## The Michigan Mathematical Journal

### (p - 1)th Roots of unity mod pn, generalized Heilbronn sums, Lind-Lehmer constants, and Fermat quotients

#### Article information

Source
Michigan Math. J., Volume 66, Issue 1 (2017), 203-219.

Dates
First available in Project Euclid: 3 March 2017

https://projecteuclid.org/euclid.mmj/1488510033

Digital Object Identifier
doi:10.1307/mmj/1488510033

Mathematical Reviews number (MathSciNet)
MR3619743

Zentralblatt MATH identifier
06723017

#### Citation

Cochrane, Todd; De Silva, Dilum; Pinner, Christopher. ( p - 1)th Roots of unity mod p n , generalized Heilbronn sums, Lind-Lehmer constants, and Fermat quotients. Michigan Math. J. 66 (2017), no. 1, 203--219. doi:10.1307/mmj/1488510033. https://projecteuclid.org/euclid.mmj/1488510033

#### References

• J. Bourgain and M.-C. Chang, Exponential sum estimates over subgroups and almost subgroups of $Z^*Q$, where $Q$ is composite with few prime factors, Geom. Funct. Anal. 16 (2006), no. 2, 327–366.
• J. Bourgain, K. Ford, S. V. Konyagin, and I. E. Shparlinski, On the divisibility of Fermat quotients, Michigan Math. J. 59 (2010), no. 2, 313–328.
• J. Bourgain, S. V. Konyagin, and I. E. Shparlinski, Product sets of rationals, multiplicative translates of subgroups in residue rings, and fixed points of the discrete logarithm, Int. Math. Res. Not. IMRN (2008), 1–29, Art. ID rnn 090.
• D. W. Boyd, Speculations concerning the range of Mahler's measure, Canad. Math. Bull. 24 (1981), 453–469.
• D. De Silva and C. Pinner, The Lind–Lehmer constant for $\mathbb Z_p^n$, Proc. Amer. Math. Soc. 142 (2014), no. 6, 1935–1941.
• A. Granville, Some conjectures related to Fermat's last theorem, Number theory, Banff, 1988, pp. 177–192, de Gruyter, New York, 1990.
• D. R. Heath-Brown, An estimate for Heilbronn's exponential sum, analytic number theory, Proceedings of a conference in honor of Heini Halberstam, pp. 451–463, Birkhäuser, Boston, 1996.
• D. R. Heath-Brown and S. V. Konyagin, New bounds for Gauss sums derived from $k{\rm th}$ powers, and for Heilbronn's exponential sum, Q. J. Math. 51 (2000), no. 2, 221–235.
• S. Konyagin and C. Pomerance, On primes recognizable in deterministic polynomial time, the mathematics of Paul Erdös, I, 176–198, Algorithms Combin., 13, Springer, Berlin, 1997.
• S. V. Konyagin and I. E. Shparlinski, Character sums with exponential functions and their applications, Cambridge Tracts in Math., 136, Cambridge University Press, Cambridge, 1999.
• D. H. Lehmer, Factorization of certain cyclotomic functions, Math. Ann. 34 (1933), 461–479.
• H. W. Lenstra, Miller's primality test, Inform. Process. Lett. 8 (1979), no. 2, 86–88.
• D. Lind, Lehmer's problem for compact Abelian groups, Proc. Amer. Math. Soc. 133 (2005), 1411–1416.
• Y. V. Malykhin, Estimates of trigonometric sums modulo $p^r$, Mat. Zametki 80 (2006), no. 5, 748–752, translated from, 80, 5, 2006, 793–796.
• H. L. Montgomery and R. C. Vaughan, Exponential sums with multiplicative coefficients, Invent. Math. 43 (1977), no. 1, 69–82.
• I. D. Shkredov, On Heilbronn's exponential sum, Q. J. Math. 64 (2012), no. 4, 1221–1230.
• I. D. Shkredov, On exponential sums over multiplicative subgroups of medium size, Finite Fields Appl. 30 (2014), 72–87.
• I. D. Shkredov, E. V. Solodkova, and I. V. Vyugin, Intersections of multiplicative subgroups and Heilbronn's exponential sum, 6, pp. 1–22, 2015, May, arXiv:1302.3839v3 [math.NT].
• I. E. Shparlinski, On the value set of Fermat quotients, Proc. Amer. Math. Soc. 140 (2012), no. 4, 1199–1206.
• Y. N. Shteinikov, Divisibility of Fermat quotients, Translation of Mat. Zametki 92 (2012), no. 1, 116–122, 92, 2012, 1, 108–114.
• Y. N. Shteinikov, Estimates of trigonometric sums over subgroups and some of their applications, Translation of Mat. Zametki 98 (2015), no. 4, 606–625, 98, 2015, 4, 667–684.