On asymptotics of some large character sums.
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Added Monday January 1, 2018:
FOREWORD:
I now consider what follows this foreword as numerical data on some large character sums. The implied asymptotic constant , 2*(exp(Euler)/Pi)^2 , is not backed by any theoretical work. I had a few email exchanges with Andrew Granville about asymptotics of character sums, and Conjecture 1 in the joint paper of Granville and Soundararajan, “LARGE CHARACTER SUMS: PRETENTIOUS CHARACTERS AND THE POLYA-VINOGRADOV THEOREM” at
https://arxiv.org/abs/math/0503113v1 .
I now believe, thanks to the emails, that their conjecture is backed by some solid theoretical computations, therefore a priori, I accept their proportionality constant of exp(Euler)/Pi ; indeed, the o(1) term in their Conjecture 1 could go to zero very very slowly.
Besides the obervations in the post (calculations), and the connection with large values of L(1, Chi_d) , I have no insights to offer on large character sums.
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Continued from sci.math disussion thread:
“large deviations in sums of Legendre symbols (a,p) for fixed prime p”.
For q = 7,462,642,151 Max(Chi_q) = 235920.
Indeed, | sum_{k=1… 3589625338} (k/q) | = 235920.
Then:
2*((exp(Euler)/Pi)^2)*sqrt(q)*( log(log(q)) + log(log(log(q))) ) =
236724.49
and
Max(Chi_q)/236724.49 = 235920/236724.49 = 0.996601576.
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The sources I consulted are mentioned in the sci.math discussion thread. Among these are articles of Andrew Granville and K. Soundararajan on large values of L(1, Chi_d), some on character sums. Others by mathematicians at the University of Calgary include tables on record values of L(1, Chi_d) for very large |d| .
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Added: Friday August 18, 2017 at 2:45 pm :
The mathematicians are M.J. Jacobson, Jr., S. Ramachandran, and H.C. Williams with the Department of Computer Science at the University of Calgary.They have published a document/paper entitled:
“Supplementary Tables for “Numerical Results on Class Groups of Imaginary Quadratic Fields” ” , and in it Table 1 gives successive discriminants | Delta | with record L(1, Delta) for Delta with Delta == 1 (mod 8).
For the prime 7979490791 with (-7979490791) = 1 (mod 8) , I find that Max(Chi_d) = 243561 with d = 7979490791 and Chi_d(n) = (n/d).
Indeed, sum_{n = 1… 3958330077} Chi_d(n) = 243561.
The formula under testing is
Max(Chi_d) ~< 2*((exp(Euler)/Pi)^2)*sqrt(d)*( log(log(d)) + log(log(log(d))) ) .
For d = 7979490791, this gives
Max(Chi_d) ~= 245007.758796 with
243561/245007.758796 = 0.994095, so a good fit.
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Added: Friday August 18, 11:15 pm
12,123,145,319 5,480,325,966 304302
d = q = 12,123,145,319 .
Max(Chi_d) = 304302
sum_{k = 1… 5480325966} (k/q) = 304302.
? 304302/(2*sqrt(q)*((exp(Euler)/Pi)^2)*( log(log(q)) + log(log(log(q))) ))
= 1.0020087094650505195701326191442298267
P.S.: The least quadratic non-residue of 12123145319 is 73.
P.P.S. :
Astonishingly,
? round((33/73.0)*12123145319)
= 5480325966
with d = q = 12123145319,
5480325966 the largest ‘k’ to include in the character sum so as to
get Max(Chi_d),
73 being the least quadratic nonredidue of q.
Where does 33 come from ?
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Added: Saturday August 19, 9:35 am
17833071959 7945428096 371104
d = q = 17833071959 , a prime for which L(1, Chi_d) is large.
Max(Chi_d) = 371104.
sum_{k = 1… 7945428096} (k/q) = 371104 // evaluating Max(Chi_d)
by exhaustive search.
? 371104/(2*sqrt(q)*((exp(Euler)/Pi)^2)*( log(log(q)) + log(log(log(q))) ))
= 1.0024544704579438919019720740240736961 .
The least quadratic non-residue of 17833071959 is 101.
Once more, 7945428096 is very close to integer multiple
of q/101:
7945428096.0/(q/101) = 44.999999974261305 .
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Added: Saturday August 19, 11 pm
29414861999 14643206407 479131
d = q = 29414861999 , a prime for which L(1, Chi_d) is large.
Max(Chi_d) = 479131.
sum_{k = 1… 14643206407} (k/q) = 479131 // evaluating Max(Chi_d)
by exhaustive search.
