of Zm. We prove the Khintchine–Groshev and Jarník type theorems for the mixed affine forms and Jarník type theorem for the classical affine forms. We further prove that the sets of badly approximable affine forms, in both the classical and mixed settings, are hyperplane winning. The latter result, for the classical setting, answers a question raised by Kleinbock (1999).

Let d(A) be the asymptotic density (if it exists) of a sequence of integers A. For any real numbers 0≤α≤β≤1, we solve the question of the existence of a sequence A of positive integers such that d(A)=α and d(A+A)=β. More generally we study the set of k-tuples (d(iA))1≤i≤k for A⊂N. This leads us to introduce subsets defined by diophantine constraints inside a random set of integers known as the set of “pseudo sth powers”. We consider similar problems for subsets of the circle R/Z, that is, we partially determine the set of k-tuples (μ(iA))1≤i≤k for A⊂R/Z.

In 2016, Jang and Kim stated a conjecture about the norms of indecomposable integers in real quadratic number fields Q(D) where D>1 is a squarefree integer. Their conjecture was later disproved by Kala for D≡2mod4. We investigate such indecomposable integers in greater detail. In particular, we find the minimal D in each congruence class D≡1,2,3mod4 that provides a counterexample to the Jang-Kim Conjecture; provide infinite families of such counterexamples; and state a refined version of the Jang-Kim Conjecture. Lastly, we prove a slightly weaker version of our refined conjecture that is of the correct order of magnitude, showing the Jang-Kim Conjecture is only wrong by at most O(D).

It has been known since Erdős that the sum of 1/(nlog⁡n) over numbers n with exactly k prime factors (with repetition) is bounded as k varies. We prove that as k tends to infinity, this sum tends to 1. Banks and Martin have conjectured that these sums decrease monotonically in k, and in earlier papers this has been shown to hold for k up to 3. However, we show that the conjecture is false in general, and in fact a global minimum occurs at k=6.

In this paper we obtain a new constant in the Pólya-Vinogradov inequality. Our argument follows previously established techniques which use the Fourier expansion of an interval to reduce to Gauss sums. Our improvement comes from approximating an interval by a function with slower decay on the edges and this allows for a better estimate of the ℓ1 norm of the Fourier transform. This approximation induces an error for our original sums which we deal with by combining some ideas of Hildebrand with Garaev and Karatsuba concerning long character sums.

Let W be a nonempty subset of the set of integers Z. A nonempty subset C of Z is said to be an asymptotic complement to W if W+C contains almost all the integers except a set of finite size. The set C is said to be a minimal asymptotic complement to W if C is an asymptotic complement to W, but C∖{c} is not an asymptotic complement to W for every c∈C. Asymptotic complements have been studied in the context of representations of integers since the time of Erdős, Hanani, Lorentz and others, while the notion of minimal asymptotic complements is due to Nathanson. In this article, we study minimal asymptotic complements in Z and deal with a problem of Nathanson on their existence and their inexistence.

We study a generalization of the classical Dedekind sum that incorporates two Dirichlet characters and develop properties that generalize those of the classical Dedekind sum. By calculating the Fourier transform of this generalized Dedekind sum, we obtain an explicit formula for its second moment. Finally, we derive upper and lower bounds for the second moment with nearly identical orders of magnitude.

We define X-base Fibonacci-Wieferich primes, which generalize Wieferich primes, where X is a finite set of algebraic numbers. We show that there are infinitely many non-X-base Fibonacci-Wieferich primes, assuming the abc-conjecture of Masser-Oesterlé-Szpiro for number fields. We also provide a new conjecture concerning the rank of the free part of the abelian group generated by all elements in X and give some heuristics that support the conjecture.

We give an exact formula for the number of G-extensions of local function fields Fq((t)) for finite abelian groups G up to a conductor bound. As an application we give a lower bound for the corresponding counting problem by discriminant.