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The primary purpose of this paper is to generalize the classical Riemann zeta function to the setting of Krull monoids with torsion class groups. We provide a first study of the same generalization by extending Euler's classical product formula to the more general scenario of Krull monoids with torsion class groups. In doing so, the Decay Theorem is fundamental as it allows us to use strong atoms instead of primes to obtain a weaker version of the Fundamental Theorem of Arithmetic in the more general setting of Krull monoids with torsion class groups. Several related examples are exhibited throughout the paper, in particular, algebraic number fields for which the generalized Riemann zeta function specializes to the Dedekind zeta function. [ABSTRACT FROM AUTHOR]

A new algorithm for computing Hecke operators for SL n was introduced in [14]. The algorithm uses tempered perfect lattices , which are certain pairs of lattices together with a quadratic form. These generalize the perfect lattices of Voronoi [17]. The present paper is the first step in characterizing tempered perfect lattices. We obtain a complete classification in the plane, where the Hecke operators are for SL 2 (Z) and its arithmetic subgroups. The results depend on the class field theory of orders in imaginary quadratic number fields. [ABSTRACT FROM AUTHOR]

The power classes of a field are well-known for their ability to parameterize elementary p -abelian Galois extensions. These classical objects have recently been reexamined through the lens of their Galois module structure. Module decompositions have been computed in several cases, providing deep new insight into absolute Galois groups. The surprising result in each case is that there are far fewer isomorphism types of indecomposables than one would expect generically, with summands predominately free over associated quotient rings. Though non-free summands are the exception both in their form and prevalence, they play the critical role in controlling arithmetic conditions in the field which allow the rest of the decomposition to be so simple. Suppose m , n ∈ N and p is prime. In a recent paper, a surprising and elegant decomposition for p m th power classes has been computed when the underlying Galois group is a cyclic group of order p n. As with previous module decompositions, at most one non-free summand appears. Outside of a particular special case when p = 2 , the structure of this exceptional summand was determined by a vector a ∈ { − ∞ , 0 , ... , n } m and a natural number d. In this paper we give field-theoretic interpretations for a and d , showing they are related to the solvability of a family of Galois embedding problems and the cyclotomic character associated to K / F. [ABSTRACT FROM AUTHOR]

We develop a uniform method to derive Chudnovsky-Ramanujan type formulae for triangle groups based on a generalization of a method of Chudnovsky and Chudnovsky; in particular, we carry out the method systematically for non-compact arithmetic triangle groups and one non-Fuchsian covering. As a result, we derive all rational Ramanujan type series given by Chan-Cooper for levels 1-4, as well as two additional rational series of a similar form prescribed by Chan-Cooper for these levels, but not found in the paper of Chan-Cooper. These two additional series were first found by Z.-W. Sun in a slightly different form. We also derive additional rational series of a similar form, but not found in the papers of Chan-Cooper nor Z.-W. Sun. As an ingredient in the method, we give an algorithm to rigorously confirm the singular values of normalized Eisenstein series of weight 2 and give examples in higher class number, which may be of independent interest. We also prove some partial completeness results for Clausen type single summation series. [ABSTRACT FROM AUTHOR]

Suppose that O L is the ring of integers of a number field L , and suppose that f (z) = ∑ n = 1 ∞ a f (n) q n ∈ S k (Γ 0 (N) +) ∩ O L [ [ q ] ] is a normalized Hecke eigenform for Γ 0 (N) +. We say that f is non-ordinary at p if there is a prime ideal p ⊂ O L above p for which a f (p) ≡ 0 (mod p). In the authors' previous paper with Ken Ono [10] it was proved that there are infinitely many Hecke eigenforms for SL 2 (Z) such that are non-ordinary at any given finite set of primes. In this paper, we extend this result to some genus 0 subgroups of SL 2 (R) , namely, the normalizers Γ 0 (N) + of the congruence subgroups Γ 0 (N). Our result also generalizes some of Choi and Kim's result in [2]. [ABSTRACT FROM AUTHOR]

