A338926 Decimal expansion of Stechkin's constant.

4, 7, 0, 9, 2, 3, 6, 8, 5, 3, 1, 4, 5, 2, 6, 7, 9, 4, 3, 5, 8, 0, 2, 3, 9, 9, 4, 7, 4, 4, 5, 7, 7, 8, 0, 8, 8, 3, 9, 2, 0, 2, 7, 1, 2, 9, 7, 6, 4, 3, 4, 1, 7, 9, 4, 7, 8, 6, 7, 0, 9, 6, 4, 8, 4, 6, 4, 8, 4, 3, 6, 2, 1, 3, 7, 6, 7, 9, 3, 9, 6, 0, 3, 6, 8, 7, 6

Offset

1,1

Comments

This constant was named by Banks and Shparlinski (2016) after the Soviet mathematician Sergey Borisovich Stechkin (1920 - 1995).

Let S_m(k, q) = Sum_{j=0..q-1} e(k*j^m/q), where q>=1, m>=2, and k are integers, and e(t) = exp(2*Pi*i*t) (i is the imaginary unit). This constant is A = sup_{m,q} max_{gcd(k,q)=1} |S_m(k, q)|/q^(1-1/m). Stechkin (1975) conjectured and Shparlinskii (1991) proved that A is finite, and Banks and Shparlinski (2016) determined its value.

References

Sergei Konyagin and Igor Shparlinski, Character sums with exponential functions and their applications, Cambridge University Press, 2004, chapter 6, pp. 37-45.

Links

Table of n, a(n) for n=1..87.

William D. Banks and Igor E. Shparlinski, On Gauss sums and the evaluation of Stechkin's constant, Mathematics of Computation, Vol. 85, No. 301 (2016), pp. 2569-2581; arXiv preprint, arXiv:1310.5669 [math.NT], 2013.

Alexandru Ciolan, Equidistribution and inequalities for partitions into powers, arXiv preprint arXiv:2002.05682 [math.NT], 2020.

I. E. Shparlinskii, Estimates of Gaussian sums, Matematicheskie Zametki, Vol. 50, No. 1 (1991), pp. 122-130 (in Russian); English translation, Mathematical Notes of the Academy of Sciences of the USSR, Vol. 50 (1991), pp. 740-746.

S. B. Stechkin, An estimate for Gaussian sums, Matematicheskie Zametki, Vol. 17, No. 4 (1975), pp. 579-588 (in Russian); English translation, Mathematical Notes of the Academy of Sciences of the USSR, Vol. 17, (1975), pp. 342-349.

Formula

Equals abs(Sum_{j=0..q-1} exp(2*Pi*i*4787*j^6/q))/q^(5/6), where q = 4606056.

Example

4.70923685314526794358023994744577808839202712976434...

Mathematica

s[k_, q_, m_] := Sum[Exp[2*Pi*I*k*j^m/q], {j, 0, q-1}]; RealDigits[Abs[s[4787, 4606056, 6]]/4606056^(5/6), 10, 90][[1]]

Crossrefs

Sequence in context: A020833 A197147 A153117 * A322166 A244499 A303173

Adjacent sequences: A338923 A338924 A338925 * A338927 A338928 A338929

Keyword

nonn,cons

Author

Amiram Eldar, Nov 15 2020

Status

approved