On certain sums over primes and the Riesz function

Alexander Patkowski

doi:10.30970/ms.65.1.3-9

Alexander Patkowski None

DOI: https://doi.org/10.30970/ms.65.1.3-9

Keywords: prime numbers; Riemann zeta function; Riesz function.

Abstract

We offer some comments on series involving the M$\ddot{o}$bius function which approximate sums over primes. To accomplish this, we utilize the derivative of the Gram series by applying Riemann-Stieltjes integration. We offer a new formula connecting the derivative of the Gram series to the Riesz function, which we then use to obtain general integral relationships.

References

B.C. Berndt, Ramanujan’s Notebooks, Part IV, New York: Springer–Verlag, 1994.

H.M. Edwards, Riemann’s Zeta function, Dover Publications, 1974.

A. Erde’lyi, (ed.) Tables of integral transforms, V.1, 1954, McGraw-Hill, New York.

M. Garg, B. Maji, Hardy–Littlewood–Riesz type equivalent criteria for the generalized Riemann hypothesis, Monatsh. Math., 201 (3) (2023), 771–788. https://doi.org/10.1007/s00605-023-01857-8

S. Gupta, A. Vatwani, Riesz-type criteria for L-functions in the Selberg class, Canadian Journal of Mathematics, 76 (3) (2024), 1062–1088. https://doi.org/10.4153/S0008414X23000354

R.B. Paris, D. Kaminski, Asymptotics and Mellin–Barnes integrals, Cambridge University Press, 2001.

A.E. Patkowski, On some instances of Fox’s integral equation, Archivum Mathematicum, 55 (3) (2019), 195–201. https://doi.org/10.5817/AM2019-3-195

H. Riesel, Prime numbers and computer methods for factorization, Birkhäuser, Boston, MA, 1985.

M. Riesz, Sur l’hypoth’ese de Riemann, Acta Math., 40 (1916), 185–190.

E.C. Titchmarsh, The theory of the Riemann zeta function, Oxford University Press, 2nd edition, 1986.