In 2007, Andrews and Paule introduced the broken k -diamond partition function Δ k (n). Many researches on the arithmetic properties for Δ k (n) have been done. In this paper, we prove that D 3 log ⁡ Δ 1 (n − 1) > 0 for n ≥ 5 and D 3 log ⁡ Δ 2 (n − 1) > 0 for n ≥ 7 , where D is the difference operator with respect to n. We also conjecture that for any k ≥ 1 and r ≥ 1 , there exists a positive integer n k (r) such that for n ≥ n k (r) , (− 1) r D r log ⁡ Δ k (n) > 0. This is analogous to the positivity of finite differences of the logarithm of the partition function, which has been proved by Chen, Wang, and Xie. Furthermore, we obtain that both { Δ 1 (n) } n ≥ 0 and { Δ 2 (n) } n ≥ 0 satisfy the higher order Turán inequalities for n ≥ 6. [ABSTRACT FROM AUTHOR]

Triple correlation sums problem concerns the non-trivial power-saving bounds for the correlation of three objects. It is conjectured that these sums are non-trivial in any fixed but arbitrarily given ranges. In this paper, the uniform non-trivial bounds for the triple correlation sums ∑ m ≥ 1 , n ≥ 1 λ π (1 , m) λ ⋆ (n) λ f (m + p n) U (m / X) V (n / H) in the level aspect are derived, where π is any G L 3 -Maaß cuspidal form, f ∈ B k ⁎ (p) , any Hecke newform of prime level p and weight k ∈ N + , and λ ⋆ (n) , n ≥ 1 , are certain coefficients of arithmetic interest. As a result, we show that sums of this type follow the 1/3- significance level. We study the strength of the result by specifying ⋆ being the cusp forms on G L 3 and G L 2 , respectively, and further obtain the more significant cancellations in these sums. [ABSTRACT FROM AUTHOR]

In this paper we study couples of finite separable extensions of the function field F q (T) which are arithmetically equivalent, i.e. such that prime ideals of F q [ T ] decompose with the same inertia degrees in the two fields, up to finitely many exceptions. In the first part of this work, we extend previous results by Cornelissen, Kontogeorgis and Van der Zalm to the case of non-geometric extensions of F q (T) , which are fields such that their field of constants may be bigger than F q. In the second part, we explicitly produce examples of non-geometric extensions of F 2 (T) which are equivalent and non-isomorphic over F 2 (T) and non-equivalent over F 4 (T) , solving a particular Inverse Galois Problem. [ABSTRACT FROM AUTHOR]

This paper is a survey on sheaves of arithmetic differential operators with congruence level on formal schemes. We present first results about these sheaves and discuss some examples. [ABSTRACT FROM AUTHOR]

A basic problem in transcendental number theory is to determine the arithmetic properties of values of special functions. Many special functions, such as Bessel functions and certain hypergeometric functions, are E -functions which are a natural generalization of the exponential function and satisfy certain linear differential equations. In this case, there exists an algorithm which determines if f (α) is transcendental or algebraic if f (z) is an E -function and α ∈ Q ‾ ⁎ is a non-zero algebraic number. In this paper, we consider the analogous question when f (z) satisfies an integral equation, in particular, a Fredholm integral equation of the first or second kind where the kernel and forcing term satisfy strong arithmetic properties. We show that in both periodic and non-periodic cases, there exists no algorithm to determine if f (0) ∈ Q is rational. Our results are an application of the undecidability of the Generalized Collatz Problem due to Conway [6]. [ABSTRACT FROM AUTHOR]

