Using the cohomology theory of Dwork, as developed by Adolphson and Sperber, we exhibit a deterministic algorithm to compute the zeta function of a nondegenerate hypersurface defined over a finite field. This algorithm is particularly well suited to work with polynomials in small characteristic that have few monomials (relative to their dimension). Our method covers toric, affine, and projective hypersurfaces, and also can be used to compute the L-function of an exponential sum.

In this paper we investigate the zeros of the Estermann zeta function $E(s; k/ \ell , \alpha )= { \mathop{\sum }\nolimits}_{n= 1}^{\infty } {\sigma }_{\alpha } (n) \exp (2\pi ink/ \ell ){n}^{- s} $ as a function of a complex variable $s$, where $k$ and $\ell $ are coprime integers and ${\sigma }_{\alpha } (n)= {\mathop{\sum }\nolimits}_{d\vert n} {d}^{\alpha } $ is the generalized divisor function with a fixed complex number $\alpha $. In particular, we study the question on how the zeros of $E(s; k/ \ell , \alpha )$ depend on the parameters $k/ \ell $ and $\alpha $. It turns out that for some zeros there is a continuous dependency whereas for other zeros there is not.

We obtain lower bounds of the correct order of magnitude for the 2kth moment of the Riemann zeta function for all k≥1. Previously such lower bounds were known only for rational values of k, with the bounds depending on the height of the rational number k. Our new bounds are continuous in k, and thus extend also to the case when k is irrational. The method is a refinement of an approach of Rudnick and Soundararajan, and applies also to moments of L-functions in families.

We investigate the behaviour of the function $L_{\alpha }(x) = \sum _{n\leq x}\lambda (n)/n^{\alpha }$, where $\lambda (n)$ is the Liouville function and $\alpha $ is a real parameter. The case where $\alpha =0$ was investigated by Pólya; the case $\alpha =1$, by Turán. The question of the existence of sign changes in both of these cases is related to the Riemann hypothesis. Using both analytic and computational methods, we investigate similar problems for the more general family $L_{\alpha }(x)$, where $0\leq \alpha \leq 1$, and their relationship to the Riemann hypothesis and other properties of the zeros of the Riemann zeta function. The case where $\alpha =1/2$is of particular interest.

For an abelian surface A over a number field k, we study the limiting distribution of the normalized Euler factors of the L-function of A. This distribution is expected to correspond to taking characteristic polynomials of a uniform random matrix in some closed subgroup of USp(4); this Sato–Tate group may be obtained from the Galois action on any Tate module of A. We show that the Sato–Tate group is limited to a particular list of 55 groups up to conjugacy. We then classify A according to the Galois module structure on the ℝ-algebra generated by endomorphisms of (the Galois type), and establish a matching with the classification of Sato–Tate groups; this shows that there are at most 52 groups up to conjugacy which occur as Sato–Tate groups for suitable A and k, of which 34 can occur for k=ℚ. Finally, we present examples of Jacobians of hyperelliptic curves exhibiting each Galois type (over ℚ whenever possible), and observe numerical agreement with the expected Sato–Tate distribution by comparing moment statistics.

It is well known that Hurwitz zeta-functions with algebraically independent parameters over the field of rational numbers are universal in the sense that their shifts approximate simultaneously any collection of analytic functions. In this paper we introduce some classes of universal composite functions of a collection of Hurwitz zeta-functions.

We investigate the first and second moments of shifted convolutions of the generalized divisor function d3(n).

We prove a sharp upper bound on the L2 norm of a GL3 Maass form restricted to GL2×ℝ+.

We study various Dirichlet series of the form ∑n≥1f(πnα)/ns, where α is an irrational number and f(x) is a trigonometric function like cot(x), 1/sin(x) or 1/sin2(x). The convergence is slow and strongly depends on the Diophantine properties of α. We provide necessary and sufficient convergence conditions using the continued fraction of α. We also show that any one of our series is equal to a related series, which converges much faster, defined in term of iterations of the continued fraction operator α↦{1/α}.

In this paper we present new explicit simultaneous rational approximations which converge subexponentially to the values of the Bell polynomials at the points where m=1,2,…,a, a∈ℕ, γ is Euler’s constant and ζ is the Riemann zeta function.

