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We establish a near dichotomy between randomness and structure for the point counts of arbitrary projective cubic threefolds over finite fields. Certain "special" subvarieties, not unlike those in the Manin conjectures, dominate. We also…
We study the behavior of partially twisted multiple zeta-functions. We give new closed and explicit formulas for special values at non-positive integer points of such zeta-functions. Our method is based on a result of M. de Crisenoy on the…
We give asymptotic formulas for some average values of the Euler function on shifted smooth numbers. The result is based on various estimates on the distribution of smooth numbers in arithmetic progressions which are due to A. Granville and…
We investigate the analogues of certain classical estimates of Littlewood for the Riemann zeta-function in the context of quadratic Dirichlet $L$-functions over function fields. In some situations, we are actually able to establish finer…
Using numerical, theoretical and general methods, we construct evaluation formulas for the Jacobi $\theta$ functions. Some of our results are conjectures, but are verified numerically.
In this paper, we establish a mean value result of arithmetic functions over shorter intervals with the Selberg-Delange method using the Hooley-Huxley contour.
We study Whitney-type estimates for approximation of convex functions in the uniform norm on various convex multivariate domains while paying a particular attention to the dependence of the involved constants on the dimension and the…
In this short note we study uniform approximations to the normal distributions by Jacobi theta functions. We shall show that scaled theta functions approach to a normal distribution exponentially fast.
We show that the distributions occurring in the geometric and spectral side of the twisted Arthur-Selberg trace formula extend to non-compactly supported test functions. The geometric assertion is modulo a hypothesis on root systems proven…
We prove a Burgess-like subconvex bound for twisted L-functions of a fixed irreducible cuspidal automorphic representation of GL(2) over a totally real number field. The proof is based on a spectral decomposition of shifted convolution sums…
In this paper, we give an explicit formula of the Shintani double zeta functions with any ramification in the most general setting of adeles over an arbitrary number field. Three applications of the explicit formula are given. First, we…
The coefficients of twisted Alexander polynomials of a knot induce regular functions of the $SL_2(\mathbb{C})$-character variety. We prove that the function of the highest degree has a finite value at an ideal point which gives a minimal…
Partial zeta functions of algebraic varieties over finite fields generalize the classical zeta function by allowing each variable to be defined over a possibly different extension field of a fixed finite field. Due to this extra variation…
We prove the Sato--Tate distribution of Kloosterman sums over function fields with explicit error terms, when the places vary in arithmetic progressions or short intervals. A joint Sato--Tate distribution of two ``different" exponential…
We introduce some generalizations of the Euler-Kronecker constant of a number field and study their arithmetic nature.
A wide range of natural and social phenomena result in observables whose distributions can be well approximated by a power-law decay. The well-known Hill estimator of the tail exponent provides results which are in many respects superior to…
Under certain general conditions, an explicit formula to compute the greatest delta-epsilon function of a continuous function is given. From this formula, a new way to analyze the uniform continuity of a continuous function is given.…
We establish sharp upper bounds on shifted moments of quadratic Dirichlet $L$-functions over function fields. As an application, we prove some bounds for moments of quadratic Dirichlet character sums over function fields.
We continue the investigation of the distribution of $\ell^{\infty}$-Selmer groups in degree $\ell$ twist families of Galois modules over number fields begun in the previous paper. Building off the work on higher Selmer groups in that part,…
We consider the alternating zeta function and the alternating $L$-function of a graph $G$, and express them by using the Ihara zeta function of $G$. Next, we define a generalized alternating zeta function of a graph, and express the…