1lcalc(1) General Commands Manual lcalc(1)
2
6 lcalc - Compute zeros and values of L-functions
7
9 lcalc [OPTIONS]
10
13 lcalc is a command-line program for computing the zeros and values of
14 L-functions. Several L-functions are built-in, and the default is the
15 Riemann zeta function. You can also specify your own L-function in the
16 form of a data file that describes it; see the DATA FILE FORMAT section
17 below.
18
21 Given an L-function (by default, the Riemann zeta), lcalc can compute
22 its zeros using either the --zeros or --zeros-interval flags. For exam‐
23 ple,
24 $ lcalc --zeros 1000
26 will compute the first thousand zeros of the Riemann zeta function,
28 while checking the (generalized) Riemann hypothesis and making sure
29 that no zeros are missed. The output consists only of the imaginary
30 parts of the zeros, since all of the real parts should be 0.5.
31 The (generalized) Riemann hypothesis is confirmed by comparing the num‐
33 ber of zeros found to the main term in the formula for N(T). The dif‐
34 ference, S(T), should be small on average. If not, then missing zeros
35 are detected and the program backtracks looking for sign changes in
36 more refined steps. This is repeated until the missing zeros are found
37 within reasonable time. Otherwise, the program exits.
38 The found zeros are also verified before being output, using the ex‐
40 plicit formula to compare sums over the zeros to sums over primes. The
41 program uses this comparison to dynamically set the output precision,
42 and exits if both sides do not agree to at least two places after the
43 decimal. More precisely, the Riemann-Weil explicit formula is checked
44 repeatedly, as zeros are outputted. Let the non-trivial zeros of L(s)
45 be 1/2 + i gamma. Let f be a function of the form f(x) = exp(-A(x-
46 x_0)^2), where A = log(10)/Digits and Digits is the working number of
47 digits of precision (fifteen for built in doubles), and x_0 varies as
48 explained shortly.
49 Then, the sums over the zeros
51 -----
53 \
54 ) f(gamma)
55 /
56 -----
57 gamma
58 are computed in two ways to working precision: first using the numeri‐
60 cally computed zeros, and second using the explicit formula to express
61 the sum over zeros as a sum over primes powers. These two are compared,
62 and the program uses the amount of agreement between the two to help
63 decide the output precision of the zeros. If the two do not agree to at
64 least two digits after the decimal, then the program quits.
65 The value x_0 is chosen to coincide with a zero of L(s), say gamma_j,
67 but translated by 0.5 so as to desymmetrize the function f (not trans‐
68 lating would leave it an even function, and it would be quadratically,
69 rather than linearly, sensitive to inaccuracies in the zeros).
70 At first, a separate test function f is used for each of the zeros,
72 with x_0 coinciding with those zeros. After a while, however, a new f
73 is taken only once every five zeros, with x_0 coinciding similarly.
74 You can also start after the Nth zero. For example,
76 $ lcalc --zeros 5 --N 5
78 outputs the sixth through tenth zeros. Caution: the --N option evalu‐
80 ates N(T) by using the formula involving the main term (from the Gamma
81 factors) and S(T). S(T) is computed by looking at the change of arg of
82 L(s) as one goes from 1/2 - iT to infinity - iT and then up to infinity
83 + iT and to 1/2 + iT. This can take a long time if T is near the ordi‐
84 nate of a zero. One could improve this by implementing Turing's method
85 for finding the Nth zero (only looking at sign changes on the critical
86 line), or by tweaking the starting point and suppressing a correspond‐
87 ing number of zeros should the program take too long, but for now, only
88 the contour integral is computed.
89 A variant of Turing's method which only looks on the critical line is
91 used for the --zeros option, to test that all zeros have been found.
92 However, with regards to the --N option, S(T) is initially computed via
93 contour integration to zoom in on the Nth zero.