In this paper, we consider certain double series analogous to Tornheim’s double series and real analytic Eisenstein series. By computing double integrals in two ways, we express the double series as a sum of products of polylogarithms. The technique generalises one given by Kanemitsu, Tanigawa and Yoshimoto. Evaluating the double series at particular points gives new evaluations for certain double series in terms of values of the Riemann zeta function and the dilogarithm which are analogues of formulas of Mordell and Goncharov.

We study the Witten multiple zeta function associated with the Lie algebra . Our main result shows that its special values at nonnegative integers are always expressible by alternating Euler sums. More precisely, every such special value of weight w at least 2 is a finite ℚ-linear combination of alternating Euler sums of weight w and depth at most 2, except when the only nonzero argument is one of the two last variables, in which case ζ(w−1) is needed.

In Gun and Ramakrishnan [‘On special values of certain Dirichlet L-functions’, Ramanujan J.15 (2008), 275–280], we gave expressions for the special values of certain Dirichlet L-function in terms of finite sums involving Jacobi symbols. In this note we extend our earlier results by giving similar expressions for two more special values of Dirichlet L-functions, namely L(−1,χm) and L(−2,χ−m′), where m,m′ are square-free integers with m≡1 mod 8 and m′≡3 mod 8 and χD is the Kronecker symbol . As a consequence, using the identities of Cohen [‘Sums involving the values at negative integers of L-functions of quadratic characters’, Math. Ann.217 (1975), 271–285], we also express the finite sums with Jacobi symbols in terms of sums involving divisor functions. Finally, we observe that the proof of Theorem 1.2 in Gun and Ramakrishnan (as above) is a direct consequence of Equation (24) in Gun, Manickam and Ramakrishnan [‘A canonical subspace of modular forms of half-integral weight’, Math. Ann.347 (2010), 899–916].

In this paper, we prove some one level density results for the low-lying zeros of families of L-functions. More specifically, the families under consideration are that of L-functions of holomorphic Hecke eigenforms of level 1 and weight k twisted with quadratic Dirichlet characters and that of cubic and quartic Dirichlet L-functions.

For two real characters ψ,ψ′ of conductor dividing 8 define where and the subscript 2 denotes the fact that the Euler factor at 2 has been removed. These double Dirichlet series can be extended to possessing a group of functional equations isomorphic to D12. The convexity bound for Z(s,w;ψ,ψ′) is |sw(s+w)|1/4+ε for ℜs=ℜw=1/2. It is proved that Moreover, the following mean square Lindelöf-type bound holds: for any Y1,Y2≥1.

Rosen gave a determinant formula for relative class numbers for cyclotomic function fields, which may be regarded as an analogue of the classical Maillet determinant. In this paper, we give a determinant formula for relative congruence zeta functions for cyclotomic function fields. Our formula may be regarded as a generalization of the determinant formula for the relative class number.

We prove a mean-value result for derivatives of L-functions at the center of the critical strip for a family of forms obtained by twisting a fixed form by quadratic characters with modulus which can be represented as sum of two squares. Such a family of forms is related to elliptic fibrations given by the equation q(t)y2=f(x) where q(t)=t2+1 and f(x) is a cubic polynomial. The aim of the paper is to establish a prototype result for such quadratic families. Though our method can be generalized to prove similar results for any positive definite quadratic form in place of sum of two squares, we refrain from doing so to keep the presentation as clear as possible.

We obtain the formula for the twisted harmonic second moment of the L-functions associated with primitive Hecke eigenforms of weight 2. A consequence of our mean-value theorem is reminiscent of recent results of Conrey and Young on the reciprocity formula for the twisted second moment of Dirichlet L-functions.

We study the asymptotical behaviour of the moduli space of morphisms of given anticanonical degree from a rational curve to a split toric variety, when the degree goes to infinity. We obtain in this case a geometric analogue of Manin’s conjecture about rational points of bounded height on varieties defined over a global field. The study is led through a generating series whose coefficients lie in a Grothendieck ring of motives, the motivic height zeta function. In order to establish convergence properties of this function, we use a notion of motivic Euler product. It relies on a construction of Denef and Loeser which associates a virtual motive to a first order logic ring formula.

We define a topological space over the p-adic numbers, in which Euler products and Dirichlet series converge. We then show how the classical Riemann zeta function has a (p-adic) Euler product structure at the negative integers. Finally, as a corollary of these results, we derive a new formula for the non-Archimedean Euler–Mascheroni constant.