104 research outputs found
Applications of some exponential sums on prime powers: a survey
A survey paper on some recent results on additive problems with prime powers
Efficient computation of the Euler-Kronecker constants of prime cyclotomic fields
We introduce a new algorithm, which is faster and requires less computing
resources than the ones previously known, to compute the Euler-Kronecker
constants for the prime cyclotomic fields
, where is an odd prime and is a primitive
-root of unity. With such a new algorithm we evaluated and
, where is the Euler-Kronecker constant of
the maximal real subfield of , for some very large primes
thus obtaining two new negative values of :
and We also evaluated and for
every odd prime , thus enlarging the size of the previously known
range for and . Our method also reveals that
difference can be computed in a much
simpler way than both its summands, see Section 3.4. Moreover, as a by-product,
we also computed
for every odd prime , where are the Dirichlet
-functions, run over the non trivial Dirichlet characters mod and
is the trivial Dirichlet character mod . As another by-product of
our computations, we will also provide more data on the generalised Euler
constants in arithmetic progressions. The programs used to performed the
computations here described and the numerical results obtained are available at
the following web address:
\url{http://www.math.unipd.it/~languasc/EK-comput.html}.Comment: 25 pages, 6 tables, 4 figures. Third known example of negative values
for Ek(q) inserted. Complete set of computation of Ek(q) and Ek(q)^+ for
every prime up to 10^6; computation of max|L'/L(1,chi)| for the same primes
inserted. Two references added, typos correcte
Sums of four prime cubes in short intervals
We prove that a suitable asymptotic formula for the average number of
representations of integers , where
are prime numbers, holds in intervals shorter than the the
ones previously known.Comment: Unconditional result improved by using a Robert-Sargos estimate
(lemmas 6-7); more detailed proof of Lemma 5 inserted. Correction of a typo.
10 page
A Diophantine problem with prime variables
We study the distribution of the values of the form , where , and
are non-zero real number not all of the same sign, with irrational, and , and are prime numbers. We prove
that, when , these value approximate rather closely any
prescribed real number.Comment: submitte
A Ces\`aro Average of Goldbach numbers
Let be the von Mangoldt function and be the counting function for the Goldbach
numbers. Let be an integer. We prove that
for , where , with or without subscripts, runs over the
non-trivial zeros of the Riemann zeta-function .Comment: submitte
On the constant in the Mertens product for arithmetic progressions. I. Identities
The aim of the paper is the proof of new identities for the constant in the
Mertens product for arithmetic progressions. We deal with the problem of the
numerical computation of these constants in another paper.Comment: References added, misprints corrected. 9 page
The number of Goldbach representations of an integer
We prove the following result: Let and assume the Riemann
Hypothesis (RH) holds. Then where
runs over the non-trivial zeros of the Riemann zeta function
A Ces\`aro Average of Hardy-Littlewood numbers
Let be the von Mangoldt function and be the counting function for the
Hardy-Littlewood numbers. Let be a sufficiently large integer. We prove
that for , where runs over the
non-trivial zeros of the Riemann zeta-function and
denotes the Bessel function of complex order and real argument .Comment: submitte
Sum of one prime and two squares of primes in short intervals
Assuming the Riemann Hypothesis we prove that the interval
contains an integer which is a sum of a prime and two squares of primes
provided that , where is an effective constant.Comment: removed unconditional case; other minor changes inserte
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