94 If you aren't concerned about verifying the (generalized) Riemann hy‐
96 pothesis, or if you don't expect it to hold (for instance, if you plan
97 to study Dirichlet series without Euler products) then you can search
98 for zeros in an "interval" using the --zeros-interval option. For exam‐
99 ple,
100 $ lcalc --zeros-interval --x=10 --y=100 --stepsize=0.1
102 searches for zeros of the Riemann zeta function in the interval from
104 1/2 + 10i to 1/2 + 100i, checking for sign changes advancing in steps
105 of size 0.1. The first column of the output contains the imaginary part
106 of the zero, and the second column contains a quantity related to S(T)
107 --- it increases roughly by two whenever a sign change, that is, a pair
108 of zeros, is missed. Higher up the critical strip you should use a
109 smaller stepsize so as not to miss zeros. The --zeros-interval options
110 makes sense if you want to output zeros as they are found.
111 The --zeros option, which does verify the (generalized) Riemann hypoth‐
113 esis, looks for several dozen zeros to make sure none have been missed
114 before outputting any zeros at all; as a result, it takes longer than
115 --zeros-interval to output the first few zeros. For collecting more
116 than just a handful of zeros, or to make certain that no zeros have
117 been missed, one should use the --zeros option.
118
121 The lcalc program can also compute the values of an L-function by using
122 the --value flag. For example,
123 $ lcalc --value --x=0.5 --y=100
125 will compute the value of the (default) Riemann zeta function at the
127 point x + yi = 0.5 + 100i. Both real and complex parts are output, sep‐
128 arated by a space; in this case, the output is approximately
129 2.692619886 -0.0203860296 which represents zeta(x+yi) = 2.692619886 -
130 0.0203860296i.
131 You can also compute values along a line segment at equally spaced
133 points:
134 $ lcalc --value-line-segment --x=0.5 --X=0.5 --y=0 --Y=10 --number-samples=100
136 computes the values of the Riemann zeta function from 0.5 + 0i through
138 0.5 + 100i at 1000 equally-spaced points. The output contains four col‐
139 umns: the real and imaginary parts of the current point (two columns),
140 followed by the real and imaginary parts of the function value (two
141 columns).
142
145 If lcalc was built with PARI support, the --elliptic-curve option can
146 be combined with the --a1 through --a6 flags to specify an elliptic
147 curve L-function. For example,
148 $ lcalc --zeros=5 --elliptic-curve --a1=0 --a2=0 --a3=0 --a4=0 --a6=1
150 computes the first five zeros of the L-function associated with the el‐
152 liptic curve y^2 = x^3 + 1.
153
156 Twists by Dirichlet characters are currently available for all zeta and
157 cusp-form L-functions. For example,
158 $ lcalc --value --x=0.5 --y=0 --twist-quadratic --start -100 --finish 100
160 will output L(1/2, chi_d) for d between -100 and 100, inclusive. Other
162 twisting options are available. For example
163 $ lcalc --zeros=200 --twist-primitive --start 3 --finish 100
165 gives the first two-hundred zeros of all primitive L(s,chi) with a con‐
167 ductor between 3 and 100, inclusive.
168 Notice that with the --twist-quadratic option one is specifying the
170 discriminant which can be negative, while with the --twist-primitive
171 option one is specifying the conductor which should be positive.
172 When using the various twisting options, other than --twist-quadratic,
174 the second output column is a label for the character modulo n. It is
175 an integer between 1 and phi(n). If one is restricting to primitive
176 charcters, then only a subset of these integers appear in the second
177 column.
178 One can obtain a table of characters using the --output-character op‐
180 tion. This doesn't work with the --twist-quadratic option, but does
181 with the other twisting options.
182 The --output-character option takes an argument of either 1 or 2. An
184 argument of 1, for example, will print a comprehensive table of chi(n)
185 for all gcd(n,conductor) = 1.