A Diophantine m -tuple with elements in the field K is a set of m non-zero (distinct) elements of K with the property that the product of any two distinct elements is one less than a square in K. Let X : (x 2 − 1) (y 2 − 1) (z 2 − 1) = k 2 , be an affine variety over K. Its K -rational points parametrize Diophantine triples over K such that the product of the elements of the triple that corresponds to the point (x , y , z , k) ∈ X (K) is equal to k. We denote by X ‾ the projective closure of X and for a fixed k by X k a variety defined by the same equation as X. In this paper, we try to understand what can the geometry of varieties X k , X and X ‾ tell us about the arithmetic of Diophantine triples. First, we prove that the variety X ‾ is birational to P 3 which leads us to a new rational parametrization of the set of Diophantine triples. Next, specializing to finite fields, we find a correspondence between a K3 surface X k for a given k ∈ F p × in the prime field F p of odd characteristic and an abelian surface which is a product of two elliptic curves E k × E k where E k : y 2 = x (k 2 (1 + k 2) 3 + 2 (1 + k 2) 2 x + x 2). We derive an explicit formula for N (p , k) , the number of Diophantine triples over F p with the product of elements equal to k. Moreover, we show that the variety X ‾ admits a fibration by rational elliptic surfaces and from it we derive the formula for the number of points on X ‾ over an arbitrary finite field F q. Using it we reprove the formula for the number of Diophantine triples over F q from [DK21]. Curiously, from the interplay of the two (K3 and rational) fibrations of X ‾ , we derive the formula for the second moment of the elliptic surface E k (and thus confirming Steven J. Miller's Bias conjecture in this particular case) which we describe in terms of Fourier coefficients of a rational newform generating S 4 (Γ 0 (8)). Finally, in the Appendix, Luka Lasić defines circular Diophantine m -tuples, and describes the parametrization of these sets. For m = 3 this method provides an elegant parametrization of Diophantine triples. [ABSTRACT FROM AUTHOR]

In 1999, Frank Schmidt noted that the number of partitions of integers with distinct parts in which the first, third, fifth, etc., summands add to n is equal to p (n) , the number of partitions of n. The object of this paper is to provide a context for this result which leads directly to many other theorems of this nature and which can be viewed as a continuation of our work on elongated partition diamonds. Again generating functions are infinite products built by the Dedekind eta function which, in turn, lead to interesting arithmetic theorems and conjectures for the related partition functions. [ABSTRACT FROM AUTHOR]

Text Let K be a complete, algebraically closed, non-Archimedean valued field, and let ϕ ∈ K ( z ) with deg ⁡ ( ϕ ) ≥ 2 . In this paper we consider the family of functions ord Res ϕ n ( x ) , which measure the resultant of ϕ n at points x in P K 1 , the Berkovich projective line, and show that they converge locally uniformly to the diagonal values of the Arakelov–Green's function g μ ϕ ( x , x ) attached to the canonical measure of ϕ . Following this, we are able to prove an equidistribution result for Rumely's crucial measures ν ϕ n , each of which is a probability measure supported at finitely many points whose weights are determined by dynamical properties of ϕ . Video For a video summary of this paper, please visit https://youtu.be/YCCZD1iwe00 . [ABSTRACT FROM AUTHOR]

Text Two infinite sequences A and B of nonnegative integers are called additive complements , if their sum contains all sufficiently large integers. We also say that B is an additive complement of A if A and B are additive complements. In this paper, we consider a problem of Ben Green on additive complements of the squares: S = { 1 2 , 2 2 , … } . The following result is proved: if B = { b n } n = 1 ∞ with b n ≥ π 2 16 n 2 − 0.57 n 1 2 log ⁡ n − β n 1 2 for all positive integers n and any given constant β , then B is not an additive complement of S . In particular, B = { ⌊ π 2 16 n 2 ⌋ | n = 1 , 2 , … } is not an additive complement of S . Video For a video summary of this paper, please visit https://youtu.be/cVXWCP4Igp8 . [ABSTRACT FROM AUTHOR]

