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Papers: List Papers; (with Abstracts); Curriculum (in Italian): long version ; short version; in English: long version. Google Scholar profile. ResearchGate page. Orcid ID. Scopus Author ID. Web of Science Researcher ID, Mathematical Reviews page, Zentralblatt page, IRIS-CINECA bibliometric parameters (italian ASN) [2023]. English C1 badge.

EK-direct-final.gp: Pari/GP script. It can be used via gp2c. The function to be run is:
global_eulerkronecker_direct (r₁,r₂,defaultprecision).
Input: 2< r₁ < r₂, two integers; defaultprecision is the number of digits requested.
Output: 𝔊_q and 𝔊_q⁺ for every odd prime q such that r₁≤q≤r₂ and the running times.
Comment: it uses the lfun command of Pari/GP and the Conrey description of Dirichlet characters. Examples on how to use the function and computational results are collected towards the end of the file.

EK-T-final.gp: Pari/GP script. It can be used via gp2c. The function to be run is:
global_eulerkroneckerT (r₁,r₂,defaultprecision,flag).
Input: 2< r₁ < r₂, two integers; defaultprecision is the number of digits requested; flag (defalult =0): if nonzero computes also the Euler generalised constant γ₀(a,q) and γ₁(a,q), 1≤a≤q.
Output: 𝔊_q and 𝔊_q⁺ for every odd prime q such that r₁≤q≤r₂ and the running times.
Comments: it computes first the needed values of ψ and T at the a/q points, see [1], and then obtain 𝔊_q and 𝔊_q⁺ with a trivial implementation of the sum over a, 1≤a≤q-1. Examples on how to use the function and computational results are collected towards the end of the file.

EK-S-DIF-final.gp: Pari/GP script. It can be used via gp2c. The function to be run is:
global_eulerkroneckerS_DIF (r₁,r₂,defaultprecision).
Input: 2< r₁ < r₂, two integers; defaultprecision is the number of digits requested.
Output: 𝔊_q and 𝔊_q⁺ for every odd prime q such that r₁≤q≤r₂ and the running times.
Comments: it computes first the values of log(Γ) and the decimated in frequency values of S at the a/q points, see [1], and then obtain 𝔊_q and 𝔊_q⁺ with a trivial implementation of the sum over a, 1≤a≤q-1. Examples on how to use the function and computational results are collected towards the end of the file.

Gen-Euler-constants.gp: Pari/GP script. It can be used via gp2c. The function to be run is:
gk(k,r₁,r₂,defaultprecision).
Input: k≥1 is an integer,r₁ ≤ r₂ two integers; defaultprecision is the number of digits requested.
Output: γ_k(a,q) for every q such that r₁≤q≤r₂ and 1≤a≤q.
Comments: it uses the values of T at the a/q points and then obtain γ_k(a,q) as described in [1]. Examples on how to use the function and computational results are collected towards the end of the file.

greedy_seq.gp: Pari/GP script. It can be used via gp2c. The function to be run is:
greedy()
whose output is the first 2088 terms of the greedy sequence of prime offsets.
Such values are used to evaluate which candidates (q prime) can have small, possibly negative, values of 𝔊_q. An exhaustive search for every prime q up to 10^(10) were performed to towards the end a list of q with evalutation greater than 1.2; for very few primes such a condition holds.

Max-direct-final.gp: Pari/GP script. It can be used via gp2c. The function to be run is:
max_direct (r₁,r₂,defaultprecision).
Input: 2< r₁ < r₂, two integers; defaultprecision is the number of digits requested.
Output: the value M_q (distinguishing between even and odd characters) for every odd prime q such that r₁≤q≤r₂ and the running times.
Comment: it uses the lfun command of Pari/GP and the Conrey description of Dirichlet characters. Examples on how to use the function and computational results are collected towards the end of the file.

Max-S-final.gp: Pari/GP script. It can be used via gp2c. The function to be run is:
maxS_DIF (r₁,r₂,defaultprecision).
Input: 2< r₁ < r₂, two integers; defaultprecision is the number of digits requested.
Output: the value M_q (distinguishing between even and odd characters) for every odd prime q such that r₁≤q≤r₂ and the running times.
Comments: it computes first the values of log(Γ) and the decimated in frequency values of S at the a/q points, see [1], and then obtain M_q with a trivial implementation of the sum over a, 1≤a≤q-1. Examples on how to use the function and computational results are collected towards the end of the file.

