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The standard twist
in the Selberg class has several interesting properties and plays a central role in the Selberg class theory. It is therefore natural to study its finer analytic properties, for example the functional equation. Here we deal with a special case, where
satisfies a functional equation with the same
-factor of the
-functions associated with the cusp forms of half-integral weight; for simplicity we present our results directly for such
-functions. We show that the standard twist
satisfies a functional equation reflecting
, whose shape is not far from a Riemann-type functional equation of degree 2 and may be regarded as a degree 2 analog of the Hurwitz–Lerch functional equation. We also deduce some results on the growth on vertical strips and on the distribution of zeros of
We prove the analog of Cramér’s short intervals theorem for primes in arithmetic progressions and prime ideals, under the relevant Riemann hypothesis. Both results are uniform in the data of the underlying structure. Our approach is based mainly on the inertia property of the counting functions of primes and prime ideals.
Introduction As an illustrative example of their celebrated circle method, Hardy and Littlewood were able to show that subject to the truth of the Generalized Riemann Hypothesis, almost all even natural numbers are the sum of two primes, the yet unproven hypothesis being removed later as a consequence of Vinogradov's work. Natural numbers which are representable as the sum of two primes are called Goldbach numbers, and it is still not known whether all, or at least all but finitely many, even positive integers ≥ 4 are of this form. The best estimate for the number of possible exceptions is due to Montgomery and Vaughan . They showed that all but O(X1–δ) even natural numbers not exceeding X are Goldbach numbers, for some small δ > 0.
More information about possible exceptions can be obtained by considering thin subsequences of the even numbers, with the aim of showing that almost all numbers in the subsequence are Goldbach numbers. In this direction, short intervals have been treated by various authors. It is now known that almost all even numbers in the interval [X, X + X11/160+ε] are Goldbach numbers (see Baker, Harman and Pintz ). Perelli  has shown that almost all even positive values of an integer polynomial satisfying some natural arithmetical conditions are Goldbach numbers.
In this paper we give further examples of sequences with this property. They arise, roughly speaking, as integer approximations to values of real-valued functions at integers points whose fractional parts are uniformly distributed modulo one. We need some notation to make this precise.
Let k ≥ 2 be an integer. Let Ek(N) be the number of natural numbers not exceeding N which are not the sum of a prime and a k-th power of a natural number. Assuming the Riemann Hypothesis for all Dirichlet L-functions it is shown that Ek(N) ≪ N1-1/25k.
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