## Explicit formulae for averages of Goldbach representations

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- by Jörg Brüdern, Jerzy Kaczorowski and Alberto Perelli PDF
- Trans. Amer. Math. Soc.
**372**(2019), 6981-6999 Request permission

## Abstract:

We prove an explicit formula, analogous to the classical explicit formula for $\psi (x)$, for the Cesàro–Riesz mean of any order $k>0$ of the number of representations of $n$ as a sum of two primes. Our approach is based on a double Mellin transform and the analytic continuation of certain functions arising therein.## References

- Gautami Bhowmik and Jan-Christoph Schlage-Puchta,
*Mean representation number of integers as the sum of primes*, Nagoya Math. J.**200**(2010), 27–33. MR**2747876**, DOI 10.1215/00277630-2010-010 - Harold Davenport,
*Multiplicative number theory*, 2nd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York-Berlin, 1980. Revised by Hugh L. Montgomery. MR**606931**, DOI 10.1007/978-1-4757-5927-3 - Akio Fujii,
*An additive problem of prime numbers. II*, Proc. Japan Acad. Ser. A Math. Sci.**67**(1991), no. 7, 248–252. MR**1137920** - D. A. Goldston and L. Yang,
*The average number of Goldbach representations*, Prime numbers and representation theory, Ye Tian and Yangbo Ye (eds.), Science Press, Beijing, 2017, pp. 1–12. - Alessandro Languasco,
*Applications of some exponential sums on prime powers: a survey*, Riv. Math. Univ. Parma (N.S.)**7**(2016), no. 1, 19–37. MR**3675401** - Alessandro Languasco and Alessandro Zaccagnini,
*The number of Goldbach representations of an integer*, Proc. Amer. Math. Soc.**140**(2012), no. 3, 795–804. MR**2869064**, DOI 10.1090/S0002-9939-2011-10957-2 - Alessandro Languasco and Alessandro Zaccagnini,
*A Cesàro average of Goldbach numbers*, Forum Math.**27**(2015), no. 4, 1945–1960. MR**3365783**, DOI 10.1515/forum-2012-0100 - Fritz Oberhettinger,
*Tables of Mellin transforms*, Springer-Verlag, New York-Heidelberg, 1974. MR**0352890**, DOI 10.1007/978-3-642-65975-1 - Reinhold Remmert,
*Classical topics in complex function theory*, Graduate Texts in Mathematics, vol. 172, Springer-Verlag, New York, 1998. Translated from the German by Leslie Kay. MR**1483074**, DOI 10.1007/978-1-4757-2956-6

## Additional Information

**Jörg Brüdern**- Affiliation: Mathematisches Institut, Bunsenstrasse 3-5, 37073 Göttingen, Germany
- Email: bruedern@uni-math.gwdg.de
**Jerzy Kaczorowski**- Affiliation: Faculty of Mathematics and Computer Science, Adam Mickiewicz University, 61-614 Poznań, Poland; and Institute of Mathematics of the Polish Academy of Sciences, 00-956 Warsaw, Poland
- MR Author ID: 96610
- Email: kjerzy@amu.edu.pl
**Alberto Perelli**- Affiliation: Dipartimento di Matematica, Università di Genova, via Dodecaneso 35, 16146 Genova, Italy
- MR Author ID: 137910
- Email: perelli@dima.unige.it
- Received by editor(s): January 22, 2018
- Received by editor(s) in revised form: November 12, 2018
- Published electronically: March 20, 2019
- Additional Notes: This research was partially supported by a grant from Deutsche Forschungsgemeinschaft, by the grant PRIN-2015 “Number Theory and Arithmetic Geometry”, and by grant 2017/25/B/ST1/00208 “Analytic methods in number theory” from the National Science Centre, Poland. The third author is a member of the GNAMPA group of INdAM
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**372**(2019), 6981-6999 - MSC (2010): Primary 11P32, 11N05
- DOI: https://doi.org/10.1090/tran/7799
- MathSciNet review: 4024544