Nonlinear exponential twists of the Liouville function
Let λ(n) be the Liouville function. We find a nontrivial upper bound for the sum The main tool we use is Vaughan’s identity for λ(n).
Nonvanishing of Fourier coefficients of newforms in progressions
Numbers with a large prime factor
O zakonomernostyakh upravlyayushchikh khaotichnostyu povedeniya funkcii i ee proizvodnyh
On a fundamental result in van der Corput's method of estimating exponential sums
On a multiplicative hybrid problem
On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions
The paper deals with lower bounds for the remainder term in asymptotics for a certain class of arithmetic functions. Typically, these are generated by a Dirichlet series of the form ζ 2(s)ζ(2s−1)ζ M(2s)H(s), where M is an arbitrary integer and H(s) has an Euler product which converges absolutely for R s > σ0, with some fixed σ0 < 1/2.
On an Effective Order Estimate of the Riemann Zeta Function in the Critical Strip.
On an estimate by Weyl and Vinogradov.
On an exponential sum.
On certain arithmetic functions involving the greatest common divisor
The paper deals with asymptotics for a class of arithmetic functions which describe the value distribution of the greatest-common-divisor function. Typically, they are generated by a Dirichlet series whose analytic behavior is determined by the factor ζ2(s)ζ(2s − 1). Furthermore, multivariate generalizations are considered.
On Certain Divisor Functions in Short Intervals
On exponential sums over smooth numbers.
On exponential sums with multiplicative coefficients, II
On Hecke eigenvalues at primes of the form [g(n)]
We study the average of the Fourier coefficients of a holomorphic cusp form for the full modular group at primes of the form [g(n)].
On -free numbers over Beatty sequences
We consider -free numbers over Beatty sequences. New results are given. In particular, for a fixed irrational number of finite type and any constant , we can show that where is the set of positive -free integers and the implied constant depends only on ...
On lattice points in large convex bodies
On Linnik's approximation to Goldbach's problem, I
On Linnik's theorem on Goldbach numbers in short intervals and related problems
Linnik proved, assuming the Riemann Hypothesis, that for any , the interval contains a number which is the sum of two primes, provided that is sufficiently large. This has subsequently been improved to the same assertion being valid for the smaller gap , the added new ingredient being Selberg’s estimate for the mean-square of primes in short intervals. Here we give another proof of this sharper result which avoids the use of Selberg’s estimate and is therefore more in the spirit of Linnik’s...
On polynomials in primes and J. Bourgain's circle method approach to ergodic theorems II