Approximate Functional Equations for Dirichlet Functions

Lavrik, A. F.

doi:10.1070/im1968v002n01abeh000634

Cited by 23 publications

(21 citation statements)

References 15 publications

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“…Following Lavrik [12], Karatsuba [9] and Turganaliev [21], we continue the function Γ (s, z) holomorphically to the right half plane Re(z) > 0 by making a change of variable t → zt and rotating the line of integration by the angle arg z. Then we have…”

Section: Introduction and The Statement Of Resultsmentioning

confidence: 99%

Calculation of values of 𝐿-functions associated to elliptic curves

Akiyama

Tanigawa

1999

Math. Comp.

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Abstract. We calculated numerically the values of L-functions of four typical elliptic curves in the critical strip in the range Im(s) ≤ 400. We found that all the non-trivial zeros in this range lie on the critical line Re(s) = 1 and are simple except the one at s = 1. The method we employed in this paper is the approximate functional equation with incomplete gamma functions in the coefficients. For incomplete gamma functions, we continued them holomorphically to the right half plane Re(s) > 0, which enables us to calculate for large Im(s). Furthermore we remark that a relation exists between Sato-Tate conjecture and the generalized Riemann Hypothesis.

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Section: Introduction and The Statement Of Resultsmentioning

confidence: 99%

Calculation of values of 𝐿-functions associated to elliptic curves

Akiyama

Tanigawa

1999

Math. Comp.

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“…Now, using the approximate functional equation for the L-functions, in the form given by Lavrik [7], in the same way as in [10], we see that we may assume, for 1 _< i < 4,…”

Section: Nimentioning

confidence: 98%

“…In [10] we used Heath-Brown's identity [2] and the approximate functional equation for the L-functions [7]. The method given there did not furnish any positive value of r for 0<3/5, hence for 7/12<0<3/5 only the result ~ = 3(0-7/12)-e, e>0 arbitrary, (1.3) proved in [4] using density theorems, seems to be available.…”

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confidence: 95%