In this paper, we deal with the étale cohomology of a proper regular arithmetic scheme X with Z p (r) and Q p (r) -coefficients, where the coefficients are complexes of étale sheaves that the author introduced in [SH]. We will prove that the étale cohomology of X with Q p (r) -coefficients agrees with the Selmer group of Bloch-Kato for any r ≧ dim (X). Using this fundamental result, we further discuss an approach to the study of zeta values (or residue) at s = r , via the étale cohomology with Z p (r) -coefficients, relating Tamagawa number conjecture of Bloch-Kato with a zeta value formula. As a consequence, we will obtain an unconditional example of an arithmetic surface for which the residue of its zeta function at s = 2 is computed modulo rational numbers prime to p , for infinitely many p 's. [ABSTRACT FROM AUTHOR]

In this paper, we study arithmetic properties of weighted Catalan numbers. Previously, Postnikov and Sagan found conditions under which the 2-adic valuations of the weighted Catalan numbers are equal to the 2-adic valuations of the Catalan numbers. We obtain the same result under weaker conditions by considering a map from a class of functions to 2-adic integers. These methods are also extended to q -weighted Catalan numbers, strengthening a previous result by Konvalinka. Finally, we prove some results on the periodicity of weighted Catalan numbers modulo an integer and apply them to the specific case of the number of combinatorial types of Morse links. Many open questions are mentioned. [ABSTRACT FROM AUTHOR]

We construct, for specific imaginary quadratic number fields with class number 1, an arithmetic site of Connes-Consani type. The main difficulty here is that their constructions and part of their results strongly rely on the natural order existing on real numbers which is compatible with basic arithmetic operations. We first define what we call an arithmetic site for such number fields, we then calculate the points of those arithmetic sites and we express them in terms of the adèle class space considered by Connes to give a spectral interpretation of zeroes of Hecke L functions of number fields. We get that the points of our arithmetic site are related not only to the zeroes of the Dedekind zeta function of the number field considered but also to other Hecke L functions involving non-trivial characters at the archimedean place. We then study the relation between the spectrum of the ring of integers of the number field and the arithmetic site. An appendix by Alain Connes shows that the compatibility with the Frobenius yields the correct notion of morphisms of semirings in this context. For a video summary of this paper, please visit https://www.youtube.com/watch?v=lWF9gbfvcYY&feature=youtu.be. [ABSTRACT FROM AUTHOR]

We study the topos of sets equipped with an action of the monoid of regular 2 × 2 matrices over the integers. In particular, we show that the topos-theoretic points are given by the double quotient GL 2 (Z ˆ) \ M 2 (A f) / GL 2 (Q) , so they classify the groups Z 2 ⊆ A ⊆ Q 2 up to isomorphism. We determine the topos automorphisms and then point out the relation with Conway's big picture and the work of Connes and Consani on the Arithmetic Site. As an application to number theory, we show that classifying extensions of Q by Z up to isomorphism relates to Goormaghtigh conjecture. For a video summary of this paper, please visit https://youtu.be/5cA1MOG26ng. [ABSTRACT FROM AUTHOR]

In this paper, we study twisted arithmetic divisors on the modular curve X 0 (N) with N square-free. For each pair (Δ , r) , where Δ ≡ r 2 mod 4 N and Δ is a fundamental discriminant, we construct a twisted arithmetic theta function ϕ ˆ Δ , r (τ) which is a generating function of arithmetic twisted Heegner divisors. We prove that the arithmetic pairing 〈 ϕ ˆ Δ , r (τ) , ω ˆ N 〉 is equal to the special value, rather than the derivative, of some Eisenstein series, thanks to some cancellation, where ω ˆ N is a normalized metric Hodge line bundle. We also prove the modularity of ϕ ˆ Δ , r (τ). [ABSTRACT FROM AUTHOR]

