Skip to main content Accessibility help


  • Gautami Bhowmik (a1), Karin Halupczok (a2), Kohji Matsumoto (a3) and Yuta Suzuki (a4)


Assuming a conjecture on distinct zeros of Dirichlet $L$ -functions we get asymptotic results on the average number of representations of an integer as the sum of two primes in arithmetic progression. On the other hand the existence of good error terms gives information on the location of zeros of $L$ -functions. Similar results are obtained for an integer in a congruence class expressed as the sum of two primes.



Hide All

The fourth author is supported by Grant-in-Aid for JSPS Research Fellow (Grant Number: JP16J00906) and had the partial aid of CEMPI for his stay at Lille.



Hide All
1. Bauer, C., Goldbach’s conjecture in arithmetic progressions: number and size of exceptional prime moduli. Arch. Math. 108 2016, 159172.
2. Bhowmik, G. and Ruzsa, I. Z., Average Goldbach and the Quasi-Riemann hypothesis. Anal. Math. 44(1) 2018, 5156.
3. Bhowmik, G. and Schlage-Puchta, J.-C., Mean representation number of integers as the sum of primes. Nagoya Math. J. 200 2010, 2733.
4. Bhowmik, G. and Schlage-Puchta, J.-C., Meromorphic continuation of the Goldbach generating function. Funct. Approx. Comment. Math. 45 2011, 4353.
5. Conrey, J. B., The Riemann hypothesis. Notices Amer. Math. Soc. 50 2003, 341353.
6. Egami, S. and Matsumoto, K., Convolutions of the von Mangoldt function and related Dirichlet series. In Number Theory. Sailing on the Sea of Number Theory (Series on Number Theory and its Applications 2 ) (eds Kanemitsu, S. and Liu, J.-Y.), World Scientific (2007), 123.
7. Ford, K., Soundararajan, K. and Zaharescu, A., On the distribution of imaginary parts of zeros of the Riemann zeta function II. Math. Ann. 343 2009, 487505.
8. Fujii, A., An additive problem of prime numbers. Acta Arith. 58 1991, 173179.
9. Gallagher, P. X., A large sieve density estimate near 𝜎 = 1. Invent. Math. 11 1970, 329339.
10. Gonek, S. M., An explicit formula of Landau and its applications to the theory of the zeta-function. Contemp. Math. 143 1993, 395413.
11. Granville, A., Refinements of Goldbach’s conjecture, and the generalized Riemann hypothesis. Funct. Approx. Comment. Math. 37 2007, 159173; Corrigendum, ibid. 38 (2008), 235–237.
12. Hardy, G. H. and Littlewood, J. E., Some problems of ‘Partitio Numerorum’; III: On the expression of a number as a sum of primes. Acta Math. 44 1923, 170.
13. Liu, M.-C. and Zhan, T., The Goldbach problem with primes in arithmetic progressions. In Analytic Number Theory (London Mathematical Society Lecture Note Series 247 ) (ed. Motohashi, Y.), Cambridge University Press (Cambridge, 1997), 227251.
14. Montgomery, H. L., Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis (Conference Board of the Mathematical Sciences 84 ), American Mathematical Society (Providence, RI, 1994).
15. Montgomery, H. L. and Vaughan, R. C., Multiplicative Number Theory I. Classical Theory, Cambridge University Press (Cambridge, 2007).
16. Rüppel, F., Convolutions of the von Mangoldt function over residue classes. Šiauliai Math. Semin. 7(15) 2012, 135156.
17. Suzuki, Y., A mean value of the representation function for the sum of two primes in arithmetic progressions. Int. J. Number Theory 13(4) 2017, 977990.
MathJax is a JavaScript display engine for mathematics. For more information see

MSC classification


  • Gautami Bhowmik (a1), Karin Halupczok (a2), Kohji Matsumoto (a3) and Yuta Suzuki (a4)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed