Sums of singular series along arithmetic progressions and with smooth weights

article Sums of singular series along arithmetic progressions and with smooth weights published yes Vivian Zieve Kuperberg author c3bac823-112d-11f0-a3f5-c264f852e697 Sums of the singular series constants that appear in the Hardy–Littlewood k-tuples conjectures have long been studied in connection to the distribution of primes. We study constrained sums of singular series, where the sum is taken over sets whose elements are specified modulo r or weighted by smooth functions. We show that the value of the sum is governed by incidences modulo r of elements of the set in the case of arithmetic progressions and by pairings of the smooth functions in the case of weights. These sums shed light on sums of singular series in other formats. World Scientific Publishing2025 eng International Journal of Number Theory 1793-0421 1793-7310 2301.0609510.1142/s1793042125500046 210153-74 yes Kuperberg, V. Z. (2025). Sums of singular series along arithmetic progressions and with smooth weights. International Journal of Number Theory. World Scientific Publishing. <a href="https://doi.org/10.1142/s1793042125500046">https://doi.org/10.1142/s1793042125500046</a> Kuperberg, Vivian Zieve. “Sums of Singular Series along Arithmetic Progressions and with Smooth Weights.” International Journal of Number Theory, vol. 21, no. 01, World Scientific Publishing, 2025, pp. 53–74, doi:<a href="https://doi.org/10.1142/s1793042125500046">10.1142/s1793042125500046</a>. Kuperberg VZ. Sums of singular series along arithmetic progressions and with smooth weights. International Journal of Number Theory. 2025;21(01):53-74. doi:<a href="https://doi.org/10.1142/s1793042125500046">10.1142/s1793042125500046</a> V.Z. Kuperberg, International Journal of Number Theory 21 (2025) 53–74. Kuperberg, Vivian Zieve. “Sums of Singular Series along Arithmetic Progressions and with Smooth Weights.” International Journal of Number Theory. World Scientific Publishing, 2025. <a href="https://doi.org/10.1142/s1793042125500046">https://doi.org/10.1142/s1793042125500046</a>. Kuperberg VZ. 2025. Sums of singular series along arithmetic progressions and with smooth weights. International Journal of Number Theory. 21(01), 53–74. V. Z. Kuperberg, “Sums of singular series along arithmetic progressions and with smooth weights,” International Journal of Number Theory, vol. 21, no. 01. World Scientific Publishing, pp. 53–74, 2025. 221952026-06-29T12:57:46Z2026-07-14T11:04:58Z