Abstract: This paper is devoted to (discrete) p-adic dynamical systems, an important domain of algebraic and arithmetic dynamics. We consider the following open problem from theory of p-adic dynamical systems. Given continuous function . Let us represent it via special convergent series, namely van der Put series. How can one specify whether this function is measure-preserving or not for an arbitrary p? In this paper, for any prime p, we present a complete description of all compatible measure-preserving functions in the additive form representation. In addition we prove the criterion in terms of coefficients with respect to the van der Put basis determining whether a compatible function preserves the Haar measure. [Copyright &y& Elsevier]

Abstract: In the present paper we investigate the sum-of-digits function for canonical number systems. We are interested in its re-partition in arithmetic progressions and similar results for sum sets. The original problem goes back to Gelfond, who proved the independence of the distribution of the digits and their sum of digits. The present paper extends results by Thuswaldner and Mauduit and Sárközy to this kind of numeration systems. [Copyright &y& Elsevier]

Abstract: Text: In this paper we investigate higher order dimensional Dedekind–Rademacher sums given by the expression where is a positive integer, are positive integers prime to and are non-negative integers. We study arithmetical properties of these sums. For instance, we prove that these sums are rational numbers and we explicit good bounds for their denominators. A reciprocity law is given generalizing a theorem of Rademacher for the classical Dedekind sums and a theorem of Zagier for higher dimensional Dedekind–Rademacher sums. On the other hand, our reciprocity results can be viewed as complements to the Beck reciprocity theorem. Taking , we derive the reciprocity and rationality theorems of Zagier. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=J1_5H28fgAg. [Copyright &y& Elsevier]

Abstract: Text: This paper proposes new explicit formulas for the doubling and addition steps in Miller''s algorithm to compute the Tate pairing on elliptic curves in Weierstrass and in Edwards form. For Edwards curves the formulas come from a new way of seeing the arithmetic. We state the first geometric interpretation of the group law on Edwards curves by presenting the functions which arise in addition and doubling. The Tate pairing on Edwards curves can be computed by using these functions in Miller''s algorithm. Computing the sum of two points or the double of a point and the coefficients of the corresponding functions is faster with our formulas than with all previously proposed formulas for pairings on Edwards curves. They are even competitive with all published formulas for pairing computation on Weierstrass curves. We also improve the formulas for Tate pairing computation on Weierstrass curves in Jacobian coordinates. Finally, we present several examples of pairing-friendly Edwards curves. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=nideQo-K9ME/. [Copyright &y& Elsevier]

Abstract: In this paper, by using an analogue of theorems of Iwasawa (Kenkichi Iwasawa Collected Papers, vol. 2, Springer, Berlin, 2001, pp. 862–870) we give a sufficient condition for Leopoldt''s conjecture (J. Reine Angew. Math. 209 (1962) 54) on the non-vanishing of the p-adic regulator of an algebraic number field. Using this sufficient condition we are able to prove Leopoldt''s conjecture for several non-Galois extensions over the rational number field . [Copyright &y& Elsevier]

In a recent paper, B. Kim and P.-C. Toh conjectured a sign pattern of a certain arithmetic function on the crank function of cubic partition pairs. We confirm their conjecture by obtaining its asymptotic formula. [ABSTRACT FROM AUTHOR]

Define D n ( x ) = ∑ k = 0 n ( n k ) 2 x k ( x + 1 ) n − k for n = 0 , 1 , 2 , … and s n ( x ) = ∑ k = 1 n 1 n ( n k ) ( n k − 1 ) x k − 1 ( x + 1 ) n − k for n = 1 , 2 , 3 , … . Then D n ( 1 ) is the n -th central Delannoy number D n , and s n ( 1 ) is the n -th little Schröder number s n . In this paper we obtain some surprising arithmetic properties of D n ( x ) and s n ( x ) . We show that 1 n ∑ k = 0 n − 1 D k ( x ) s k + 1 ( x ) ∈ Z [ x ( x + 1 ) ] for all n = 1 , 2 , 3 , … . Moreover, for any odd prime p and p -adic integer x ≢ 0 , − 1 ( mod p ) , we establish the supercongruence ∑ k = 0 p − 1 D k ( x ) s k + 1 ( x ) ≡ 0 ( mod p 2 ) . As an application we confirm Conjecture 5.5 in [S14a] , in particular we prove that 1 n ∑ k = 0 n − 1 T k M k ( − 3 ) n − 1 − k ∈ Z for all n = 1 , 2 , 3 , … , where T k is the k -th central trinomial coefficient and M k is the k -th Motzkin number. [ABSTRACT FROM AUTHOR]