Max-T-final.gp: Pari/GP script. It can be used via gp2c. The function to be run is:
maxT (r₁,r₂,defaultprecision).
Input: 2< r₁ < r₂, two integers; defaultprecision is the number of digits requested.
Output: the value M_q (distinguishing between even and odd characters) for every odd prime q such that r₁≤q≤r₂ and the running times.
Comments: it computes first the needed values of ψ and T at the a/q points, see [1], and then obtain M_q with a trivial implementation of the sum over a, 1≤a≤q-1. Examples on how to use the function and computational results are collected towards the end of the file.

Precomputations:
EKmain.c: C program.
input: an odd prime q.
output: the ascii file primroot.res contains q and g, a primitive root mod q.

m_precpsi.c: C program. Computes the needed values of ψ(a/q), a =g^j mod q, g=primroot, for j in a given interval [x,y]. Saves the results on an ASCII file.
input: the file primroot.res (reads q and g), 0≤ x < y ≤ q-2: integers.
output: the ascii file precψ--x-y.res contains q in its first line and then, line by line, the values of ψ(g^j/q), x≤j≤y.

DIFm_precS.c: C program. Computes the needed decimated in frequency values of -(S(a/q)+S(1-a/q)), a =g^j mod q, g=primroot, 1-a/q = g^j+m/q, m=(q-1)/2, for j in a given interval [x,y]. Saves the results on an ASCII file.
input: the file primroot.res (reads q and g), 0≤ x < y ≤ q-2: integers.
output: the ascii file precomp_S-DIF--x-y.res contains q in its first line and then, line by line, the decimated in frequency values of S. i.e. the values of -(S(g^j/q)+S(1-g^j/q)), m=(q-1)/2, x≤j≤y.

m_precT.c: C program. Computes the needed values of -T(a/q), a =g^j mod q, g=primroot, for j in a given interval [x,y]. Saves the results on an ASCII file.
input: the file primroot.res (reads q and g), 0≤ x < y ≤ q-2: integers.
output: the ascii file precT--x-y.res contains q in its first line and then, line by line, the values of -T(g^j/q), x≤j≤y.

FFT step: (using the FFTW library): All the following programs use the FFTW-guru64 interface (which also allows to transform sequences having dimension larger than 2³¹-1).

Euler-Kronecker constants computation:

EK-S-fftwl.c: C program.
It computes 𝔊_q⁺ and 𝔊_q via FFT; it needs the fftw library. It's the long double precision version. Values of log(Γ) are computed using the internal function of the C programming language. Uses the decimated in frequency values for S and the sequence g^k mod q.
input: the ascii files primroot.res, precomp_S-DIF.res.
output: the values 𝔊_q⁺ and 𝔊_q and their difference together with the running times.

EK-T-fftwl.c: C program.
It computes 𝔊_q⁺ and 𝔊_q via FFT; it needs the fftw library. It's the long double precision version. Double precision values of ψ are computed using the GSL library.
input: the ascii files primroot.res, precomp_T.res.
output: the values 𝔊_q⁺ and 𝔊_q and their difference together with the running times.

An alternative version of EK-T-fftwl.c (which uses the ψ function values generated by the program m_precpsi previously described; precomputed and stored on an external file as for T and S) is also available on request together with the quadruple precision versions of all the previously mentioned programs.

Maximum over non trivial characters for the logarithmic derivative at 1 of Dirichlet L-functions modulo q:

Max-S-fftwl.c: C program.
It computes M_q via FFT; it needs the fftw library. It's the long double precision version. Values of log(Γ) are computed using the internal function of the C programming language. Uses the decimated in frequency values for S and the sequence g^k mod q.
input: the ascii files primroot.res, precomp_S-DIF.res.
output: the value M_q (distinguishing between even and odd characters) and the running times.

Max-T-fftwl.c: C program.
It computes M_q via FFT; it needs the fftw library. It's the long double precision version. Double precision values of ψ are computed using the GSL library.
input: the ascii files primroot.res, precomp_T.res.
output: the value M_q (distinguishing between even and odd characters) and the running times.

[1] A. Languasco - Efficient computation of the Euler-Kronecker constants for prime cyclotomic fields - Research in Number Theory 7 (2021), no. 1, Paper no. 2.

[2] K. Ford; F. Luca; P. Moree - Values of the Euler phi-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields  - Math. Comp. 83 (2014), 1457-1476.

[3] P. Moree - Irregular Behaviour of Class Numbers and Euler-Kronecker Constants of Cyclotomic Fields: The Log Log Log Devil at Play, in Irregularities in the Distribution of Prime Numbers. From the Era of Helmut Maier's Matrix Method and Beyond (J. Pintz and M.Th. Rassias, eds.), Springer, 2018, pp. 143-163.

[4] A. Languasco, L. Righi - A fast algorithm to compute the Ramanujan-Deninger gamma function and some number-theoretic applications - Mathematics of Computation 90 (2021), 2899--2921.