How many signatures are required to start the formal decision making stage of a direct democracy mechanism?

Constitution of the Republic of Latvia (last amended in 2005)

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 72. The President has the right to suspend the proclamation of a law for a period of two months. The President shall suspend the proclamation of a law if so requested by not less than one-third of the members of the Saeima. This right may be exercised by the President, or by one-third of the members of the Saeima, within ten days of the adoption of the law by the Saeima. The law thus suspended shall be put to a national referendum if so requested by not less than one-tenth of the electorate. If no such request is received during the aforementioned two-month period, the law shall then be proclaimed after the expiration of such period. A national referendum shall not take place, however, if the Saeima again votes on the law and not less than three-quarters of all members of the Saeima vote for the adoption of the law.

Law on National Referendums, Legislative Initiatives and European Citizens? Initiative (last amended in 2012)

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Article 11(1) If the Saeima has not adopted without change as to its content a draft law or a draft amendment to the Constitution submitted by at least one-tenth of the electorate, this draft law or draft amendment to the Constitution must be put to a national referendum.

Article 22Not fewer than 10,000 Latvian citizens eligible to vote, upon indicating their full name and personal identity number, shall have the right to submit to the Central Election Commission a fully elaborated draft law or a draft amendment to the Constitution. Not earlier than 12 months before the submission of the draft law or the draft amendment to the Constitution, each signature must be certified by a sworn notary public or a local government authority that performs notarial functions.

?Both legislative and constitutional initiatives have two stages. During the first stage, the signatures of at least 10,000 citizens entitled to vote must be collected; their collection is entirely up to the proponents of the initiative. The initiative does not have to be registered. The initiative group does not have any formal status. When the required 10,000 signatures have been collected, the initiators submit the draft to the Central Electoral Commission. After the registration of the initiative by the CEC, the formal signature collection begins to collect signatures of at least 10% of the electorate.?

See: Direct Democracy in Europe. Edited by B.Kaufmann and M.D.Waters. 2004. Carolina Academic Press

See sources for details.