In this paper, we iterate the algebraic étale-Brauer set for any nice variety X over a number field k with π 1 ét ( X ‾ ) finite and we show that the iterated set coincides with the original algebraic étale-Brauer set. This provides some evidence towards the conjectures by Colliot-Thélène on the arithmetic of rational points on nice geometrically rationally connected varieties over k and by Skorobogatov on the arithmetic of rational points on K3 surfaces over k ; moreover, it gives a partial answer to an “algebraic” analogue of a question by Poonen about iterating the descent set. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the expected number of points in F q 2 of multiplicity m , for 0 ≤ m ≤ q + 1 , with respect to a randomly chosen arrangement of q + 1 lines with different slopes, is ( 1 / ( m ! e ) ) q 2 + O ( q ) , as q → ∞ . We further state that the distance between the number of such points in a randomly chosen arrangement and ( 1 / ( m ! e ) ) q 2 is lower than q ln ⁡ q with probability close to 1 for large q . [ABSTRACT FROM AUTHOR]

Text We develop geometric methods to study the generating weights of free modules of vector-valued modular forms of half-integral weight, taking values in a complex representation of the metaplectic group. We then compute the generating weights for modular forms taking values in the Weil representation attached to cyclic quadratic modules of order 2 p r , where p ≥ 5 is a prime. We also show that the generating weights approach a simple limiting distribution as p grows, or as r grows and p remains fixed. Video For a video summary of this paper, please visit https://youtu.be/QNbPSXXKot4 . [ABSTRACT FROM AUTHOR]

In this paper, we study the polynomial Gauss sums over finite fields, and present an analogue of Davenport–Hasse's theorem for the polynomial Gauss sums, which is a generalization of the previous result obtained by Hayes. [ABSTRACT FROM AUTHOR]

We investigate level p Eisenstein congruences for GSp 4 , generalisations of level 1 congruences predicted by Harder. By studying the associated Galois and automorphic representations we see conditions that guarantee the existence of a paramodular form satisfying the congruence. This provides theoretical justification for computational evidence found in the author's previous paper. [ABSTRACT FROM AUTHOR]

In this paper we focus on finding all the factorials expressible as a product of a fixed number of 2 k -nacci numbers with k ≥ 2 . We derive the 2-adic valuation of the 2 k -nacci sequence and use it to establish bounds on the solutions of the initial equation. In addition, we specify a more general family of sequences, for which we can perform a similar procedure. We also investigate a possible connection of these results with p -regular sequences. [ABSTRACT FROM AUTHOR]

In this paper we present an asymptotic expansion for a family of multiple integrals connected with relatives of the Dickman function. The coefficients of this expansion have a similar arithmetic structure as those appearing in Soundararajan's work on an analogous expansion for the Dickman function. [ABSTRACT FROM AUTHOR]

In this paper we introduce the notion of global equivariant hom-Lie algebra. This is a Lie algebra-like structure associated with twisted derivations. We prove several results on the structure of modules of twisted derivations and how they form global equivariant hom-Lie algebras. Particular emphasis is put on examples and results in arithmetic geometry. [ABSTRACT FROM AUTHOR]

The theory of differential equations has an arithmetic analogue in which derivatives of functions are replaced by Fermat quotients of numbers. Many classical differential equations (Riccati, Weierstrass, Painlevé, etc.) were previously shown to possess arithmetic analogues. The paper introduces and studies an arithmetic analogue of the Euler differential equations for the rigid body. [ABSTRACT FROM AUTHOR]

