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### Local problems with primes I.

Journal für die reine und angewandte Mathematik

### Exponential sums and mean-value theorems connected with Ramanujan's T -function.

Seminaire de Théorie des Nombres de Bordeaux

### On the prime number theorem for the coefficients of certain modular forms

Banach Center Publications

### Goldbach numbers represented by polynomials.

Revista Matemática Iberoamericana

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.

Journal für die reine und angewandte Mathematik

### Sieve methods and class-number problems III.

Journal für die reine und angewandte Mathematik

### On the exceptional set for the $2k$-twin primes problem

Compositio Mathematica

### Sieve methods and class-number problems. I.

Journal für die reine und angewandte Mathematik

### On Linnik's theorem on Goldbach numbers in short intervals and related problems

Annales de l'institut Fourier

Linnik proved, assuming the Riemann Hypothesis, that for any $ϵ>0$, the interval $\left[N,N+{log}^{3+ϵ}N\right]$ contains a number which is the sum of two primes, provided that $N$ is sufficiently large. This has subsequently been improved to the same assertion being valid for the smaller gap $C\phantom{\rule{0.166667em}{0ex}}{log}^{2}\phantom{\rule{0.166667em}{0ex}}N$, 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

Annales de l'institut Fourier

We show that for almost all $n\in \mathbf{N}$, the inequality $|{p}_{1}+{p}_{2}-\mathrm{exp}\left(\left(\mathrm{log}\phantom{\rule{0.166667em}{0ex}}n{\right)}^{\gamma }\right)|<1$ has solutions with odd prime numbers ${p}_{1}$ and ${p}_{2}$, provided $1<\gamma <\frac{3}{2}$. 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

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

### Zeros and poles of Dirichlet series

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Under certain mild analytic assumptions one obtains a lower bound, essentially of order $r$, for the number of zeros and poles of a Dirichlet series in a disk of radius $r$. 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 $L$-functions

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In questo lavoro vengono studiati gli zeri reali di una classe di serie di Dirichlet, che generalizzano le funzioni $L(s,\chi)$, 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 $L$-functions

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

In questo lavoro vengono studiati gli zeri reali di una classe di serie di Dirichlet, che generalizzano le funzioni $L(s,\chi)$, 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 $L$-functions, I

Journal of the European Mathematical Society

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 $\le 1/d$ of the $L$-functions of any degree $d\ge 1$ in the extended Selberg class. In particular, this solves the resonance problem in all such cases.

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