Local problems with primes I.
Exponential sums and mean-value theorems connected with Ramanujan's T -function.
On the prime number theorem for the coefficients of certain modular forms
Goldbach numbers represented by polynomials.
Let N be a large positive real number. It is well known that almost all even integers in the interval [N, 2N] are Goldbach numbers, i.e. a sum of two primes. The same result also holds for short intervals of the form [N, N+H], see Mikawa [4], Perelli-Pintz [7] and Kaczorowski-Perelli-Pintz [3] for the choice of admissible values of H and the size of the exceptional set in several problems in this direction. One may ask if similar results hold for thinner sequences of integers in [N,...
Sieve methods and class-number probelms. II.
Sieve methods and class-number problems III.
On the exceptional set for the -twin primes problem
Sieve methods and class-number problems. 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...
Goldbach numbers in sparse sequences
We show that for almost all , the inequality has solutions with odd prime numbers and , provided . Moreover, we give a rather sharp bound for the exceptional set. This result provides almost-all results for Goldbach numbers in sequences rather thinner than the values taken by any polynomial.
Exponential sums and additive problems involving square-free numbers
Zeros and poles of Dirichlet series
Under certain mild analytic assumptions one obtains a lower bound, essentially of order , for the number of zeros and poles of a Dirichlet series in a disk of radius . A more precise result is also obtained under more restrictive assumptions but still applying to a large class of Dirichlet series.
Real zeros of general -functions
In questo lavoro vengono studiati gli zeri reali di una classe di serie di Dirichlet, che generalizzano le funzioni , definite in [8], Combinando le tecniche elementari di Pintz [9] con alcuni metodi analitici si ottiene l’estensione dei classici teoremi di Hecke e Siegel.
Real zeros of general -functions
In questo lavoro vengono studiati gli zeri reali di una classe di serie di Dirichlet, che generalizzano le funzioni , definite in [8], Combinando le tecniche elementari di Pintz [9] con alcuni metodi analitici si ottiene l’estensione dei classici teoremi di Hecke e Siegel.
Twists and resonance of -functions, I
We obtain the basic analytic properties, i.e. meromorphic continuation, polar structure and bounds for the order of growth, of all the nonlinear twists with exponents of the -functions of any degree in the extended Selberg class. In particular, this solves the resonance problem in all such cases.
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