Let B s , t ( n ) denote the number of ( s , t ) -regular bipartitions. Recently, Dou discovered an infinite family of congruences modulo 11 for B 3 , 11 ( n ) . She also presented several conjectures on B s , t ( n ) . In this paper, utilizing an theta function identity appeared in Berndt's book, we confirm three conjectures on B 3 , 7 ( n ) given by Dou. Moreover, we prove several infinite families of congruences modulo 3 and 5 for B 3 , s ( n ) and B 5 , s ( n ) . In addition, we prove many infinite families of congruences modulo 7 for B 3 , 7 ( n ) by employing an identity of Newman. [ABSTRACT FROM AUTHOR]

Zagier defined lifts of weakly holomorphic modular functions to weakly holomorphic modular forms of weight 3/2. Duke and Jenkins extended Zagier-lifts for weakly holomorphic modular forms of negative-integral weights. In this paper, we show that half-integral weight harmonic weak Maass forms can be interpreted as Zagier-lifts of negative-integral weight harmonic weak Maass forms. [ABSTRACT FROM AUTHOR]

In this paper we study the structure of the algebra of Drinfeld modular forms for the principal congruence subgroup Γ ( T ) of the full modular group GL ( 2 , F q [ T ] ) . To this end and based on work of Cornelissen, we introduce a new class of Eisenstein series and describe their fundamental properties. These so-called modified Eisenstein series allow for a simplification of prior results and are an indispensable tool for future work on Drinfeld modular forms of level T . [ABSTRACT FROM AUTHOR]

The sequence of derangements is given by the formula D 0 = 1 , D n = n D n − 1 + ( − 1 ) n , n > 0 . It is a classical object appearing in combinatorics and number theory. In this paper we consider such arithmetic properties of the sequence of derangements as: periodicity modulo d , where d ∈ N + , p -adic valuations and prime divisors. Next, we use them to establish arithmetic properties of the sequences of even and odd derangements. [ABSTRACT FROM AUTHOR]

In 1935 Carlitz introduced Bernoulli–Carlitz numbers as analogues of Bernoulli numbers for the rational function field F r ( T ) . In this paper, we introduce Cauchy–Carlitz numbers as analogues of Cauchy numbers. By using Stirling–Carlitz numbers, we give their arithmetical and combinatorial properties and relations with Bernoulli–Carlitz numbers for F r ( T ) . Several new identities are also obtained by using Hasse–Teichmüller derivatives. [ABSTRACT FROM AUTHOR]

A new class of partitions, partitions with designated summands, was introduced by Andrews, Lewis and Lovejoy. Let PD ( n ) denote the number of partitions of n with designated summands. Andrews, Lewis and Lovejoy established many congruences modulo 3 and powers of 2 for PD ( n ) by using the theory of modular forms. In this paper, we prove several infinite families of congruences modulo 9 and 27 for PD ( n ) by employing the generating functions of PD ( 3 n ) and PD ( 3 n + 1 ) which were discovered by Chen, Ji, Jin and Shen. For example, we prove that for n ≥ 0 and k ≥ 1 , PD ( 2 18 k − 1 ( 12 n + 1 ) ) ≡ 0 ( mod 27 ) . Furthermore, using some results due to Newman, we find some strange congruences modulo 27 for PD ( n ) . For example, we prove that for k ≥ 0 , PD ( 13 9 k ( 75 p + 2 ) ) ≡ 0 ( mod 27 ) and PD ( 2 × 13 9 k + 8 ) ≡ 0 ( mod 27 ) , where p is a prime and p ≡ 1 ( mod 12 ) . [ABSTRACT FROM AUTHOR]

Tachiya investigated a class of infinite products of rational functions arithmetically and established that their values at certain algebraic points are algebraic numbers if and only if the infinite products are rational functions. In this paper we prove further arithmetical results for the values of these infinite products both qualitatively and quantitatively, which can be carried out by studying these infinite products as formal power series carefully. [ABSTRACT FROM AUTHOR]