? 479131/(2*sqrt(q)*((exp(Euler)/Pi)^2)*( log(log(q)) + log(log(log(q))) ))
= 1.001343083765517939368188581285160618 .
? 14643206407.0/29414861999 – 114.0/229
= -1.0139543579001317403965968380240395634 E-10
The denominator, 229, is a prime and
a quadratic non-residue of 29414861999.
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Added: Monday August 21, 2:45 pm
35535649679 17250315378 528854
d = q = 35535649679, is a prime for which L(1, Chi_d) is large.
Max(Chi_d) = 528854.
sum_{k = 1… 17250315378} (k/q) = 528854 // evaluating Max(Chi_d)
by exhaustive search.
? 528854/(2*sqrt(q)*((exp(Euler)/Pi)^2)*( log(log(q)) + log(log(log(q))) ))
= 1.0032054837039944918878497593626545 .
(a good fit)
? 17250315378.0/35535649679 – 50.0/103
= -4.3713793677182068003299876219157301046 E-12
The denominator, 103, is a prime and
a quadratic non-residue of 35535649679.
83, 89 and 103 are the three smallest non-residues
of 35535649679.
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Added Tuesday August 22, 2017 at 10 pm
There’s a preprint of Jacobson et al on class groups and class numbers for imaginary quadratic number fields with discriminant Delta satifying | Delta | < 2^40, which is approximately 10^12.
Their Table 6 includes six new record values of L(1, Chi_Delta). The Delta which give large L(1, Chi_Delta) are appropriate for looking for large Max(Chi_Delta), empirically.
I’m looking at Delta = -210015218111 the first of six in Table 6. For such a large |Delta|, it’s no longer feasible to compute Max(Chi_|Delta|) using exhaustive search, i.e. adding up the (k/210015218111) Legendre symbols from k = 1 on up. However, character sums for this modulus, d = 210015218111 can be approximated using a Fourier-series type expansion.
For x = round( (48/97)*210015218111) = 103925056385,
sum_{ k <= x} (k/210015218111) ~~= 1328010 .
Then,
1328010/ (2*sqrt(q)*((exp(Euler)/Pi)^2)*( log(log(q)) + log(log(log(q))) ))
~= 1.0146778
which is not a bad fit. Five more record discriminants.
The preprint is at location:
https://arxiv.org/abs/1502.07953
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Added Wednesday August 23, 2017 3 pm
I estimate Max(Chi_|Delta|) ~= 1681470 for Delta = -q ,
q = 332323080311.
Then,
1681470/(2*sqrt(q)*((exp(Euler)/Pi)^2)*( log(log(q)) + log(log(log(q))) ))
~= 1.0161065
(a good fit).
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Added Friday August 25, 2017 1:40 am
I estimate Max(Chi_|Delta|) ~= 2073587 for Delta = -q ,
q = 503494619759.
Then,
2073587/(2*sqrt(q)*((exp(Euler)/Pi)^2)*( log(log(q)) + log(log(log(q))) ))
~= 1.013417
(a good fit).
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Added Sunday August 27, 2017 5 am
I estimate Max(Chi_q) ~= 2450593 for q = 685122125399.
Then,
2450593/(2*sqrt(q)*((exp(Euler)/Pi)^2)*( log(log(q)) + log(log(log(q))) ))
~= 1.02333827 .
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Added Monday August 28, 2017 11 am
I’m not so clear about things when q is a composite modulus. I’m more confident in the calculation when q is an odd prime modulus.
q=210015218111 is prime . Character sums Chi_d or (./d) for
d = 210015218111 were mentioned above Tuesday August 22.
It has the largest Upper Littlewood Index (ULI)
among prime |Delta| in Table 6 of :
https://arxiv.org/abs/1502.07953 (Mosunov, Jacobson).
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Added Wednesday December 20, 2017
I now consider the above as numerical data on some large character sums. The implied asymptotic constant , 2*(exp(Euler)/Pi)^2 , is not backed by any theoretical work. I had a few email exchanges with Andrew Granville about asymptotics of character sums, and Conjecture 1 in the joint paper of Granville and Soundararajan, “LARGE CHARACTER SUMS: PRETENTIOUS CHARACTERS AND THE POLYA-VINOGRADOV THEOREM” at
https://arxiv.org/abs/math/0503113v1
I now believe, thanks to the emails, that their conjecture is backed by some solid theoretical computations, therefore a priori, I accept their proportionality constant of exp(Euler)/Pi ; the o(1) term in their Conjecture 1 could go to zero very very slowly.
Besides the obervations in the post (calculations), and the connection with large values of L(1, Chi_d) , I have no insights to offer on large character sums.
