CONTENTS -- Preface -- On Jacobi Forms with Levels Hiroki Aoki -- 1. Definitions and basic properties -- 1.1. Elliptic modular forms -- 1.2. Siegel modular forms of degree 2 -- 1.3. Jacobi forms -- 1.4. Jacobi forms with levels -- 1.5. Theta decomposition -- 1.6. Structure theorem for weak Jacobi forms -- 2. Applications -- 2.1. Structure of Siegel modular forms -- 2.2. Formal series of Jacobi forms -- 2.3. Index-Level change of Jacobi forms -- Acknowledgements -- References -- Additive Representation in Thin Sequences, VIII: Diophantine Inequalities in Review Jorg Brudern, Koichi Kawada and Trevor D. Wooley -- 1. Theme and results -- 1.1. Diophantine inequalities -- 1.2. Additive cubic forms -- 1.3. Linear forms in primes -- 1.4. Further applications -- 1.5. A related diophantine inequality -- 2. The Fourier transform method -- 2.1. Some classical integrals -- 2.2. Counting solutions of diophantine inequalities -- 2.3. Weighted counting -- 2.4. The central interval -- 2.5. The interference principle -- 3. Classical mean square methods -- 3.1. Plancherel's identity -- 3.2. Some mean values -- 3.3. The amplification technique -- 3.4. Linear forms in primes -- 3.5. Bessel's inequality -- 4. Semi-classical averaging -- 4.1. Another mean square approach -- 4.2. Exponential sums over test sequences -- 4.3. Potential applications -- 5. Fourier analysis of exceptional sets -- 5.1. An illustrative example -- 5.2. A quadratic average -- 5.3. Some brief heckling -- 5.4. An inequality involving quadratic polynomials -- 5.5. An application of Vinogradov's method -- 5.6. Linear forms in primes, yet again -- 6. Outstanding arts -- 6.1. Smooth cubic Weyl sums -- 6.2. Senary cubic forms -- 6.3. Two technical estimates -- 6.4. The lower bound variant -- 6.5. An auxiliary inequality -- 6.6. Additive forms of large degree -- 6.7. Proof of Theorem 1.8.

6.8. Proof of Theorem 1.9 -- 7. An appendix on inhomogeneous polynomials -- 7.1. The counting integral -- 7.2. The central interval -- 7.3. The complementary compositum -- Acknowledgements -- References -- Annexe to the Gallery: An Addendum to "Additive Representation in Thin Sequences, VIII: Diophantine Inequalities in Review" Jorg Brudern, Koichi Kawada and Trevor D. Wooley -- 11. Downloading updates -- References -- A Note on the Distribution of Primes in Arithmetic Progressions Zhen Cui and Boqing Xue -- 1. Introduction -- 2. Notations and Some Lemmas -- 3. Proofs -- 4. Further Remark -- Acknowledgements -- References -- Matrices of Finite Abelian Groups, Finite Fourier Transform and Codes Shigeru Kanemitsu and Michel Waldschmidt -- 1. The matrix of a finite abelian group -- 1.1. Matrix of a finite group -- 1.2. Matrix of a finite abelian group -- 1.3. Matrix of a cyclic group -- 1.4. The group ring of a cyclic group and the algebra of circulants -- 1.5. The group ring F[G] of a finite abelian group G -- 2. Finite Fourier Transform associated with a finite abelian group -- 2.1. Generalized Finite Fourier Transform -- 2.2. Case of a cyclic group: Finite Fourier Transform -- 3. Hamming weight and Generalized Finite Fourier Transform -- 4. The matrix of a finite group -- 4.1. An example: 3 -- 4.2. The general case -- 4.3. Frobenius -- Acknowledgements -- References -- A Remark on a Result of Eichler Yoshiyuki Kitaoka -- 1. Introduction -- 2. Quadratic forms -- 3. Even positive definite quaternary quadratic form with prime discriminant -- 4. A modification of Eichler's sum -- References -- On Weyl Sums over Primes in Short Intervals Angel V. Kumchev -- 1. Introduction -- 2. Auxiliary results -- 3. Multilinear Weyl sums -- 4. Proof of Theorem 1.2 -- Acknowledgements -- References.

On Congruences for Certain Binomial Coefficients of E. Lehmer's Type Takako Kuzumaki and Jerzy Urbanowicz -- 1. Introductions -- 2. Theorems -- References -- Sign Changes of the Coefficients of Automorphic L-Functions Yuk-Kam Lau, Jianya Liu and Jie Wu -- 1. Introduction -- 2. The least quadratic non-residue -- 2.1. Estimates for character sums and individual bounds for nx -- 2.2. Linnik's large sieve and almost bound for nx -- 3. Automorphic L-functions of GLm(AK) -- 3.1. The analytic strong multiplicity one theorem for GLm(AK) -- 3.2. Sign changes of the coefficients of L-functions for GLm(AQ) -- 4. Classic modular forms -- 4.1. The first negative Hecke eigenvalue -- 4.2. Statistic study of the first sign-change -- 4.3. Recognition of newforms by signs of Hecke eigenvalues -- 4.4. Recognition of newforms by values of Hecke eigenvalues -- 4.5. Matching signs of Hecke eigenvalues of two newforms -- 4.6. The number of Hecke eigenvalues of same signs -- 5. Symmetric square L-functions -- 5.1. The first negative coefficient of symmetric square L-functions -- 5.2. Positive and negative values over a positive density of primes -- 5.3. ±-results over integers -- 5.4. The number of coefficients of L(s, sym2f) of same signs -- 6. Half-integral weight case -- 6.1. Theta functions -- 6.2. Half-integral weight forms and Shimura's theory -- 6.3. Sign-change problem of half-integral weight cusp forms -- Acknowledgements -- References -- On Fourier Coefficients of Automorphic Forms Guangshi Lu -- 1. Integral power sums of cusp forms -- 2. Symmetric power L-functions and their Rankin-Selberg L-functions -- 3. Main ingredients in the proof -- 4. Further refinement -- 5. Further Discussion -- Acknowledgements -- References -- The Twists of Hessian Elliptic Curves over Splitting Fields of Cubic Polynomials and the Related Elliptic 3-Folds Katsuya Miyake.

1. The twists of Hessian elliptic curves -- 2. The proof of Theorem 1.2 -- 3. The twists over cyclic cubic fields -- 4. The elliptic 3-folds -- 4.1. The general case -- 4.2. The case of the twists over relative cyclic cubic extensions -- References -- Asymptotic Voronoi's Summation Formulas and Their Duality for SL3(Z) Xiumin Ren and Yangbo Ye -- 1. Introduction -- 2. Proof of Theorem 1 -- 3. Proof of Theorem 2 -- 4. Applications and duality -- Acknowledgements -- References -- Jerzy Urbanowicz's Work in Pure Mathematics Andrzej Schinzel -- 1. Congruences -- 2. Identities -- 3. Diophantine equations -- 4. Algebraic K-theory -- Acknowledgements -- References -- Conjectures Involving Arithmetical Sequences Zhi-Wei Sun -- 1. Introduction -- 2. Conjectures on number-theoretic sequences -- 2.1. Conjectures on sequences involving primes -- 2.2. Conjectures on other number-theoretic sequences -- 3. Conjectures on combinatorial sequences -- Acknowledgements -- References -- Index.