Let f ( n ) be a multiplicative function satisfying | f ( n ) | ≤ 1 , q ( ≤ N 2 ) be a prime number and a be an integer with ( a , q ) = 1 , χ be a non-principal Dirichlet character modulo q . In this paper, we shall prove that ∑ n ≤ N f ( n ) χ ( n + a ) ≪ N q 1 4 log ⁡ log ⁡ ( 6 N ) + q 1 4 N 1 2 log ⁡ ( 6 N ) + N log ⁡ log ⁡ ( 6 N ) and that ∑ n ≤ N f ( n ) χ ( n + a 1 ) ⋯ χ ( n + a t ) ≪ N q 1 4 log ⁡ log ⁡ ( 6 N ) + q 1 4 N 1 2 log ⁡ ( 6 N ) + N log ⁡ log ⁡ ( 6 N ) , where t ≥ 2 , a 1 , … , a t are pairwise distinct integers modulo q . [ABSTRACT FROM AUTHOR]

Text We prove that the arboreal Galois representation attached to a large class of quadratic polynomials defined over a field of rational functions k ( t ) in characteristic zero has finite index in the full automorphism group of the associated preimage tree. Moreover, we show that in most cases, the index is bounded independently of the polynomial. Video For a video summary of this paper, please visit http://youtu.be/loO7qn4ncWs . [ABSTRACT FROM AUTHOR]

A breakthrough in developing a theory of hypercomplex analytic modular forms over Clifford algebras has been the proof of the existence of non-trivial cusp forms for important discrete arithmetic subgroups of the Ahlfors–Vahlen group. Hypercomplex analytic modular forms in turn also include Maaß forms associated to particular eigenvalues as special cases. In this paper we establish a Selberg trace formula for this new class of automorphic forms. In particular, we show that the dimension of the space of hypercomplex-analytic cusp forms is finite. Finally, we describe the space of Eisenstein series and give a dimension formula for the complete space of k -holomorphic Cliffordian modular forms. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we obtain a new estimate of an irrationality measure of the number . We have with an “arithmetical method”. The previous results were by G. Rhin in 1987 and by V.H. Salikhov in 2007. [Copyright &y& Elsevier]

Abstract: Let f be a Drinfeld modular form for . From such a form, one can obtain two forms for the full modular group : by taking the trace or the norm from to . In this paper we show some connections between the arithmetic modulo of the coefficients of the u-series expansion of f and those of a form closely related to its trace, and of the coefficients of f and those of its norm. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the arithmetic properties of bipartitions with 3-core. Let denote the number of bipartitions with 3-core of n. We will prove one infinite family of congruences modulo 5 for . We also establish one surprising congruence modulo 14 for . Finally, we prove that, if denotes the number of representations of a nonnegative integer n in the form with , then . [Copyright &y& Elsevier]

Abstract: In this paper we study additive functions on arithmetic progressions with large moduli. We are able to improve some former results given by Elliott. [Copyright &y& Elsevier]

Abstract: We construct a new system of three terms arithmetic geometric mean (we say AGM). Our system is an extension of the classical Gauss AGM. It is induced from an isogeny formula of the modular forms defined on the complex 2-dimensional hyperball with respect to a principal congruence subgroup of the Picard modular group . Our AGM function is expressed via the Appell hypergeometric function. These results are the extension of the properties discovered by Gauss himself. Our result is based on the work of K. Matsumoto in 1989 [K. Matsumoto, On modular functions in 2 variables attached to a family of hyperelliptic curves of genus 3, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 16 (4) (1989) 557–578]. And our present paper is a continuation of the previous work [K. Koike, H. Shiga, Isogeny formulas for the Picard modular form and a three terms arithmetic geometric mean, J. Number Theory 124 (2007) 123–141. ]. [Copyright &y& Elsevier]