186 The characters are constructed multiplicatively in terms of generators
188 of the cyclic groups mod the prime powers that divide n, and there's no
189 simple formula to go from the label to the character. As a result,
190 --output-character will also output a table for each character before
191 outputting the zeros of the corresponding L-function (but not with the
192 --twist-quadratic option).
193 These tables contain six columns:
195 1 the value of n
198 2 a label for the character, an integer between 1 and phi(n)
200 3 the conductor of the inducing character (same as the first col‐
202 umn, if primitive).
203 4 column 4: m, an integer that is coprime with n
205 5 Re(chi(m))
207 6 Im(chi(m))
209 If, instead, --output-character=2 is given, then only the value of
212 chi(-1) (that is, whether chi is even or odd) and whether chi is primi‐
213 tive or not will be printed.
214
217 For basic program usage, run
218 $ lcalc --help
220 Most of lcalc's options are sufficiently explained by its output. Here
222 we collect some further information about specific options.
223 --rank-compute, -r
226 Compute the analytic rank. Analytic rank works well even for
227 high rank since the method used does not compute derivatives,
228 but rather looks at the behaviour of the L-function near the
229 critical point.
230 --derivative=<n>, -d <n>
233 Compute the nth derivative. Presently the derivative option uses
234 numeric differentiation (taking linear combinations of L(s + mh)
235 for integers m, to pull out the relevant Taylor coefficient of
236 L(s)). To get good results with higher derivatives, one should
237 build libLfunction/lcalc with more precision.
238
241 • Compute the first thousand zeros of the Riemann zeta function using
242 the --zeros option, and output their imaginary parts (after verifying
243 that their real parts are all one-half):
244 $ lcalc --zeros 1000
246 • Compute the value of the Riemann zeta function at x + iy using the
248 --value, --x, and --y options:
249 $ lcalc --value --x=0.5 --y=14.134725141738
251 • Compute the first 10 zeros of the Ramanujan tau L-function using the
253 --tau and --zeros (-z) options:
254 $ lcalc --tau --zeros=10
256 • Compute the first zero of the real quadratic Dirichlet L-function of
258 conductor 4 after using the --twist-quadratic, --start, and --finish
259 options to specify the L-function:
260 $ lcalc --twist-quadratic --start=-4 --finish=-4 --zeros=1
262
265 The lcalc -F option allows you to load L-function data from a file.
266 Here we explain the format of this file. Basically it must contain the
267 functional equation, Dirichlet coefficients, and some other helpful in‐
268 formation.
269 The first line should contain an integer, either 1, 2, or 3: 1 speci‐
271 fies that the Dirichlet coefficients are to be given as integers (up to
272 thirty-two bits long), 2 that they are floating point numbers, 3 that
273 they are complex numbers. So, for example, the first line would be a 2
274 for cusp form or Maass form L-function (we normalize the L-functional
275 so that the functional equation is s <-> 1-s, so the normalized Dirich‐
276 let coefficients for a cusp form L-function are not integers).
277 The second line specifies info that the calculator can exploit (or will
279 exploit at some future date). It is an integer that is assigned to dif‐
280 ferent types of L-functions. Currently:
281 • -2 for L(s,chi) but where the number of coefficients computed is less
283 than the period
284 • -1 for zeta
286 • 0 for unknown
288 • 1 for periodic, including L(s,chi)
290 • 2 for cusp form (in S_K(Gamma_0(N))
292 • 3 for Maass form for SL_2(Z)
294 • other integers reserved for future types.
296 The third line is an integer that specifies how many Dirichlet coef‐
298 ficients to read from the file; there should be at least this many
299 Dirichlet coefficients in the file. This is a useful quantity to
300 specify sincethe data file might have, say, a million coefficients,
301 but one might want to read in just ten thousand of them.
302 The fourth line is either 0, if the Dirichlet coefficients are not
304 periodic, or a positive integer specifying the period otherwise (this
305 happens in the case of Dirichlet L-functions). For a Maass form it
306 should be 0.
