An inequality for ratios of gamma functions

Let Γ(x) denote Euler's gamma function. The following inequality is proved: for y>0 and x>1 we have

     

On the product of two primitive elements of maximal subfields of a finite field

Let denote a finite field with r elements. Let q be a power of a prime, and p₁,p₂, p₃ be distinct primes. Put

     

On consecutive happy numbers

Let e?1 and b?2 be integers. For a positive integer with 0?aj<b, define

     

Exceptional congruences for the coefficients of certain eta-product newforms

Let F(z)=∑_n=1^∞a(n)qⁿ denote the unique weight 16 normalized cuspidal eigenform on . In the early 1970s, Serre and Swinnerton-Dyer conjectured that

     

Some curious congruences modulo primes

Let n be a positive odd integer and let p>n+1 be a prime. We mainly derive the following congruence:

     

Congruences involving Bernoulli and Euler numbers

Let [x] be the integral part of x. Let p>5 be a prime. In the paper we mainly determine , , and in terms of Euler and Bernoulli numbers. For example, we have

     

Openness of the Galois image for τ-modules of dimension 1

Let C be a smooth projective absolutely irreducible curve over a finite field , F its function field and A the subring of F of functions which are regular outside a fixed point ∞ of C. For every place ? of A, we denote the completion of A at ? by .In [Pi2], Pink proved the Mumford-Tate conjecture for Drinfeld modules. Let φ be a Drinfeld module of rank r defined over a finitely generated field K containing F. For every place ? of A, we denote by Γ_? the image of the representation

     

Symmetry and specializability in the continued fraction expansions of some infinite products

Let f(x)∈Z[x]. Set f₀(x)=x and, for n?1, define fn(x)=f(fn−1(x)). We describe several infinite families of polynomials for which the infinite product

     

On certain exponential sums over primes

Let f(x) be a real valued polynomial in x of degree k?4 with leading coefficient α. In this paper, we prove a non-trivial upper bound for the quantity

     

On character sums over flat numbers

Let q?2 be an integer, χ be any non-principal character mod q, and H=H(q)?q. In this paper the authors prove some estimates for character sums of the form

     

On the residue classes of integer sequences satisfying a linear three term recurrence formula

In this paper we investigate linear three-term recurrence formulae with sequences of integers (T(n))_n?0 and (U(n))_n?0, which are ultimately periodic modulo m, e.g.

     

On the number of representations of certain integers as sums of 11 or 13 squares

Let r_k(n) denote the number of representations of an integer n as a sum of k squares. We prove that for odd primes p,

     

A problem of Chowla revisited

In 1964, S. Chowla asked if there is a non-zero integer-valued function f with prime period p such that f(p)=0 and

     

The average value of Fourier coefficients of cusp forms in arithmetic progressions

Recently Blomer showed that if α(n) denote the normalized Fourier coefficients of any holomorphic cusp form f with integral weight, then

     

Diophantine equations with products of consecutive values of a quadratic polynomial

Let a, b, c, d be given nonnegative integers with a,d?1. Using Chebyshev?s inequalities for the function π(x) and some results concerning arithmetic progressions of prime numbers, we study the Diophantine equation

     

Galois closure and Lagrangian varieties

Let X be a complex projective variety and consider the morphism