307 The fifth line is a positive integer, the quasi-degree. This is the
309 number of gamma factors of the form Gamma(gamma s + lambda) in the
310 functional equation, where gamma is either 0.5 or 1, and lambda is a
311 complex number with Re(lambda) >= 0. For example, it is 1 for Dirich‐
312 let L-functions, 1 for cusp form L-functions, 2 for Maass form L-
313 functions, etc. Note that the "1" for cusp form L-functions could be
314 a "2" if you wish to split the gamma factor up using the Legendre du‐
315 plication formula. But it's better not to.
316 Next come the gamma factors, with two lines for each gamma factor.
318 The first of each pair of lines contains gamma (either 0.5 or 1), and
319 the second line contains a pair of floating point numbers separated
320 by a space specifying the real and imaginary parts of lambda (even if
321 purely real, lambda should be given as a pair of numbers, the second
322 one then being 0).
323 Next you specify the functional equation. Let
325 a
327 --------'
328 ' | |
329 | |
330 Lambda(s) = Q^s | | Gamma(gamma_j*s + lambda_j)*L(s)
331 | |
332 j = 1
333 satisfy Lambda(s) = omega*conj(Lambda(1-conj(s))), where Q is a real
335 number, omega is complex,and "conj" denotes the complex conjugate.
336 Notice that the functional equation is s into 1-s; that is, the gamma
337 factors and Dirichlet coefficients of L(s) should be normalized cor‐
338 rectly so as to have critical line Re(s) = 1/2.
339 The next line in the datafile is Q giving as a floating point number,
341 and the one after that gives omega as a pair x y of floating point
342 numbers, specifying the real and imaginary parts of omega (even when
343 omega is unity, it should be given as a pair of floating points, for
344 example 1 0 to indicate 1 + 0i).
345 We need to allow for the possibility of poles. For example, if L(s) =
347 zeta(s) then Lambda(s) has simple poles with residue 1 at s=1 and
348 residue -1 at s=0. To take into account such possibilities I assume
349 that Lambda(s) has at most simple poles. So, the next line specifies
350 the number of poles (usually 0) and then, for each pole there are two
351 lines. The first line in each pair gives the pole x+iy as a pair x y
352 of floating point numbers. The second line specifies the residue at
353 that pole, also as a pair of floating point numbers.
354 Finally the Dirichlet coefficients. The remaining lines in the file
356 contain a list of Dirichlet coefficients, at least as many as indi‐
357 cated in line three of the datafile. The coefficients can be inte‐
358 gers, real, or complex. If complex they should, as usual, be given as
359 a pair x y of floating point numbers separated by a space. Otherwise
360 they should be given as single column of integers or floating point
361 numbers respectively.
362 The datafile should only contain numbers and no comments. The com‐
364 ments below are for the sake of understanding the structure of the
365 datafile.
366 An example data file for the Maass form for SL_2(Z) associated to the
368 eigenvalue with R = 13.779751351891 is given below (beware: the
369 Dirichlet coefficients for this particular L-function are only accu‐
370 rate to a handful of decimal places, especially near the tail of the
371 file).
372 2 the Dirichlet coefficients are real
374 3 this is a Maass form L-function
375 29900 use this many of the Dirichlet coefficients
376 0 zero since the coefficients are not periodic
377 2 two gamma factors
378 .5 the first gamma factor
379 0 6.88987567594535 the first lambda
380 .5 the second gamma factor
381 0 -6.88987567594535 the second lambda
382 .3183098861837906715 the Q in the functional equation
383 1 0 the omega 1 + 0i in the functional equation
384 0 the number of poles of Lambda(s)
385 1 the first Dirichlet coefficient
386 1.549304477941 the second Dirichlet coefficient
387 0.246899772454 the third Dirichlet coefficient
388 1.400344365369 the fourth Dirichlet coefficient
389 ...
390 Several such example data files are included with lcalc.
392
395 Report bugs to https://gitlab.com/sagemath/lcalc/issues
396 lcalc(1)